Rosa's TurbineBenchmark

Percentage Accurate: 84.8% → 99.7%

Time: 12.4s

Alternatives: 10

Speedup: 1.4×

• 51.8% 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: 84.8% 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: 99.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (+
   3.0
   (-
    (/ 2.0 (* r r))
    (/
     (* 0.125 (+ 3.0 (* -2.0 v)))
     (* (/ 1.0 (* r w)) (/ (- 1.0 v) (* r w))))))
  -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))) / ((1.0 / (r * w)) * ((1.0 - v) / (r * w)))))) + -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))) / ((1.0d0 / (r * w)) * ((1.0d0 - v) / (r * w)))))) + (-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))) / ((1.0 / (r * w)) * ((1.0 - v) / (r * w)))))) + -4.5;
}

Python

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

MATLAB

function tmp = code(v, w, r)
	tmp = (3.0 + ((2.0 / (r * r)) - ((0.125 * (3.0 + (-2.0 * v))) / ((1.0 / (r * w)) * ((1.0 - v) / (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[(0.125 * N[(3.0 + N[(-2.0 * v), $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] + -4.5), $MachinePrecision]

Derivation

1. Initial program 83.4%

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

3. Simplified 85.4%

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

   1. associate-*r* 97.2%

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

   2. *-commutative 97.2%

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

   3. *-un-lft-identity 97.2%

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

   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%

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

Alternative 2: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := 0.125 \cdot \left(3 + -2 \cdot v\right)\\ \mathbf{if}\;v \leq -26500000 \lor \neg \left(v \leq 0.0073\right):\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \frac{t_1}{\frac{\frac{-v}{r \cdot w}}{r \cdot w}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \frac{t_1}{\frac{1}{r \cdot w} \cdot \frac{\frac{1}{r}}{w}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (* 0.125 (+ 3.0 (* -2.0 v)))))
   (if (or (<= v -26500000.0) (not (<= v 0.0073)))
     (+ -4.5 (+ 3.0 (- t_0 (/ t_1 (/ (/ (- v) (* r w)) (* r w))))))
     (+ -4.5 (+ 3.0 (- t_0 (/ t_1 (* (/ 1.0 (* r w)) (/ (/ 1.0 r) w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = 0.125 * (3.0 + (-2.0 * v));
	double tmp;
	if ((v <= -26500000.0) || !(v <= 0.0073)) {
		tmp = -4.5 + (3.0 + (t_0 - (t_1 / ((-v / (r * w)) / (r * w)))));
	} else {
		tmp = -4.5 + (3.0 + (t_0 - (t_1 / ((1.0 / (r * w)) * ((1.0 / r) / w)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = 0.125d0 * (3.0d0 + ((-2.0d0) * v))
    if ((v <= (-26500000.0d0)) .or. (.not. (v <= 0.0073d0))) then
        tmp = (-4.5d0) + (3.0d0 + (t_0 - (t_1 / ((-v / (r * w)) / (r * w)))))
    else
        tmp = (-4.5d0) + (3.0d0 + (t_0 - (t_1 / ((1.0d0 / (r * w)) * ((1.0d0 / r) / w)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = 0.125 * (3.0 + (-2.0 * v));
	double tmp;
	if ((v <= -26500000.0) || !(v <= 0.0073)) {
		tmp = -4.5 + (3.0 + (t_0 - (t_1 / ((-v / (r * w)) / (r * w)))));
	} else {
		tmp = -4.5 + (3.0 + (t_0 - (t_1 / ((1.0 / (r * w)) * ((1.0 / r) / w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = 0.125 * (3.0 + (-2.0 * v))
	tmp = 0
	if (v <= -26500000.0) or not (v <= 0.0073):
		tmp = -4.5 + (3.0 + (t_0 - (t_1 / ((-v / (r * w)) / (r * w)))))
	else:
		tmp = -4.5 + (3.0 + (t_0 - (t_1 / ((1.0 / (r * w)) * ((1.0 / r) / w)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(0.125 * Float64(3.0 + Float64(-2.0 * v)))
	tmp = 0.0
	if ((v <= -26500000.0) || !(v <= 0.0073))
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(t_1 / Float64(Float64(Float64(-v) / Float64(r * w)) / Float64(r * w))))));
	else
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(t_1 / Float64(Float64(1.0 / Float64(r * w)) * Float64(Float64(1.0 / r) / w))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = 0.125 * (3.0 + (-2.0 * v));
	tmp = 0.0;
	if ((v <= -26500000.0) || ~((v <= 0.0073)))
		tmp = -4.5 + (3.0 + (t_0 - (t_1 / ((-v / (r * w)) / (r * w)))));
	else
		tmp = -4.5 + (3.0 + (t_0 - (t_1 / ((1.0 / (r * w)) * ((1.0 / r) / w)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -26500000.0], N[Not[LessEqual[v, 0.0073]], $MachinePrecision]], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(t$95$1 / N[(N[((-v) / N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(t$95$1 / N[(N[(1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / r), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -2.65e7 or 0.00730000000000000007 < v

   1. Initial program 79.9%

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

   3. Simplified 84.2%

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

      1. associate-*r* 98.1%

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

      2. *-commutative 98.1%

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

      3. *-un-lft-identity 98.1%

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

      4. associate-*r* 99.7%

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

      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. Step-by-step derivation

      1. associate-*l/ 99.8%

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

      2. *-un-lft-identity 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in v around inf 99.8%

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

      1. neg-mul-1 99.8%

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

      2. distribute-neg-frac 99.8%

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

   10. Simplified 99.8%

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

   if -2.65e7 < v < 0.00730000000000000007

   1. Initial program 86.4%

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

   3. Simplified 86.5%

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

      1. associate-*r* 96.4%

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

      2. *-commutative 96.4%

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

      3. *-un-lft-identity 96.4%

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

      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. Taylor expanded in v around 0 99.6%

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

      1. associate-/r* 99.6%

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

   8. Simplified 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -26500000 \lor \neg \left(v \leq 0.0073\right):\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{\frac{-v}{r \cdot w}}{r \cdot w}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1}{r \cdot w} \cdot \frac{\frac{1}{r}}{w}}\right)\right)\\ \end{array} \]

Alternative 3: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := 0.125 \cdot \left(3 + -2 \cdot v\right)\\ \mathbf{if}\;v \leq -28000000 \lor \neg \left(v \leq 0.0073\right):\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \frac{t_1}{\frac{\frac{-v}{r \cdot w}}{r \cdot w}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \frac{t_1}{\frac{\frac{1}{r \cdot w}}{r \cdot w}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (* 0.125 (+ 3.0 (* -2.0 v)))))
   (if (or (<= v -28000000.0) (not (<= v 0.0073)))
     (+ -4.5 (+ 3.0 (- t_0 (/ t_1 (/ (/ (- v) (* r w)) (* r w))))))
     (+ -4.5 (+ 3.0 (- t_0 (/ t_1 (/ (/ 1.0 (* r w)) (* r w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = 0.125 * (3.0 + (-2.0 * v));
	double tmp;
	if ((v <= -28000000.0) || !(v <= 0.0073)) {
		tmp = -4.5 + (3.0 + (t_0 - (t_1 / ((-v / (r * w)) / (r * w)))));
	} else {
		tmp = -4.5 + (3.0 + (t_0 - (t_1 / ((1.0 / (r * w)) / (r * w)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = 0.125d0 * (3.0d0 + ((-2.0d0) * v))
    if ((v <= (-28000000.0d0)) .or. (.not. (v <= 0.0073d0))) then
        tmp = (-4.5d0) + (3.0d0 + (t_0 - (t_1 / ((-v / (r * w)) / (r * w)))))
    else
        tmp = (-4.5d0) + (3.0d0 + (t_0 - (t_1 / ((1.0d0 / (r * w)) / (r * w)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = 0.125 * (3.0 + (-2.0 * v));
	double tmp;
	if ((v <= -28000000.0) || !(v <= 0.0073)) {
		tmp = -4.5 + (3.0 + (t_0 - (t_1 / ((-v / (r * w)) / (r * w)))));
	} else {
		tmp = -4.5 + (3.0 + (t_0 - (t_1 / ((1.0 / (r * w)) / (r * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = 0.125 * (3.0 + (-2.0 * v))
	tmp = 0
	if (v <= -28000000.0) or not (v <= 0.0073):
		tmp = -4.5 + (3.0 + (t_0 - (t_1 / ((-v / (r * w)) / (r * w)))))
	else:
		tmp = -4.5 + (3.0 + (t_0 - (t_1 / ((1.0 / (r * w)) / (r * w)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(0.125 * Float64(3.0 + Float64(-2.0 * v)))
	tmp = 0.0
	if ((v <= -28000000.0) || !(v <= 0.0073))
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(t_1 / Float64(Float64(Float64(-v) / Float64(r * w)) / Float64(r * w))))));
	else
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(t_1 / Float64(Float64(1.0 / Float64(r * w)) / Float64(r * w))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = 0.125 * (3.0 + (-2.0 * v));
	tmp = 0.0;
	if ((v <= -28000000.0) || ~((v <= 0.0073)))
		tmp = -4.5 + (3.0 + (t_0 - (t_1 / ((-v / (r * w)) / (r * w)))));
	else
		tmp = -4.5 + (3.0 + (t_0 - (t_1 / ((1.0 / (r * w)) / (r * w)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -28000000.0], N[Not[LessEqual[v, 0.0073]], $MachinePrecision]], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(t$95$1 / N[(N[((-v) / N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(t$95$1 / N[(N[(1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -2.8e7 or 0.00730000000000000007 < v

   1. Initial program 79.9%

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

   3. Simplified 84.2%

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

      1. associate-*r* 98.1%

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

      2. *-commutative 98.1%

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

      3. *-un-lft-identity 98.1%

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

      4. associate-*r* 99.7%

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

      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. Step-by-step derivation

      1. associate-*l/ 99.8%

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

      2. *-un-lft-identity 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in v around inf 99.8%

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

      1. neg-mul-1 99.8%

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

      2. distribute-neg-frac 99.8%

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

   10. Simplified 99.8%

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

   if -2.8e7 < v < 0.00730000000000000007

   1. Initial program 86.4%

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

   3. Simplified 86.5%

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

      1. associate-*r* 96.4%

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

      2. *-commutative 96.4%

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

      3. *-un-lft-identity 96.4%

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

      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. Step-by-step derivation

      1. associate-*l/ 99.8%

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

      2. *-un-lft-identity 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in v around 0 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -28000000 \lor \neg \left(v \leq 0.0073\right):\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{\frac{-v}{r \cdot w}}{r \cdot w}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{\frac{1}{r \cdot w}}{r \cdot w}}\right)\right)\\ \end{array} \]

Alternative 4: 89.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.125 \cdot \left(3 + -2 \cdot v\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2.1 \cdot 10^{-126}:\\ \;\;\;\;-4.5 + \left(3 + \left(t_1 - \frac{t_0}{\frac{\frac{1}{r \cdot w}}{r \cdot w}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(t_1 - \frac{t_0}{\frac{\frac{1 - v}{r}}{w \cdot \left(r \cdot w\right)}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* 0.125 (+ 3.0 (* -2.0 v)))) (t_1 (/ 2.0 (* r r))))
   (if (<= r 2.1e-126)
     (+ -4.5 (+ 3.0 (- t_1 (/ t_0 (/ (/ 1.0 (* r w)) (* r w))))))
     (+ -4.5 (+ 3.0 (- t_1 (/ t_0 (/ (/ (- 1.0 v) r) (* w (* r w))))))))))

C

double code(double v, double w, double r) {
	double t_0 = 0.125 * (3.0 + (-2.0 * v));
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.1e-126) {
		tmp = -4.5 + (3.0 + (t_1 - (t_0 / ((1.0 / (r * w)) / (r * w)))));
	} else {
		tmp = -4.5 + (3.0 + (t_1 - (t_0 / (((1.0 - v) / r) / (w * (r * w))))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.125d0 * (3.0d0 + ((-2.0d0) * v))
    t_1 = 2.0d0 / (r * r)
    if (r <= 2.1d-126) then
        tmp = (-4.5d0) + (3.0d0 + (t_1 - (t_0 / ((1.0d0 / (r * w)) / (r * w)))))
    else
        tmp = (-4.5d0) + (3.0d0 + (t_1 - (t_0 / (((1.0d0 - v) / r) / (w * (r * w))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 0.125 * (3.0 + (-2.0 * v));
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.1e-126) {
		tmp = -4.5 + (3.0 + (t_1 - (t_0 / ((1.0 / (r * w)) / (r * w)))));
	} else {
		tmp = -4.5 + (3.0 + (t_1 - (t_0 / (((1.0 - v) / r) / (w * (r * w))))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 0.125 * (3.0 + (-2.0 * v))
	t_1 = 2.0 / (r * r)
	tmp = 0
	if r <= 2.1e-126:
		tmp = -4.5 + (3.0 + (t_1 - (t_0 / ((1.0 / (r * w)) / (r * w)))))
	else:
		tmp = -4.5 + (3.0 + (t_1 - (t_0 / (((1.0 - v) / r) / (w * (r * w))))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(0.125 * Float64(3.0 + Float64(-2.0 * v)))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 2.1e-126)
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_1 - Float64(t_0 / Float64(Float64(1.0 / Float64(r * w)) / Float64(r * w))))));
	else
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_1 - Float64(t_0 / Float64(Float64(Float64(1.0 - v) / r) / Float64(w * Float64(r * w)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 0.125 * (3.0 + (-2.0 * v));
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 2.1e-126)
		tmp = -4.5 + (3.0 + (t_1 - (t_0 / ((1.0 / (r * w)) / (r * w)))));
	else
		tmp = -4.5 + (3.0 + (t_1 - (t_0 / (((1.0 - v) / r) / (w * (r * w))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 2.1e-126], N[(-4.5 + N[(3.0 + N[(t$95$1 - N[(t$95$0 / N[(N[(1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(3.0 + N[(t$95$1 - N[(t$95$0 / N[(N[(N[(1.0 - v), $MachinePrecision] / r), $MachinePrecision] / N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if r < 2.0999999999999999e-126

   1. Initial program 83.4%

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

   3. Simplified 85.6%

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

      1. associate-*r* 95.9%

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

      2. *-commutative 95.9%

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

      3. *-un-lft-identity 95.9%

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

      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. Step-by-step derivation

      1. associate-*l/ 99.8%

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

      2. *-un-lft-identity 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in v around 0 87.0%

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

   if 2.0999999999999999e-126 < r

   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 85.0%

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

      1. associate-*r* 99.8%

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

      2. *-commutative 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. associate-*r* 99.7%

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

      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. Step-by-step derivation

      1. associate-/r* 98.7%

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

      2. frac-times 98.6%

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

      3. metadata-eval 98.6%

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

      4. times-frac 98.6%

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

      5. *-un-lft-identity 98.6%

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

      6. *-un-lft-identity 98.6%

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

   7. Applied egg-rr 98.6%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.1 \cdot 10^{-126}:\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{\frac{1}{r \cdot w}}{r \cdot w}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{\frac{1 - v}{r}}{w \cdot \left(r \cdot w\right)}}\right)\right)\\ \end{array} \]

Alternative 5: 87.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 1.65 \cdot 10^{-92}:\\ \;\;\;\;-4.5 + \left(3 + \left(t_0 - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{\frac{1}{r \cdot w}}{r \cdot w}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 1.65e-92)
     (+
      -4.5
      (+
       3.0
       (- t_0 (/ (* 0.125 (+ 3.0 (* -2.0 v))) (/ (/ 1.0 (* r w)) (* r w))))))
     (+
      (+ t_0 (* (/ (+ -0.375 (* v 0.25)) (- 1.0 v)) (* r (* r (* w w)))))
      -1.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 1.65e-92) {
		tmp = -4.5 + (3.0 + (t_0 - ((0.125 * (3.0 + (-2.0 * v))) / ((1.0 / (r * w)) / (r * w)))));
	} else {
		tmp = (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (r * (w * w))))) + -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 (r <= 1.65d-92) then
        tmp = (-4.5d0) + (3.0d0 + (t_0 - ((0.125d0 * (3.0d0 + ((-2.0d0) * v))) / ((1.0d0 / (r * w)) / (r * w)))))
    else
        tmp = (t_0 + ((((-0.375d0) + (v * 0.25d0)) / (1.0d0 - v)) * (r * (r * (w * w))))) + (-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 (r <= 1.65e-92) {
		tmp = -4.5 + (3.0 + (t_0 - ((0.125 * (3.0 + (-2.0 * v))) / ((1.0 / (r * w)) / (r * w)))));
	} else {
		tmp = (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (r * (w * w))))) + -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 1.65e-92:
		tmp = -4.5 + (3.0 + (t_0 - ((0.125 * (3.0 + (-2.0 * v))) / ((1.0 / (r * w)) / (r * w)))))
	else:
		tmp = (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (r * (w * w))))) + -1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 1.65e-92)
		tmp = Float64(-4.5 + Float64(3.0 + Float64(t_0 - Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) / Float64(Float64(1.0 / Float64(r * w)) / Float64(r * w))))));
	else
		tmp = Float64(Float64(t_0 + Float64(Float64(Float64(-0.375 + Float64(v * 0.25)) / Float64(1.0 - v)) * Float64(r * Float64(r * Float64(w * w))))) + -1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 1.65e-92)
		tmp = -4.5 + (3.0 + (t_0 - ((0.125 * (3.0 + (-2.0 * v))) / ((1.0 / (r * w)) / (r * w)))));
	else
		tmp = (t_0 + (((-0.375 + (v * 0.25)) / (1.0 - v)) * (r * (r * (w * w))))) + -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[LessEqual[r, 1.65e-92], N[(-4.5 + N[(3.0 + N[(t$95$0 - N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + N[(N[(N[(-0.375 + N[(v * 0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.64999999999999999e-92

   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 85.8%

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

      1. associate-*r* 96.1%

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

      2. *-commutative 96.1%

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

      3. *-un-lft-identity 96.1%

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

      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. Step-by-step derivation

      1. associate-*l/ 99.8%

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

      2. *-un-lft-identity 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in v around 0 87.2%

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

   if 1.64999999999999999e-92 < r

   1. Initial program 82.8%

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

   3. Simplified 84.5%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.65 \cdot 10^{-92}:\\ \;\;\;\;-4.5 + \left(3 + \left(\frac{2}{r \cdot r} - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{\frac{1}{r \cdot w}}{r \cdot w}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5\\ \end{array} \]

Alternative 6: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

MATLAB

function tmp = code(v, w, r)
	tmp = -4.5 + (3.0 + ((2.0 / (r * r)) - ((0.125 * (3.0 + (-2.0 * v))) / (((1.0 - v) / (r * w)) / (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[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 - v), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.4%

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

3. Simplified 85.4%

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

   1. associate-*r* 97.2%

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

   2. *-commutative 97.2%

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

   3. *-un-lft-identity 97.2%

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

   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. Step-by-step derivation

   1. associate-*l/ 99.8%

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

   2. *-un-lft-identity 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 7: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(r \cdot \left(w \cdot w\right)\right)\\ \mathbf{if}\;r \leq 5.5 \cdot 10^{-88}:\\ \;\;\;\;-1.5 + \left(2 \cdot \frac{\frac{1}{r}}{r} + t_0 \cdot \left(\frac{0.125}{v} + -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot t_0\right) + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* r (* r (* w w)))))
   (if (<= r 5.5e-88)
     (+ -1.5 (+ (* 2.0 (/ (/ 1.0 r) r)) (* t_0 (+ (/ 0.125 v) -0.25))))
     (+
      (+ (/ 2.0 (* r r)) (* (/ (+ -0.375 (* v 0.25)) (- 1.0 v)) t_0))
      -1.5))))

C

double code(double v, double w, double r) {
	double t_0 = r * (r * (w * w));
	double tmp;
	if (r <= 5.5e-88) {
		tmp = -1.5 + ((2.0 * ((1.0 / r) / r)) + (t_0 * ((0.125 / v) + -0.25)));
	} else {
		tmp = ((2.0 / (r * r)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * t_0)) + -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 = r * (r * (w * w))
    if (r <= 5.5d-88) then
        tmp = (-1.5d0) + ((2.0d0 * ((1.0d0 / r) / r)) + (t_0 * ((0.125d0 / v) + (-0.25d0))))
    else
        tmp = ((2.0d0 / (r * r)) + ((((-0.375d0) + (v * 0.25d0)) / (1.0d0 - v)) * t_0)) + (-1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = r * (r * (w * w));
	double tmp;
	if (r <= 5.5e-88) {
		tmp = -1.5 + ((2.0 * ((1.0 / r) / r)) + (t_0 * ((0.125 / v) + -0.25)));
	} else {
		tmp = ((2.0 / (r * r)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * t_0)) + -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = r * (r * (w * w))
	tmp = 0
	if r <= 5.5e-88:
		tmp = -1.5 + ((2.0 * ((1.0 / r) / r)) + (t_0 * ((0.125 / v) + -0.25)))
	else:
		tmp = ((2.0 / (r * r)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * t_0)) + -1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(r * Float64(r * Float64(w * w)))
	tmp = 0.0
	if (r <= 5.5e-88)
		tmp = Float64(-1.5 + Float64(Float64(2.0 * Float64(Float64(1.0 / r) / r)) + Float64(t_0 * Float64(Float64(0.125 / v) + -0.25))));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(Float64(-0.375 + Float64(v * 0.25)) / Float64(1.0 - v)) * t_0)) + -1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = r * (r * (w * w));
	tmp = 0.0;
	if (r <= 5.5e-88)
		tmp = -1.5 + ((2.0 * ((1.0 / r) / r)) + (t_0 * ((0.125 / v) + -0.25)));
	else
		tmp = ((2.0 / (r * r)) + (((-0.375 + (v * 0.25)) / (1.0 - v)) * t_0)) + -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 5.5e-88], N[(-1.5 + N[(N[(2.0 * N[(N[(1.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(N[(0.125 / v), $MachinePrecision] + -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(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] * t$95$0), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 5.49999999999999971e-88

   1. Initial program 83.8%

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

   3. Simplified 85.8%

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

      1. associate-/r* 85.8%

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

      2. div-inv 85.8%

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

      3. *-un-lft-identity 85.8%

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

      4. times-frac 85.8%

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

      5. metadata-eval 85.8%

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

   5. Applied egg-rr 85.8%

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

   6. Taylor expanded in v around inf 79.3%

      \[\leadsto \left(2 \cdot \frac{\frac{1}{r}}{r} + \color{blue}{\left(0.125 \cdot \frac{1}{v} - 0.25\right)} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5 \]
   7. Step-by-step derivation

      1. sub-neg 79.3%

         \[\leadsto \left(2 \cdot \frac{\frac{1}{r}}{r} + \color{blue}{\left(0.125 \cdot \frac{1}{v} + \left(-0.25\right)\right)} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5 \]

      2. associate-*r/ 79.3%

         \[\leadsto \left(2 \cdot \frac{\frac{1}{r}}{r} + \left(\color{blue}{\frac{0.125 \cdot 1}{v}} + \left(-0.25\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5 \]

      3. metadata-eval 79.3%

         \[\leadsto \left(2 \cdot \frac{\frac{1}{r}}{r} + \left(\frac{\color{blue}{0.125}}{v} + \left(-0.25\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5 \]

      4. metadata-eval 79.3%

         \[\leadsto \left(2 \cdot \frac{\frac{1}{r}}{r} + \left(\frac{0.125}{v} + \color{blue}{-0.25}\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5 \]

   8. Simplified 79.3%

      \[\leadsto \left(2 \cdot \frac{\frac{1}{r}}{r} + \color{blue}{\left(\frac{0.125}{v} + -0.25\right)} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5 \]

   if 5.49999999999999971e-88 < r

   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 84.3%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.5 \cdot 10^{-88}:\\ \;\;\;\;-1.5 + \left(2 \cdot \frac{\frac{1}{r}}{r} + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{0.125}{v} + -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + \frac{-0.375 + v \cdot 0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5\\ \end{array} \]

Alternative 8: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(r \cdot \left(w \cdot w\right)\right)\\ t_1 := 2 \cdot \frac{\frac{1}{r}}{r}\\ \mathbf{if}\;r \leq 3.2 \cdot 10^{-100}:\\ \;\;\;\;-1.5 + \left(t_1 + t_0 \cdot \left(\frac{0.125}{v} + -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_1 + -0.375 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* r (* r (* w w)))) (t_1 (* 2.0 (/ (/ 1.0 r) r))))
   (if (<= r 3.2e-100)
     (+ -1.5 (+ t_1 (* t_0 (+ (/ 0.125 v) -0.25))))
     (+ -1.5 (+ t_1 (* -0.375 t_0))))))

C

double code(double v, double w, double r) {
	double t_0 = r * (r * (w * w));
	double t_1 = 2.0 * ((1.0 / r) / r);
	double tmp;
	if (r <= 3.2e-100) {
		tmp = -1.5 + (t_1 + (t_0 * ((0.125 / v) + -0.25)));
	} else {
		tmp = -1.5 + (t_1 + (-0.375 * t_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) :: t_1
    real(8) :: tmp
    t_0 = r * (r * (w * w))
    t_1 = 2.0d0 * ((1.0d0 / r) / r)
    if (r <= 3.2d-100) then
        tmp = (-1.5d0) + (t_1 + (t_0 * ((0.125d0 / v) + (-0.25d0))))
    else
        tmp = (-1.5d0) + (t_1 + ((-0.375d0) * t_0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = r * (r * (w * w));
	double t_1 = 2.0 * ((1.0 / r) / r);
	double tmp;
	if (r <= 3.2e-100) {
		tmp = -1.5 + (t_1 + (t_0 * ((0.125 / v) + -0.25)));
	} else {
		tmp = -1.5 + (t_1 + (-0.375 * t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = r * (r * (w * w))
	t_1 = 2.0 * ((1.0 / r) / r)
	tmp = 0
	if r <= 3.2e-100:
		tmp = -1.5 + (t_1 + (t_0 * ((0.125 / v) + -0.25)))
	else:
		tmp = -1.5 + (t_1 + (-0.375 * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(r * Float64(r * Float64(w * w)))
	t_1 = Float64(2.0 * Float64(Float64(1.0 / r) / r))
	tmp = 0.0
	if (r <= 3.2e-100)
		tmp = Float64(-1.5 + Float64(t_1 + Float64(t_0 * Float64(Float64(0.125 / v) + -0.25))));
	else
		tmp = Float64(-1.5 + Float64(t_1 + Float64(-0.375 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = r * (r * (w * w));
	t_1 = 2.0 * ((1.0 / r) / r);
	tmp = 0.0;
	if (r <= 3.2e-100)
		tmp = -1.5 + (t_1 + (t_0 * ((0.125 / v) + -0.25)));
	else
		tmp = -1.5 + (t_1 + (-0.375 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[(1.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 3.2e-100], N[(-1.5 + N[(t$95$1 + N[(t$95$0 * N[(N[(0.125 / v), $MachinePrecision] + -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(t$95$1 + N[(-0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if r < 3.20000000000000017e-100

   1. Initial program 83.9%

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

   3. Simplified 86.1%

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

      1. associate-/r* 86.1%

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

      2. div-inv 86.1%

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

      3. *-un-lft-identity 86.1%

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

      4. times-frac 86.1%

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

      5. metadata-eval 86.1%

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

   5. Applied egg-rr 86.1%

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

   6. Taylor expanded in v around inf 79.4%

      \[\leadsto \left(2 \cdot \frac{\frac{1}{r}}{r} + \color{blue}{\left(0.125 \cdot \frac{1}{v} - 0.25\right)} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5 \]
   7. Step-by-step derivation

      1. sub-neg 79.4%

         \[\leadsto \left(2 \cdot \frac{\frac{1}{r}}{r} + \color{blue}{\left(0.125 \cdot \frac{1}{v} + \left(-0.25\right)\right)} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5 \]

      2. associate-*r/ 79.4%

         \[\leadsto \left(2 \cdot \frac{\frac{1}{r}}{r} + \left(\color{blue}{\frac{0.125 \cdot 1}{v}} + \left(-0.25\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5 \]

      3. metadata-eval 79.4%

         \[\leadsto \left(2 \cdot \frac{\frac{1}{r}}{r} + \left(\frac{\color{blue}{0.125}}{v} + \left(-0.25\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5 \]

      4. metadata-eval 79.4%

         \[\leadsto \left(2 \cdot \frac{\frac{1}{r}}{r} + \left(\frac{0.125}{v} + \color{blue}{-0.25}\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5 \]

   8. Simplified 79.4%

      \[\leadsto \left(2 \cdot \frac{\frac{1}{r}}{r} + \color{blue}{\left(\frac{0.125}{v} + -0.25\right)} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + -1.5 \]

   if 3.20000000000000017e-100 < r

   1. Initial program 82.3%

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

   3. Simplified 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. Step-by-step derivation

      1. associate-/r* 83.9%

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

      2. div-inv 83.9%

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

      3. *-un-lft-identity 83.9%

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

      4. times-frac 83.9%

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

      5. metadata-eval 83.9%

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

   5. Applied egg-rr 83.9%

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

   6. Taylor expanded in v around 0 81.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3.2 \cdot 10^{-100}:\\ \;\;\;\;-1.5 + \left(2 \cdot \frac{\frac{1}{r}}{r} + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{0.125}{v} + -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(2 \cdot \frac{\frac{1}{r}}{r} + -0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \]

Alternative 9: 85.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(r \cdot \left(w \cdot w\right)\right)\\ t_1 := 2 \cdot \frac{\frac{1}{r}}{r}\\ \mathbf{if}\;v \leq -4 \cdot 10^{+23}:\\ \;\;\;\;-1.5 + \left(t_1 + t_0 \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(t_1 + -0.375 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* r (* r (* w w)))) (t_1 (* 2.0 (/ (/ 1.0 r) r))))
   (if (<= v -4e+23)
     (+ -1.5 (+ t_1 (* t_0 -0.25)))
     (+ -1.5 (+ t_1 (* -0.375 t_0))))))

C

double code(double v, double w, double r) {
	double t_0 = r * (r * (w * w));
	double t_1 = 2.0 * ((1.0 / r) / r);
	double tmp;
	if (v <= -4e+23) {
		tmp = -1.5 + (t_1 + (t_0 * -0.25));
	} else {
		tmp = -1.5 + (t_1 + (-0.375 * t_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) :: t_1
    real(8) :: tmp
    t_0 = r * (r * (w * w))
    t_1 = 2.0d0 * ((1.0d0 / r) / r)
    if (v <= (-4d+23)) then
        tmp = (-1.5d0) + (t_1 + (t_0 * (-0.25d0)))
    else
        tmp = (-1.5d0) + (t_1 + ((-0.375d0) * t_0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = r * (r * (w * w));
	double t_1 = 2.0 * ((1.0 / r) / r);
	double tmp;
	if (v <= -4e+23) {
		tmp = -1.5 + (t_1 + (t_0 * -0.25));
	} else {
		tmp = -1.5 + (t_1 + (-0.375 * t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = r * (r * (w * w))
	t_1 = 2.0 * ((1.0 / r) / r)
	tmp = 0
	if v <= -4e+23:
		tmp = -1.5 + (t_1 + (t_0 * -0.25))
	else:
		tmp = -1.5 + (t_1 + (-0.375 * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(r * Float64(r * Float64(w * w)))
	t_1 = Float64(2.0 * Float64(Float64(1.0 / r) / r))
	tmp = 0.0
	if (v <= -4e+23)
		tmp = Float64(-1.5 + Float64(t_1 + Float64(t_0 * -0.25)));
	else
		tmp = Float64(-1.5 + Float64(t_1 + Float64(-0.375 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = r * (r * (w * w));
	t_1 = 2.0 * ((1.0 / r) / r);
	tmp = 0.0;
	if (v <= -4e+23)
		tmp = -1.5 + (t_1 + (t_0 * -0.25));
	else
		tmp = -1.5 + (t_1 + (-0.375 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[(1.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -4e+23], N[(-1.5 + N[(t$95$1 + N[(t$95$0 * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(t$95$1 + N[(-0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -3.9999999999999997e23

   1. Initial program 76.4%

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

   3. Simplified 84.1%

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

      1. associate-/r* 84.0%

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

      2. div-inv 84.0%

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

      3. *-un-lft-identity 84.0%

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

      4. times-frac 84.0%

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

      5. metadata-eval 84.0%

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

   5. Applied egg-rr 84.0%

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

   6. Taylor expanded in v around inf 84.0%

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

   if -3.9999999999999997e23 < v

   1. Initial program 85.1%

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

   3. Simplified 85.7%

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

      1. associate-/r* 85.7%

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

      2. div-inv 85.7%

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

      3. *-un-lft-identity 85.7%

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

      4. times-frac 85.7%

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

      5. metadata-eval 85.7%

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

   5. Applied egg-rr 85.7%

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

   6. Taylor expanded in v around 0 84.8%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -4 \cdot 10^{+23}:\\ \;\;\;\;-1.5 + \left(2 \cdot \frac{\frac{1}{r}}{r} + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(2 \cdot \frac{\frac{1}{r}}{r} + -0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \]

Alternative 10: 83.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

3. Simplified 85.4%

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

   1. associate-/r* 85.4%

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

   2. div-inv 85.4%

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

   3. *-un-lft-identity 85.4%

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

   4. times-frac 85.4%

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

   5. metadata-eval 85.4%

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

5. Applied egg-rr 85.4%

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

6. Taylor expanded in v around 0 82.5%

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

7. Final simplification 82.5%

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

Reproduce

herbie shell --seed 2023334 
(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))