Rosa's TurbineBenchmark

Percentage Accurate: 84.2% → 99.8%

Time: 11.6s

Alternatives: 8

Speedup: 1.7×

• 53.3% 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: 84.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \left(\frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(r \cdot w\right)\right)\right)\right) + -4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (+
   3.0
   (-
    (/ 2.0 (* r r))
    (* (* r w) (* (/ (fma v -0.25 0.375) (- 1.0 v)) (* r w)))))
  -4.5))

C

double code(double v, double w, double r) {
	return (3.0 + ((2.0 / (r * r)) - ((r * w) * ((fma(v, -0.25, 0.375) / (1.0 - v)) * (r * w))))) + -4.5;
}

Julia

function code(v, w, r)
	return Float64(Float64(3.0 + Float64(Float64(2.0 / Float64(r * r)) - Float64(Float64(r * w) * Float64(Float64(fma(v, -0.25, 0.375) / Float64(1.0 - v)) * Float64(r * w))))) + -4.5)
end

Wolfram

code[v_, w_, r_] := N[(N[(3.0 + N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(N[(r * w), $MachinePrecision] * N[(N[(N[(v * -0.25 + 0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision]

Derivation

1. Initial program 83.1%

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

3. Simplified 86.4%

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

   1. associate-/r/ 86.4%

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

   2. associate-*r* 82.2%

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

   3. swap-sqr 99.8%

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

   4. associate-*r* 99.8%

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

   5. +-commutative 99.8%

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

   6. distribute-rgt-in 99.8%

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

   7. *-commutative 99.8%

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

   8. associate-*l* 99.8%

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

   9. metadata-eval 99.8%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   10. metadata-eval 99.8%

       \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot -0.25 + \color{blue}{0.375}}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   11. fma-udef 99.8%

       \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{\color{blue}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

5. Applied egg-rr 99.8%

   \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\left(\frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}\right)\right) + -4.5 \]

6. Final simplification 99.8%

   \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \left(\frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(r \cdot w\right)\right)\right)\right) + -4.5 \]

Alternative 2: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return -4.5 + (3.0 + ((2.0 / (r * r)) - ((0.125 * (3.0 + (v * -2.0))) / ((1.0 / (r * w)) * ((1.0 - v) / (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 = (-4.5d0) + (3.0d0 + ((2.0d0 / (r * r)) - ((0.125d0 * (3.0d0 + (v * (-2.0d0)))) / ((1.0d0 / (r * w)) * ((1.0d0 - v) / (r * w))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

3. Simplified 86.4%

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

   1. associate-*r* 97.9%

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

   2. *-commutative 97.9%

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

   3. *-un-lft-identity 97.9%

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

   4. associate-*r* 99.8%

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

   5. times-frac 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2 \cdot 10^{-48}:\\ \;\;\;\;\left(t_0 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right) + -1.5\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \left(r \cdot w\right) \cdot \frac{r}{\frac{1 - v}{w \cdot \left(0.375 + v \cdot -0.25\right)}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 2e-48)
     (+ (+ t_0 (* (* (* r w) (* r w)) -0.375)) -1.5)
     (+
      -4.5
      (+
       3.0
       (- t_0 (* (* r w) (/ r (/ (- 1.0 v) (* w (+ 0.375 (* v -0.25))))))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2e-48) {
		tmp = (t_0 + (((r * w) * (r * w)) * -0.375)) + -1.5;
	} else {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (r / ((1.0 - v) / (w * (0.375 + (v * -0.25))))))));
	}
	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 <= 2d-48) then
        tmp = (t_0 + (((r * w) * (r * w)) * (-0.375d0))) + (-1.5d0)
    else
        tmp = (-4.5d0) + (3.0d0 + (t_0 - ((r * w) * (r / ((1.0d0 - v) / (w * (0.375d0 + (v * (-0.25d0)))))))))
    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 <= 2e-48) {
		tmp = (t_0 + (((r * w) * (r * w)) * -0.375)) + -1.5;
	} else {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (r / ((1.0 - v) / (w * (0.375 + (v * -0.25))))))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 2e-48:
		tmp = (t_0 + (((r * w) * (r * w)) * -0.375)) + -1.5
	else:
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (r / ((1.0 - v) / (w * (0.375 + (v * -0.25))))))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 2e-48)
		tmp = Float64(Float64(t_0 + Float64(Float64(Float64(r * w) * Float64(r * w)) * -0.375)) + -1.5);
	else
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(Float64(r * w) * Float64(r / Float64(Float64(1.0 - v) / Float64(w * Float64(0.375 + Float64(v * -0.25)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 2e-48)
		tmp = (t_0 + (((r * w) * (r * w)) * -0.375)) + -1.5;
	else
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (r / ((1.0 - v) / (w * (0.375 + (v * -0.25))))))));
	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, 2e-48], N[(N[(t$95$0 + N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(N[(r * w), $MachinePrecision] * N[(r / N[(N[(1.0 - v), $MachinePrecision] / N[(w * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.9999999999999999e-48

   1. Initial program 81.5%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in v around 0 77.8%

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

      1. *-commutative 77.8%

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

      2. unpow2 77.8%

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

      3. unpow2 77.8%

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

      4. swap-sqr 93.9%

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

      5. unpow2 93.9%

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

   6. Simplified 93.9%

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

      1. unpow2 93.9%

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

   8. Applied egg-rr 93.9%

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

   if 1.9999999999999999e-48 < r

   1. Initial program 88.1%

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

   3. Simplified 93.9%

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

      1. associate-/r/ 93.9%

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

      2. associate-*r* 84.4%

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

      3. swap-sqr 99.8%

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

      4. associate-*r* 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt-in 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*l* 99.8%

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

      9. metadata-eval 99.8%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      10. metadata-eval 99.8%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot -0.25 + \color{blue}{0.375}}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      11. fma-udef 99.8%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{\color{blue}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   5. Applied egg-rr 99.8%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\left(\frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}\right)\right) + -4.5 \]

   6. Taylor expanded in r around 0 95.4%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\frac{r \cdot \left(w \cdot \left(0.375 + -0.25 \cdot v\right)\right)}{1 - v}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]
   7. Step-by-step derivation

      1. associate-/l* 99.8%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\frac{r}{\frac{1 - v}{w \cdot \left(0.375 + -0.25 \cdot v\right)}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      2. +-commutative 99.8%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\frac{1 - v}{w \cdot \color{blue}{\left(-0.25 \cdot v + 0.375\right)}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      3. *-commutative 99.8%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\frac{1 - v}{w \cdot \left(\color{blue}{v \cdot -0.25} + 0.375\right)}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      4. fma-udef 99.8%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\frac{1 - v}{w \cdot \color{blue}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   8. Simplified 99.8%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\frac{r}{\frac{1 - v}{w \cdot \mathsf{fma}\left(v, -0.25, 0.375\right)}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   9. Taylor expanded in w around 0 99.8%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\color{blue}{\frac{1 - v}{w \cdot \left(0.375 + -0.25 \cdot v\right)}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-48}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right) + -1.5\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \frac{r}{\frac{1 - v}{w \cdot \left(0.375 + v \cdot -0.25\right)}}\right)\right)\\ \end{array} \]

Alternative 4: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1.5:\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \left(r \cdot w\right) \cdot \frac{r}{\frac{4}{w} + \frac{2}{v \cdot w}}\right)\right)\\ \mathbf{elif}\;v \leq 1:\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \left(r \cdot w\right) \cdot \frac{r}{\frac{4}{w}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v -1.5)
     (+ -4.5 (+ 3.0 (- t_0 (* (* r w) (/ r (+ (/ 4.0 w) (/ 2.0 (* v w))))))))
     (if (<= v 1.0)
       (+ -4.5 (+ 3.0 (- t_0 (* (* r w) (* w (* r 0.375))))))
       (+ -4.5 (+ 3.0 (- t_0 (* (* r w) (/ r (/ 4.0 w))))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= -1.5) {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (r / ((4.0 / w) + (2.0 / (v * w)))))));
	} else if (v <= 1.0) {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.375)))));
	} else {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (r / (4.0 / 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 <= (-1.5d0)) then
        tmp = (-4.5d0) + (3.0d0 + (t_0 - ((r * w) * (r / ((4.0d0 / w) + (2.0d0 / (v * w)))))))
    else if (v <= 1.0d0) then
        tmp = (-4.5d0) + (3.0d0 + (t_0 - ((r * w) * (w * (r * 0.375d0)))))
    else
        tmp = (-4.5d0) + (3.0d0 + (t_0 - ((r * w) * (r / (4.0d0 / 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 <= -1.5) {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (r / ((4.0 / w) + (2.0 / (v * w)))))));
	} else if (v <= 1.0) {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.375)))));
	} else {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (r / (4.0 / w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if v <= -1.5:
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (r / ((4.0 / w) + (2.0 / (v * w)))))))
	elif v <= 1.0:
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.375)))))
	else:
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (r / (4.0 / w)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -1.5)
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(Float64(r * w) * Float64(r / Float64(Float64(4.0 / w) + Float64(2.0 / Float64(v * w))))))));
	elseif (v <= 1.0)
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(Float64(r * w) * Float64(w * Float64(r * 0.375))))));
	else
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(Float64(r * w) * Float64(r / Float64(4.0 / w))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= -1.5)
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (r / ((4.0 / w) + (2.0 / (v * w)))))));
	elseif (v <= 1.0)
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.375)))));
	else
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (r / (4.0 / w)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -1.5], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(N[(r * w), $MachinePrecision] * N[(r / N[(N[(4.0 / w), $MachinePrecision] + N[(2.0 / N[(v * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 1.0], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(N[(r * w), $MachinePrecision] * N[(r / N[(4.0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.5

   1. Initial program 79.0%

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

   3. Simplified 84.9%

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

      1. associate-/r/ 84.9%

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

      2. associate-*r* 79.5%

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

      3. swap-sqr 99.7%

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

      4. associate-*r* 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt-in 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*l* 99.8%

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

      9. metadata-eval 99.8%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      10. metadata-eval 99.8%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot -0.25 + \color{blue}{0.375}}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      11. fma-udef 99.8%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{\color{blue}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   5. Applied egg-rr 99.8%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\left(\frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}\right)\right) + -4.5 \]

   6. Taylor expanded in r around 0 78.1%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\frac{r \cdot \left(w \cdot \left(0.375 + -0.25 \cdot v\right)\right)}{1 - v}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]
   7. Step-by-step derivation

      1. associate-/l* 82.5%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\frac{r}{\frac{1 - v}{w \cdot \left(0.375 + -0.25 \cdot v\right)}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      2. +-commutative 82.5%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\frac{1 - v}{w \cdot \color{blue}{\left(-0.25 \cdot v + 0.375\right)}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      3. *-commutative 82.5%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\frac{1 - v}{w \cdot \left(\color{blue}{v \cdot -0.25} + 0.375\right)}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      4. fma-udef 82.5%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\frac{1 - v}{w \cdot \color{blue}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   8. Simplified 82.5%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\frac{r}{\frac{1 - v}{w \cdot \mathsf{fma}\left(v, -0.25, 0.375\right)}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   9. Taylor expanded in v around inf 99.0%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\color{blue}{4 \cdot \frac{1}{w} + 2 \cdot \frac{1}{v \cdot w}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]
   10. Step-by-step derivation

       1. associate-*r/ 99.0%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\color{blue}{\frac{4 \cdot 1}{w}} + 2 \cdot \frac{1}{v \cdot w}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

       2. metadata-eval 99.0%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\frac{\color{blue}{4}}{w} + 2 \cdot \frac{1}{v \cdot w}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

       3. associate-*r/ 99.0%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\frac{4}{w} + \color{blue}{\frac{2 \cdot 1}{v \cdot w}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

       4. metadata-eval 99.0%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\frac{4}{w} + \frac{\color{blue}{2}}{v \cdot w}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

       5. *-commutative 99.0%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\frac{4}{w} + \frac{2}{\color{blue}{w \cdot v}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   11. Simplified 99.0%

       \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\color{blue}{\frac{4}{w} + \frac{2}{w \cdot v}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   if -1.5 < v < 1

   1. Initial program 83.7%

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

   3. Simplified 83.7%

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

      1. associate-/r/ 83.7%

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

      2. associate-*r* 79.4%

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

      3. swap-sqr 99.9%

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

      4. associate-*r* 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-rgt-in 99.9%

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

      7. *-commutative 99.9%

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

      8. associate-*l* 99.9%

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

      9. metadata-eval 99.9%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      10. metadata-eval 99.9%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot -0.25 + \color{blue}{0.375}}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      11. fma-udef 99.9%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{\color{blue}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   5. Applied egg-rr 99.9%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\left(\frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}\right)\right) + -4.5 \]

   6. Taylor expanded in v around 0 98.8%

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

      1. associate-*r* 98.9%

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

   8. Simplified 98.9%

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

   if 1 < v

   1. Initial program 85.9%

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

   3. Simplified 92.6%

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

      1. associate-/r/ 92.7%

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

      2. associate-*r* 89.7%

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

      3. swap-sqr 99.8%

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

      4. associate-*r* 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt-in 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*l* 99.8%

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

      9. metadata-eval 99.8%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      10. metadata-eval 99.8%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot -0.25 + \color{blue}{0.375}}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      11. fma-udef 99.8%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{\color{blue}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   5. Applied egg-rr 99.8%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\left(\frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}\right)\right) + -4.5 \]

   6. Taylor expanded in r around 0 89.9%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\frac{r \cdot \left(w \cdot \left(0.375 + -0.25 \cdot v\right)\right)}{1 - v}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]
   7. Step-by-step derivation

      1. associate-/l* 92.7%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\frac{r}{\frac{1 - v}{w \cdot \left(0.375 + -0.25 \cdot v\right)}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      2. +-commutative 92.7%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\frac{1 - v}{w \cdot \color{blue}{\left(-0.25 \cdot v + 0.375\right)}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      3. *-commutative 92.7%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\frac{1 - v}{w \cdot \left(\color{blue}{v \cdot -0.25} + 0.375\right)}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      4. fma-udef 92.7%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\frac{1 - v}{w \cdot \color{blue}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   8. Simplified 92.7%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\frac{r}{\frac{1 - v}{w \cdot \mathsf{fma}\left(v, -0.25, 0.375\right)}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   9. Taylor expanded in v around inf 99.9%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\color{blue}{\frac{4}{w}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.5:\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \frac{r}{\frac{4}{w} + \frac{2}{v \cdot w}}\right)\right)\\ \mathbf{elif}\;v \leq 1:\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \frac{r}{\frac{4}{w}}\right)\right)\\ \end{array} \]

Alternative 5: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1.25 \lor \neg \left(v \leq 1\right):\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right) + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -1.25) (not (<= v 1.0)))
     (+ -4.5 (+ 3.0 (- t_0 (* (* r w) (* w (* r 0.25))))))
     (+ (+ t_0 (* (* (* r w) (* r w)) -0.375)) -1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1.25) || !(v <= 1.0)) {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.25)))));
	} else {
		tmp = (t_0 + (((r * w) * (r * w)) * -0.375)) + -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-1.25d0)) .or. (.not. (v <= 1.0d0))) then
        tmp = (-4.5d0) + (3.0d0 + (t_0 - ((r * w) * (w * (r * 0.25d0)))))
    else
        tmp = (t_0 + (((r * w) * (r * w)) * (-0.375d0))) + (-1.5d0)
    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) || !(v <= 1.0)) {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.25)))));
	} else {
		tmp = (t_0 + (((r * w) * (r * w)) * -0.375)) + -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -1.25) or not (v <= 1.0):
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.25)))))
	else:
		tmp = (t_0 + (((r * w) * (r * w)) * -0.375)) + -1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -1.25) || !(v <= 1.0))
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(Float64(r * w) * Float64(w * Float64(r * 0.25))))));
	else
		tmp = Float64(Float64(t_0 + Float64(Float64(Float64(r * w) * Float64(r * w)) * -0.375)) + -1.5);
	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) || ~((v <= 1.0)))
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.25)))));
	else
		tmp = (t_0 + (((r * w) * (r * w)) * -0.375)) + -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -1.25], N[Not[LessEqual[v, 1.0]], $MachinePrecision]], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.25 or 1 < v

   1. Initial program 82.6%

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

   3. Simplified 88.9%

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

      1. associate-/r/ 89.0%

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

      2. associate-*r* 84.8%

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

      3. swap-sqr 99.8%

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

      4. associate-*r* 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt-in 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*l* 99.8%

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

      9. metadata-eval 99.8%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      10. metadata-eval 99.8%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot -0.25 + \color{blue}{0.375}}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      11. fma-udef 99.8%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{\color{blue}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   5. Applied egg-rr 99.8%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\left(\frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}\right)\right) + -4.5 \]

   6. Taylor expanded in v around inf 99.1%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\left(0.25 \cdot \left(r \cdot w\right)\right)} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]
   7. Step-by-step derivation

      1. associate-*r* 99.1%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\left(\left(0.25 \cdot r\right) \cdot w\right)} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   8. Simplified 99.1%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\left(\left(0.25 \cdot r\right) \cdot w\right)} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   if -1.25 < v < 1

   1. Initial program 83.7%

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

   3. Simplified 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 v around 0 79.4%

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

      1. *-commutative 79.4%

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

      2. unpow2 79.4%

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

      3. unpow2 79.4%

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

      4. swap-sqr 98.8%

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

      5. unpow2 98.8%

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

   6. Simplified 98.8%

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

      1. unpow2 98.8%

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

   8. Applied egg-rr 98.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.25 \lor \neg \left(v \leq 1\right):\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot -0.375\right) + -1.5\\ \end{array} \]

Alternative 6: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1.2 \lor \neg \left(v \leq 1\right):\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -1.2) (not (<= v 1.0)))
     (+ -4.5 (+ 3.0 (- t_0 (* (* r w) (* w (* r 0.25))))))
     (+ -4.5 (+ 3.0 (- t_0 (* (* r w) (* w (* r 0.375)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1.2) || !(v <= 1.0)) {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.25)))));
	} else {
		tmp = -4.5 + (3.0 + (t_0 - ((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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-1.2d0)) .or. (.not. (v <= 1.0d0))) then
        tmp = (-4.5d0) + (3.0d0 + (t_0 - ((r * w) * (w * (r * 0.25d0)))))
    else
        tmp = (-4.5d0) + (3.0d0 + (t_0 - ((r * w) * (w * (r * 0.375d0)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1.2) || !(v <= 1.0)) {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.25)))));
	} else {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.375)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -1.2) or not (v <= 1.0):
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.25)))))
	else:
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.375)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -1.2) || !(v <= 1.0))
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(Float64(r * w) * Float64(w * Float64(r * 0.25))))));
	else
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(Float64(r * w) * Float64(w * Float64(r * 0.375))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -1.2) || ~((v <= 1.0)))
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.25)))));
	else
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.375)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -1.2], N[Not[LessEqual[v, 1.0]], $MachinePrecision]], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.19999999999999996 or 1 < v

   1. Initial program 82.6%

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

   3. Simplified 88.9%

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

      1. associate-/r/ 89.0%

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

      2. associate-*r* 84.8%

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

      3. swap-sqr 99.8%

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

      4. associate-*r* 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt-in 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*l* 99.8%

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

      9. metadata-eval 99.8%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      10. metadata-eval 99.8%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot -0.25 + \color{blue}{0.375}}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      11. fma-udef 99.8%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{\color{blue}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   5. Applied egg-rr 99.8%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\left(\frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}\right)\right) + -4.5 \]

   6. Taylor expanded in v around inf 99.1%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\left(0.25 \cdot \left(r \cdot w\right)\right)} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]
   7. Step-by-step derivation

      1. associate-*r* 99.1%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\left(\left(0.25 \cdot r\right) \cdot w\right)} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   8. Simplified 99.1%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\left(\left(0.25 \cdot r\right) \cdot w\right)} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   if -1.19999999999999996 < v < 1

   1. Initial program 83.7%

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

   3. Simplified 83.7%

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

      1. associate-/r/ 83.7%

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

      2. associate-*r* 79.4%

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

      3. swap-sqr 99.9%

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

      4. associate-*r* 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-rgt-in 99.9%

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

      7. *-commutative 99.9%

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

      8. associate-*l* 99.9%

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

      9. metadata-eval 99.9%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      10. metadata-eval 99.9%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot -0.25 + \color{blue}{0.375}}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      11. fma-udef 99.9%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{\color{blue}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   5. Applied egg-rr 99.9%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\left(\frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}\right)\right) + -4.5 \]

   6. Taylor expanded in v around 0 98.8%

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

      1. associate-*r* 98.9%

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

   8. Simplified 98.9%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\left(\left(0.375 \cdot r\right) \cdot w\right)} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.2 \lor \neg \left(v \leq 1\right):\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1.4 \lor \neg \left(v \leq 1\right):\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \left(r \cdot w\right) \cdot \frac{r}{\frac{4}{w}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -1.4) (not (<= v 1.0)))
     (+ -4.5 (+ 3.0 (- t_0 (* (* r w) (/ r (/ 4.0 w))))))
     (+ -4.5 (+ 3.0 (- t_0 (* (* r w) (* w (* r 0.375)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1.4) || !(v <= 1.0)) {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (r / (4.0 / w)))));
	} else {
		tmp = -4.5 + (3.0 + (t_0 - ((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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-1.4d0)) .or. (.not. (v <= 1.0d0))) then
        tmp = (-4.5d0) + (3.0d0 + (t_0 - ((r * w) * (r / (4.0d0 / w)))))
    else
        tmp = (-4.5d0) + (3.0d0 + (t_0 - ((r * w) * (w * (r * 0.375d0)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1.4) || !(v <= 1.0)) {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (r / (4.0 / w)))));
	} else {
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.375)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -1.4) or not (v <= 1.0):
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (r / (4.0 / w)))))
	else:
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.375)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -1.4) || !(v <= 1.0))
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(Float64(r * w) * Float64(r / Float64(4.0 / w))))));
	else
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(Float64(r * w) * Float64(w * Float64(r * 0.375))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -1.4) || ~((v <= 1.0)))
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (r / (4.0 / w)))));
	else
		tmp = -4.5 + (3.0 + (t_0 - ((r * w) * (w * (r * 0.375)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -1.4], N[Not[LessEqual[v, 1.0]], $MachinePrecision]], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(N[(r * w), $MachinePrecision] * N[(r / N[(4.0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.3999999999999999 or 1 < v

   1. Initial program 82.6%

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

   3. Simplified 88.9%

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

      1. associate-/r/ 89.0%

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

      2. associate-*r* 84.8%

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

      3. swap-sqr 99.8%

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

      4. associate-*r* 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-rgt-in 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*l* 99.8%

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

      9. metadata-eval 99.8%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      10. metadata-eval 99.8%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot -0.25 + \color{blue}{0.375}}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      11. fma-udef 99.8%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{\color{blue}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   5. Applied egg-rr 99.8%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\left(\frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}\right)\right) + -4.5 \]

   6. Taylor expanded in r around 0 84.3%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\frac{r \cdot \left(w \cdot \left(0.375 + -0.25 \cdot v\right)\right)}{1 - v}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]
   7. Step-by-step derivation

      1. associate-/l* 87.8%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\frac{r}{\frac{1 - v}{w \cdot \left(0.375 + -0.25 \cdot v\right)}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      2. +-commutative 87.8%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\frac{1 - v}{w \cdot \color{blue}{\left(-0.25 \cdot v + 0.375\right)}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      3. *-commutative 87.8%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\frac{1 - v}{w \cdot \left(\color{blue}{v \cdot -0.25} + 0.375\right)}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      4. fma-udef 87.8%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\frac{1 - v}{w \cdot \color{blue}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   8. Simplified 87.8%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\frac{r}{\frac{1 - v}{w \cdot \mathsf{fma}\left(v, -0.25, 0.375\right)}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   9. Taylor expanded in v around inf 99.1%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \frac{r}{\color{blue}{\frac{4}{w}}} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   if -1.3999999999999999 < v < 1

   1. Initial program 83.7%

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

   3. Simplified 83.7%

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

      1. associate-/r/ 83.7%

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

      2. associate-*r* 79.4%

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

      3. swap-sqr 99.9%

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

      4. associate-*r* 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-rgt-in 99.9%

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

      7. *-commutative 99.9%

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

      8. associate-*l* 99.9%

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

      9. metadata-eval 99.9%

         \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      10. metadata-eval 99.9%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{v \cdot -0.25 + \color{blue}{0.375}}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

      11. fma-udef 99.9%

          \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \left(\frac{\color{blue}{\mathsf{fma}\left(v, -0.25, 0.375\right)}}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

   5. Applied egg-rr 99.9%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\left(\frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(r \cdot w\right)\right) \cdot \left(r \cdot w\right)}\right)\right) + -4.5 \]

   6. Taylor expanded in v around 0 98.8%

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

      1. associate-*r* 98.9%

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

   8. Simplified 98.9%

      \[\leadsto \left(3 + \left(\frac{2}{r \cdot r} - \color{blue}{\left(\left(0.375 \cdot r\right) \cdot w\right)} \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.4 \lor \neg \left(v \leq 1\right):\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \frac{r}{\frac{4}{w}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\right)\\ \end{array} \]

Alternative 8: 93.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

3. Simplified 86.4%

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

4. Taylor expanded in v around 0 77.6%

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

   1. *-commutative 77.6%

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

   2. unpow2 77.6%

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

   3. unpow2 77.6%

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

   4. swap-sqr 92.9%

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

   5. unpow2 92.9%

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

6. Simplified 92.9%

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

   1. unpow2 92.9%

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

8. Applied egg-rr 92.9%

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

9. Final simplification 92.9%

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

Reproduce

herbie shell --seed 2023315 
(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))