Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.9% → 88.3%

Time: 2.3min

Alternatives: 8

Speedup: 1905.0×

• 32.6% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = 15.798768730026065
  2. b = 8.504910518499819e+307
  3. angle = 9.48679026983315e+297
  4. x-scale = -4.174837770477229e-305
  5. y-scale = -4.482545722572893e+104

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\\ t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Initial Program: 24.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\\ t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Alternative 1: 88.3% accurate, 26.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := b\_m \cdot \left(a \cdot \frac{-4}{x-scale \cdot y-scale}\right)\\ \mathbf{if}\;b\_m \leq 1.16 \cdot 10^{-201}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(b\_m \cdot \left(a \cdot \left(b\_m \cdot a\right)\right)\right)}{x-scale \cdot y-scale}}{x-scale \cdot y-scale}\\ \mathbf{elif}\;b\_m \leq 3.6 \cdot 10^{+213}:\\ \;\;\;\;\left(b\_m \cdot t\_0\right) \cdot \frac{\frac{a}{x-scale}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{x-scale} \cdot \left(t\_0 \cdot \frac{b\_m}{y-scale}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* b_m (* a (/ -4.0 (* x-scale y-scale))))))
   (if (<= b_m 1.16e-201)
     (/
      (/ (* -4.0 (* b_m (* a (* b_m a)))) (* x-scale y-scale))
      (* x-scale y-scale))
     (if (<= b_m 3.6e+213)
       (* (* b_m t_0) (/ (/ a x-scale) y-scale))
       (* (/ a x-scale) (* t_0 (/ b_m y-scale)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b_m * (a * (-4.0 / (x_45_scale * y_45_scale)));
	double tmp;
	if (b_m <= 1.16e-201) {
		tmp = ((-4.0 * (b_m * (a * (b_m * a)))) / (x_45_scale * y_45_scale)) / (x_45_scale * y_45_scale);
	} else if (b_m <= 3.6e+213) {
		tmp = (b_m * t_0) * ((a / x_45_scale) / y_45_scale);
	} else {
		tmp = (a / x_45_scale) * (t_0 * (b_m / y_45_scale));
	}
	return tmp;
}

Fortran

b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b_m * (a * ((-4.0d0) / (x_45scale * y_45scale)))
    if (b_m <= 1.16d-201) then
        tmp = (((-4.0d0) * (b_m * (a * (b_m * a)))) / (x_45scale * y_45scale)) / (x_45scale * y_45scale)
    else if (b_m <= 3.6d+213) then
        tmp = (b_m * t_0) * ((a / x_45scale) / y_45scale)
    else
        tmp = (a / x_45scale) * (t_0 * (b_m / y_45scale))
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b_m * (a * (-4.0 / (x_45_scale * y_45_scale)));
	double tmp;
	if (b_m <= 1.16e-201) {
		tmp = ((-4.0 * (b_m * (a * (b_m * a)))) / (x_45_scale * y_45_scale)) / (x_45_scale * y_45_scale);
	} else if (b_m <= 3.6e+213) {
		tmp = (b_m * t_0) * ((a / x_45_scale) / y_45_scale);
	} else {
		tmp = (a / x_45_scale) * (t_0 * (b_m / y_45_scale));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = b_m * (a * (-4.0 / (x_45_scale * y_45_scale)))
	tmp = 0
	if b_m <= 1.16e-201:
		tmp = ((-4.0 * (b_m * (a * (b_m * a)))) / (x_45_scale * y_45_scale)) / (x_45_scale * y_45_scale)
	elif b_m <= 3.6e+213:
		tmp = (b_m * t_0) * ((a / x_45_scale) / y_45_scale)
	else:
		tmp = (a / x_45_scale) * (t_0 * (b_m / y_45_scale))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b_m * Float64(a * Float64(-4.0 / Float64(x_45_scale * y_45_scale))))
	tmp = 0.0
	if (b_m <= 1.16e-201)
		tmp = Float64(Float64(Float64(-4.0 * Float64(b_m * Float64(a * Float64(b_m * a)))) / Float64(x_45_scale * y_45_scale)) / Float64(x_45_scale * y_45_scale));
	elseif (b_m <= 3.6e+213)
		tmp = Float64(Float64(b_m * t_0) * Float64(Float64(a / x_45_scale) / y_45_scale));
	else
		tmp = Float64(Float64(a / x_45_scale) * Float64(t_0 * Float64(b_m / y_45_scale)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = b_m * (a * (-4.0 / (x_45_scale * y_45_scale)));
	tmp = 0.0;
	if (b_m <= 1.16e-201)
		tmp = ((-4.0 * (b_m * (a * (b_m * a)))) / (x_45_scale * y_45_scale)) / (x_45_scale * y_45_scale);
	elseif (b_m <= 3.6e+213)
		tmp = (b_m * t_0) * ((a / x_45_scale) / y_45_scale);
	else
		tmp = (a / x_45_scale) * (t_0 * (b_m / y_45_scale));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b$95$m * N[(a * N[(-4.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 1.16e-201], N[(N[(N[(-4.0 * N[(b$95$m * N[(a * N[(b$95$m * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 3.6e+213], N[(N[(b$95$m * t$95$0), $MachinePrecision] * N[(N[(a / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision], N[(N[(a / x$45$scale), $MachinePrecision] * N[(t$95$0 * N[(b$95$m / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.15999999999999995e-201

   1. Initial program 30.1%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 52.7

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 52.7%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 60.2%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot \color{blue}{\frac{a}{x-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 69.0%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a}{x-scale}}{y-scale}} \]
   10. Step-by-step derivation

   11. Applied rewrites 81.1%

       \[\leadsto \frac{\frac{-4 \cdot \left(b \cdot \left(a \cdot \left(a \cdot b\right)\right)\right)}{x-scale \cdot y-scale}}{\color{blue}{x-scale \cdot y-scale}} \]

   if 1.15999999999999995e-201 < b < 3.6000000000000001e213

   1. Initial program 25.9%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 60.0

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 60.0%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.8%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot \color{blue}{\frac{a}{x-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 87.2%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a}{x-scale}}{y-scale}} \]
   10. Step-by-step derivation

   11. Applied rewrites 93.4%

       \[\leadsto \left(\left(\left(a \cdot \frac{-4}{x-scale \cdot y-scale}\right) \cdot b\right) \cdot b\right) \cdot \frac{\color{blue}{\frac{a}{x-scale}}}{y-scale} \]

   if 3.6000000000000001e213 < b

   1. Initial program 0.0%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 50.0

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.7%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot \color{blue}{\frac{a}{x-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 50.7%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{\left(x-scale \cdot y-scale\right) \cdot y-scale} \cdot \frac{a}{x-scale} \]
   10. Step-by-step derivation

   11. Applied rewrites 91.9%

       \[\leadsto \left(\left(\left(a \cdot \frac{-4}{x-scale \cdot y-scale}\right) \cdot b\right) \cdot \frac{b}{y-scale}\right) \cdot \frac{\color{blue}{a}}{x-scale} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{-201}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(b \cdot \left(a \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot y-scale}}{x-scale \cdot y-scale}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{+213}:\\ \;\;\;\;\left(b \cdot \left(b \cdot \left(a \cdot \frac{-4}{x-scale \cdot y-scale}\right)\right)\right) \cdot \frac{\frac{a}{x-scale}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{x-scale} \cdot \left(\left(b \cdot \left(a \cdot \frac{-4}{x-scale \cdot y-scale}\right)\right) \cdot \frac{b}{y-scale}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.9% accurate, 29.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2.1 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(b\_m \cdot \left(a \cdot \left(b\_m \cdot a\right)\right)\right)}{x-scale \cdot y-scale}}{x-scale \cdot y-scale}\\ \mathbf{elif}\;b\_m \leq 5 \cdot 10^{+146}:\\ \;\;\;\;\frac{a \cdot \frac{a \cdot \left(-4 \cdot \left(b\_m \cdot b\_m\right)\right)}{x-scale \cdot y-scale}}{x-scale \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \left(b\_m \cdot \left(a \cdot -4\right)\right)\right) \cdot \frac{b\_m}{y-scale \cdot \left(x-scale \cdot y-scale\right)}}{x-scale}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 2.1e-158)
   (/
    (/ (* -4.0 (* b_m (* a (* b_m a)))) (* x-scale y-scale))
    (* x-scale y-scale))
   (if (<= b_m 5e+146)
     (/
      (* a (/ (* a (* -4.0 (* b_m b_m))) (* x-scale y-scale)))
      (* x-scale y-scale))
     (/
      (* (* a (* b_m (* a -4.0))) (/ b_m (* y-scale (* x-scale y-scale))))
      x-scale))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 2.1e-158) {
		tmp = ((-4.0 * (b_m * (a * (b_m * a)))) / (x_45_scale * y_45_scale)) / (x_45_scale * y_45_scale);
	} else if (b_m <= 5e+146) {
		tmp = (a * ((a * (-4.0 * (b_m * b_m))) / (x_45_scale * y_45_scale))) / (x_45_scale * y_45_scale);
	} else {
		tmp = ((a * (b_m * (a * -4.0))) * (b_m / (y_45_scale * (x_45_scale * y_45_scale)))) / x_45_scale;
	}
	return tmp;
}

Fortran

b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (b_m <= 2.1d-158) then
        tmp = (((-4.0d0) * (b_m * (a * (b_m * a)))) / (x_45scale * y_45scale)) / (x_45scale * y_45scale)
    else if (b_m <= 5d+146) then
        tmp = (a * ((a * ((-4.0d0) * (b_m * b_m))) / (x_45scale * y_45scale))) / (x_45scale * y_45scale)
    else
        tmp = ((a * (b_m * (a * (-4.0d0)))) * (b_m / (y_45scale * (x_45scale * y_45scale)))) / x_45scale
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 2.1e-158) {
		tmp = ((-4.0 * (b_m * (a * (b_m * a)))) / (x_45_scale * y_45_scale)) / (x_45_scale * y_45_scale);
	} else if (b_m <= 5e+146) {
		tmp = (a * ((a * (-4.0 * (b_m * b_m))) / (x_45_scale * y_45_scale))) / (x_45_scale * y_45_scale);
	} else {
		tmp = ((a * (b_m * (a * -4.0))) * (b_m / (y_45_scale * (x_45_scale * y_45_scale)))) / x_45_scale;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 2.1e-158:
		tmp = ((-4.0 * (b_m * (a * (b_m * a)))) / (x_45_scale * y_45_scale)) / (x_45_scale * y_45_scale)
	elif b_m <= 5e+146:
		tmp = (a * ((a * (-4.0 * (b_m * b_m))) / (x_45_scale * y_45_scale))) / (x_45_scale * y_45_scale)
	else:
		tmp = ((a * (b_m * (a * -4.0))) * (b_m / (y_45_scale * (x_45_scale * y_45_scale)))) / x_45_scale
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 2.1e-158)
		tmp = Float64(Float64(Float64(-4.0 * Float64(b_m * Float64(a * Float64(b_m * a)))) / Float64(x_45_scale * y_45_scale)) / Float64(x_45_scale * y_45_scale));
	elseif (b_m <= 5e+146)
		tmp = Float64(Float64(a * Float64(Float64(a * Float64(-4.0 * Float64(b_m * b_m))) / Float64(x_45_scale * y_45_scale))) / Float64(x_45_scale * y_45_scale));
	else
		tmp = Float64(Float64(Float64(a * Float64(b_m * Float64(a * -4.0))) * Float64(b_m / Float64(y_45_scale * Float64(x_45_scale * y_45_scale)))) / x_45_scale);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 2.1e-158)
		tmp = ((-4.0 * (b_m * (a * (b_m * a)))) / (x_45_scale * y_45_scale)) / (x_45_scale * y_45_scale);
	elseif (b_m <= 5e+146)
		tmp = (a * ((a * (-4.0 * (b_m * b_m))) / (x_45_scale * y_45_scale))) / (x_45_scale * y_45_scale);
	else
		tmp = ((a * (b_m * (a * -4.0))) * (b_m / (y_45_scale * (x_45_scale * y_45_scale)))) / x_45_scale;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 2.1e-158], N[(N[(N[(-4.0 * N[(b$95$m * N[(a * N[(b$95$m * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 5e+146], N[(N[(a * N[(N[(a * N[(-4.0 * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(b$95$m * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b$95$m / N[(y$45$scale * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.09999999999999991e-158

   1. Initial program 30.1%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 52.3

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 52.3%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 59.5%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot \color{blue}{\frac{a}{x-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.9%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a}{x-scale}}{y-scale}} \]
   10. Step-by-step derivation

   11. Applied rewrites 80.4%

       \[\leadsto \frac{\frac{-4 \cdot \left(b \cdot \left(a \cdot \left(a \cdot b\right)\right)\right)}{x-scale \cdot y-scale}}{\color{blue}{x-scale \cdot y-scale}} \]

   if 2.09999999999999991e-158 < b < 4.9999999999999999e146

   1. Initial program 27.9%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 60.3

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 60.3%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.4%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot \color{blue}{\frac{a}{x-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 92.7%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a}{x-scale}}{y-scale}} \]
   10. Step-by-step derivation

   11. Applied rewrites 94.9%

       \[\leadsto \frac{\frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot a}{\color{blue}{x-scale \cdot y-scale}} \]

   if 4.9999999999999999e146 < b

   1. Initial program 4.5%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 59.1

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 64.3%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot \color{blue}{\frac{a}{x-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 64.7%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a}{x-scale}}{y-scale}} \]
   10. Step-by-step derivation

   11. Applied rewrites 90.1%

       \[\leadsto \frac{\left(a \cdot \left(\left(a \cdot -4\right) \cdot b\right)\right) \cdot \frac{b}{y-scale \cdot \left(x-scale \cdot y-scale\right)}}{\color{blue}{x-scale}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(b \cdot \left(a \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot y-scale}}{x-scale \cdot y-scale}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+146}:\\ \;\;\;\;\frac{a \cdot \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale}}{x-scale \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \left(b \cdot \left(a \cdot -4\right)\right)\right) \cdot \frac{b}{y-scale \cdot \left(x-scale \cdot y-scale\right)}}{x-scale}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.2% accurate, 29.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.1 \cdot 10^{-158}:\\ \;\;\;\;\frac{4 \cdot \left(\left(a \cdot \left(b\_m \cdot a\right)\right) \cdot \left(-b\_m\right)\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \mathbf{elif}\;b\_m \leq 5 \cdot 10^{+146}:\\ \;\;\;\;\frac{a \cdot \frac{a \cdot \left(-4 \cdot \left(b\_m \cdot b\_m\right)\right)}{x-scale \cdot y-scale}}{x-scale \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \left(b\_m \cdot \left(a \cdot -4\right)\right)\right) \cdot \frac{b\_m}{y-scale \cdot \left(x-scale \cdot y-scale\right)}}{x-scale}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 1.1e-158)
   (/
    (* 4.0 (* (* a (* b_m a)) (- b_m)))
    (* (* x-scale y-scale) (* x-scale y-scale)))
   (if (<= b_m 5e+146)
     (/
      (* a (/ (* a (* -4.0 (* b_m b_m))) (* x-scale y-scale)))
      (* x-scale y-scale))
     (/
      (* (* a (* b_m (* a -4.0))) (/ b_m (* y-scale (* x-scale y-scale))))
      x-scale))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 1.1e-158) {
		tmp = (4.0 * ((a * (b_m * a)) * -b_m)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale));
	} else if (b_m <= 5e+146) {
		tmp = (a * ((a * (-4.0 * (b_m * b_m))) / (x_45_scale * y_45_scale))) / (x_45_scale * y_45_scale);
	} else {
		tmp = ((a * (b_m * (a * -4.0))) * (b_m / (y_45_scale * (x_45_scale * y_45_scale)))) / x_45_scale;
	}
	return tmp;
}

Fortran

b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (b_m <= 1.1d-158) then
        tmp = (4.0d0 * ((a * (b_m * a)) * -b_m)) / ((x_45scale * y_45scale) * (x_45scale * y_45scale))
    else if (b_m <= 5d+146) then
        tmp = (a * ((a * ((-4.0d0) * (b_m * b_m))) / (x_45scale * y_45scale))) / (x_45scale * y_45scale)
    else
        tmp = ((a * (b_m * (a * (-4.0d0)))) * (b_m / (y_45scale * (x_45scale * y_45scale)))) / x_45scale
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 1.1e-158) {
		tmp = (4.0 * ((a * (b_m * a)) * -b_m)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale));
	} else if (b_m <= 5e+146) {
		tmp = (a * ((a * (-4.0 * (b_m * b_m))) / (x_45_scale * y_45_scale))) / (x_45_scale * y_45_scale);
	} else {
		tmp = ((a * (b_m * (a * -4.0))) * (b_m / (y_45_scale * (x_45_scale * y_45_scale)))) / x_45_scale;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 1.1e-158:
		tmp = (4.0 * ((a * (b_m * a)) * -b_m)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))
	elif b_m <= 5e+146:
		tmp = (a * ((a * (-4.0 * (b_m * b_m))) / (x_45_scale * y_45_scale))) / (x_45_scale * y_45_scale)
	else:
		tmp = ((a * (b_m * (a * -4.0))) * (b_m / (y_45_scale * (x_45_scale * y_45_scale)))) / x_45_scale
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 1.1e-158)
		tmp = Float64(Float64(4.0 * Float64(Float64(a * Float64(b_m * a)) * Float64(-b_m))) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale)));
	elseif (b_m <= 5e+146)
		tmp = Float64(Float64(a * Float64(Float64(a * Float64(-4.0 * Float64(b_m * b_m))) / Float64(x_45_scale * y_45_scale))) / Float64(x_45_scale * y_45_scale));
	else
		tmp = Float64(Float64(Float64(a * Float64(b_m * Float64(a * -4.0))) * Float64(b_m / Float64(y_45_scale * Float64(x_45_scale * y_45_scale)))) / x_45_scale);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 1.1e-158)
		tmp = (4.0 * ((a * (b_m * a)) * -b_m)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale));
	elseif (b_m <= 5e+146)
		tmp = (a * ((a * (-4.0 * (b_m * b_m))) / (x_45_scale * y_45_scale))) / (x_45_scale * y_45_scale);
	else
		tmp = ((a * (b_m * (a * -4.0))) * (b_m / (y_45_scale * (x_45_scale * y_45_scale)))) / x_45_scale;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 1.1e-158], N[(N[(4.0 * N[(N[(a * N[(b$95$m * a), $MachinePrecision]), $MachinePrecision] * (-b$95$m)), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 5e+146], N[(N[(a * N[(N[(a * N[(-4.0 * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(b$95$m * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b$95$m / N[(y$45$scale * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.1000000000000001e-158

   1. Initial program 30.1%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 52.3

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 52.3%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 59.5%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot \color{blue}{\frac{a}{x-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.9%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a}{x-scale}}{y-scale}} \]
   10. Step-by-step derivation

   11. Applied rewrites 74.5%

       \[\leadsto \frac{4 \cdot \left(b \cdot \left(a \cdot \left(a \cdot b\right)\right)\right)}{\color{blue}{\left(x-scale \cdot \left(-y-scale\right)\right) \cdot \left(x-scale \cdot y-scale\right)}} \]

   if 1.1000000000000001e-158 < b < 4.9999999999999999e146

   1. Initial program 27.9%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 60.3

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 60.3%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 78.4%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot \color{blue}{\frac{a}{x-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 92.7%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a}{x-scale}}{y-scale}} \]
   10. Step-by-step derivation

   11. Applied rewrites 94.9%

       \[\leadsto \frac{\frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot a}{\color{blue}{x-scale \cdot y-scale}} \]

   if 4.9999999999999999e146 < b

   1. Initial program 4.5%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 59.1

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 64.3%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot \color{blue}{\frac{a}{x-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 64.7%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a}{x-scale}}{y-scale}} \]
   10. Step-by-step derivation

   11. Applied rewrites 90.1%

       \[\leadsto \frac{\left(a \cdot \left(\left(a \cdot -4\right) \cdot b\right)\right) \cdot \frac{b}{y-scale \cdot \left(x-scale \cdot y-scale\right)}}{\color{blue}{x-scale}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-158}:\\ \;\;\;\;\frac{4 \cdot \left(\left(a \cdot \left(b \cdot a\right)\right) \cdot \left(-b\right)\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+146}:\\ \;\;\;\;\frac{a \cdot \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale}}{x-scale \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \left(b \cdot \left(a \cdot -4\right)\right)\right) \cdot \frac{b}{y-scale \cdot \left(x-scale \cdot y-scale\right)}}{x-scale}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.9% accurate, 29.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.02 \cdot 10^{-133}:\\ \;\;\;\;\frac{4 \cdot \left(\left(a \cdot \left(b\_m \cdot a\right)\right) \cdot \left(-b\_m\right)\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \mathbf{elif}\;b\_m \leq 3.7 \cdot 10^{+141}:\\ \;\;\;\;a \cdot \left(\frac{-4 \cdot \left(b\_m \cdot b\_m\right)}{x-scale \cdot y-scale} \cdot \frac{a}{x-scale \cdot y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \left(b\_m \cdot \left(a \cdot -4\right)\right)\right) \cdot \frac{b\_m}{y-scale \cdot \left(x-scale \cdot y-scale\right)}}{x-scale}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 1.02e-133)
   (/
    (* 4.0 (* (* a (* b_m a)) (- b_m)))
    (* (* x-scale y-scale) (* x-scale y-scale)))
   (if (<= b_m 3.7e+141)
     (*
      a
      (*
       (/ (* -4.0 (* b_m b_m)) (* x-scale y-scale))
       (/ a (* x-scale y-scale))))
     (/
      (* (* a (* b_m (* a -4.0))) (/ b_m (* y-scale (* x-scale y-scale))))
      x-scale))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 1.02e-133) {
		tmp = (4.0 * ((a * (b_m * a)) * -b_m)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale));
	} else if (b_m <= 3.7e+141) {
		tmp = a * (((-4.0 * (b_m * b_m)) / (x_45_scale * y_45_scale)) * (a / (x_45_scale * y_45_scale)));
	} else {
		tmp = ((a * (b_m * (a * -4.0))) * (b_m / (y_45_scale * (x_45_scale * y_45_scale)))) / x_45_scale;
	}
	return tmp;
}

Fortran

b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (b_m <= 1.02d-133) then
        tmp = (4.0d0 * ((a * (b_m * a)) * -b_m)) / ((x_45scale * y_45scale) * (x_45scale * y_45scale))
    else if (b_m <= 3.7d+141) then
        tmp = a * ((((-4.0d0) * (b_m * b_m)) / (x_45scale * y_45scale)) * (a / (x_45scale * y_45scale)))
    else
        tmp = ((a * (b_m * (a * (-4.0d0)))) * (b_m / (y_45scale * (x_45scale * y_45scale)))) / x_45scale
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 1.02e-133) {
		tmp = (4.0 * ((a * (b_m * a)) * -b_m)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale));
	} else if (b_m <= 3.7e+141) {
		tmp = a * (((-4.0 * (b_m * b_m)) / (x_45_scale * y_45_scale)) * (a / (x_45_scale * y_45_scale)));
	} else {
		tmp = ((a * (b_m * (a * -4.0))) * (b_m / (y_45_scale * (x_45_scale * y_45_scale)))) / x_45_scale;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 1.02e-133:
		tmp = (4.0 * ((a * (b_m * a)) * -b_m)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))
	elif b_m <= 3.7e+141:
		tmp = a * (((-4.0 * (b_m * b_m)) / (x_45_scale * y_45_scale)) * (a / (x_45_scale * y_45_scale)))
	else:
		tmp = ((a * (b_m * (a * -4.0))) * (b_m / (y_45_scale * (x_45_scale * y_45_scale)))) / x_45_scale
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 1.02e-133)
		tmp = Float64(Float64(4.0 * Float64(Float64(a * Float64(b_m * a)) * Float64(-b_m))) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale)));
	elseif (b_m <= 3.7e+141)
		tmp = Float64(a * Float64(Float64(Float64(-4.0 * Float64(b_m * b_m)) / Float64(x_45_scale * y_45_scale)) * Float64(a / Float64(x_45_scale * y_45_scale))));
	else
		tmp = Float64(Float64(Float64(a * Float64(b_m * Float64(a * -4.0))) * Float64(b_m / Float64(y_45_scale * Float64(x_45_scale * y_45_scale)))) / x_45_scale);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 1.02e-133)
		tmp = (4.0 * ((a * (b_m * a)) * -b_m)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale));
	elseif (b_m <= 3.7e+141)
		tmp = a * (((-4.0 * (b_m * b_m)) / (x_45_scale * y_45_scale)) * (a / (x_45_scale * y_45_scale)));
	else
		tmp = ((a * (b_m * (a * -4.0))) * (b_m / (y_45_scale * (x_45_scale * y_45_scale)))) / x_45_scale;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 1.02e-133], N[(N[(4.0 * N[(N[(a * N[(b$95$m * a), $MachinePrecision]), $MachinePrecision] * (-b$95$m)), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 3.7e+141], N[(a * N[(N[(N[(-4.0 * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(a / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(b$95$m * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b$95$m / N[(y$45$scale * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.02e-133

   1. Initial program 29.8%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 53.1

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 53.1%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 60.3%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot \color{blue}{\frac{a}{x-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.9%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a}{x-scale}}{y-scale}} \]
   10. Step-by-step derivation

   11. Applied rewrites 75.2%

       \[\leadsto \frac{4 \cdot \left(b \cdot \left(a \cdot \left(a \cdot b\right)\right)\right)}{\color{blue}{\left(x-scale \cdot \left(-y-scale\right)\right) \cdot \left(x-scale \cdot y-scale\right)}} \]

   if 1.02e-133 < b < 3.7000000000000003e141

   1. Initial program 29.2%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 58.8

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 77.0%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot \color{blue}{\frac{a}{x-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 91.9%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a}{x-scale}}{y-scale}} \]
   10. Step-by-step derivation

   11. Applied rewrites 94.7%

       \[\leadsto a \cdot \color{blue}{\left(\frac{-4 \cdot \left(b \cdot b\right)}{x-scale \cdot y-scale} \cdot \frac{a}{x-scale \cdot y-scale}\right)} \]

   if 3.7000000000000003e141 < b

   1. Initial program 4.3%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 58.5

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 67.1%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot \color{blue}{\frac{a}{x-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.6%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a}{x-scale}}{y-scale}} \]
   10. Step-by-step derivation

   11. Applied rewrites 91.0%

       \[\leadsto \frac{\left(a \cdot \left(\left(a \cdot -4\right) \cdot b\right)\right) \cdot \frac{b}{y-scale \cdot \left(x-scale \cdot y-scale\right)}}{\color{blue}{x-scale}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.02 \cdot 10^{-133}:\\ \;\;\;\;\frac{4 \cdot \left(\left(a \cdot \left(b \cdot a\right)\right) \cdot \left(-b\right)\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{+141}:\\ \;\;\;\;a \cdot \left(\frac{-4 \cdot \left(b \cdot b\right)}{x-scale \cdot y-scale} \cdot \frac{a}{x-scale \cdot y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \left(b \cdot \left(a \cdot -4\right)\right)\right) \cdot \frac{b}{y-scale \cdot \left(x-scale \cdot y-scale\right)}}{x-scale}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.4% accurate, 29.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.02 \cdot 10^{-133}:\\ \;\;\;\;\frac{4 \cdot \left(\left(a \cdot \left(b\_m \cdot a\right)\right) \cdot \left(-b\_m\right)\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \mathbf{elif}\;b\_m \leq 3.7 \cdot 10^{+141}:\\ \;\;\;\;a \cdot \left(\frac{-4 \cdot \left(b\_m \cdot b\_m\right)}{x-scale \cdot y-scale} \cdot \frac{a}{x-scale \cdot y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(b\_m \cdot \left(a \cdot -4\right)\right)\right) \cdot \frac{b\_m}{x-scale \cdot \left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 1.02e-133)
   (/
    (* 4.0 (* (* a (* b_m a)) (- b_m)))
    (* (* x-scale y-scale) (* x-scale y-scale)))
   (if (<= b_m 3.7e+141)
     (*
      a
      (*
       (/ (* -4.0 (* b_m b_m)) (* x-scale y-scale))
       (/ a (* x-scale y-scale))))
     (*
      (* a (* b_m (* a -4.0)))
      (/ b_m (* x-scale (* y-scale (* x-scale y-scale))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 1.02e-133) {
		tmp = (4.0 * ((a * (b_m * a)) * -b_m)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale));
	} else if (b_m <= 3.7e+141) {
		tmp = a * (((-4.0 * (b_m * b_m)) / (x_45_scale * y_45_scale)) * (a / (x_45_scale * y_45_scale)));
	} else {
		tmp = (a * (b_m * (a * -4.0))) * (b_m / (x_45_scale * (y_45_scale * (x_45_scale * y_45_scale))));
	}
	return tmp;
}

Fortran

b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (b_m <= 1.02d-133) then
        tmp = (4.0d0 * ((a * (b_m * a)) * -b_m)) / ((x_45scale * y_45scale) * (x_45scale * y_45scale))
    else if (b_m <= 3.7d+141) then
        tmp = a * ((((-4.0d0) * (b_m * b_m)) / (x_45scale * y_45scale)) * (a / (x_45scale * y_45scale)))
    else
        tmp = (a * (b_m * (a * (-4.0d0)))) * (b_m / (x_45scale * (y_45scale * (x_45scale * y_45scale))))
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 1.02e-133) {
		tmp = (4.0 * ((a * (b_m * a)) * -b_m)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale));
	} else if (b_m <= 3.7e+141) {
		tmp = a * (((-4.0 * (b_m * b_m)) / (x_45_scale * y_45_scale)) * (a / (x_45_scale * y_45_scale)));
	} else {
		tmp = (a * (b_m * (a * -4.0))) * (b_m / (x_45_scale * (y_45_scale * (x_45_scale * y_45_scale))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 1.02e-133:
		tmp = (4.0 * ((a * (b_m * a)) * -b_m)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))
	elif b_m <= 3.7e+141:
		tmp = a * (((-4.0 * (b_m * b_m)) / (x_45_scale * y_45_scale)) * (a / (x_45_scale * y_45_scale)))
	else:
		tmp = (a * (b_m * (a * -4.0))) * (b_m / (x_45_scale * (y_45_scale * (x_45_scale * y_45_scale))))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 1.02e-133)
		tmp = Float64(Float64(4.0 * Float64(Float64(a * Float64(b_m * a)) * Float64(-b_m))) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale)));
	elseif (b_m <= 3.7e+141)
		tmp = Float64(a * Float64(Float64(Float64(-4.0 * Float64(b_m * b_m)) / Float64(x_45_scale * y_45_scale)) * Float64(a / Float64(x_45_scale * y_45_scale))));
	else
		tmp = Float64(Float64(a * Float64(b_m * Float64(a * -4.0))) * Float64(b_m / Float64(x_45_scale * Float64(y_45_scale * Float64(x_45_scale * y_45_scale)))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 1.02e-133)
		tmp = (4.0 * ((a * (b_m * a)) * -b_m)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale));
	elseif (b_m <= 3.7e+141)
		tmp = a * (((-4.0 * (b_m * b_m)) / (x_45_scale * y_45_scale)) * (a / (x_45_scale * y_45_scale)));
	else
		tmp = (a * (b_m * (a * -4.0))) * (b_m / (x_45_scale * (y_45_scale * (x_45_scale * y_45_scale))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 1.02e-133], N[(N[(4.0 * N[(N[(a * N[(b$95$m * a), $MachinePrecision]), $MachinePrecision] * (-b$95$m)), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 3.7e+141], N[(a * N[(N[(N[(-4.0 * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(a / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(b$95$m * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b$95$m / N[(x$45$scale * N[(y$45$scale * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.02e-133

   1. Initial program 29.8%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 53.1

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 53.1%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 60.3%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot \color{blue}{\frac{a}{x-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.9%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a}{x-scale}}{y-scale}} \]
   10. Step-by-step derivation

   11. Applied rewrites 75.2%

       \[\leadsto \frac{4 \cdot \left(b \cdot \left(a \cdot \left(a \cdot b\right)\right)\right)}{\color{blue}{\left(x-scale \cdot \left(-y-scale\right)\right) \cdot \left(x-scale \cdot y-scale\right)}} \]

   if 1.02e-133 < b < 3.7000000000000003e141

   1. Initial program 29.2%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 58.8

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 58.8%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 77.0%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot \color{blue}{\frac{a}{x-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 91.9%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a}{x-scale}}{y-scale}} \]
   10. Step-by-step derivation

   11. Applied rewrites 94.7%

       \[\leadsto a \cdot \color{blue}{\left(\frac{-4 \cdot \left(b \cdot b\right)}{x-scale \cdot y-scale} \cdot \frac{a}{x-scale \cdot y-scale}\right)} \]

   if 3.7000000000000003e141 < b

   1. Initial program 4.3%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 58.5

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 67.1%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot \color{blue}{\frac{a}{x-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 67.6%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a}{x-scale}}{y-scale}} \]

   11. Applied rewrites 83.7%

       \[\leadsto \frac{b}{x-scale \cdot \left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right)} \cdot \color{blue}{\left(a \cdot \left(\left(a \cdot -4\right) \cdot b\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.02 \cdot 10^{-133}:\\ \;\;\;\;\frac{4 \cdot \left(\left(a \cdot \left(b \cdot a\right)\right) \cdot \left(-b\right)\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{+141}:\\ \;\;\;\;a \cdot \left(\frac{-4 \cdot \left(b \cdot b\right)}{x-scale \cdot y-scale} \cdot \frac{a}{x-scale \cdot y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(b \cdot \left(a \cdot -4\right)\right)\right) \cdot \frac{b}{x-scale \cdot \left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.8% accurate, 29.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 6 \cdot 10^{-221}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(b\_m \cdot \left(a \cdot \left(b\_m \cdot a\right)\right)\right)}{x-scale \cdot y-scale}}{x-scale \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m \cdot \left(b\_m \cdot \left(a \cdot \frac{-4}{x-scale \cdot y-scale}\right)\right)\right) \cdot \frac{\frac{a}{x-scale}}{y-scale}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 6e-221)
   (/
    (/ (* -4.0 (* b_m (* a (* b_m a)))) (* x-scale y-scale))
    (* x-scale y-scale))
   (*
    (* b_m (* b_m (* a (/ -4.0 (* x-scale y-scale)))))
    (/ (/ a x-scale) y-scale))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 6e-221) {
		tmp = ((-4.0 * (b_m * (a * (b_m * a)))) / (x_45_scale * y_45_scale)) / (x_45_scale * y_45_scale);
	} else {
		tmp = (b_m * (b_m * (a * (-4.0 / (x_45_scale * y_45_scale))))) * ((a / x_45_scale) / y_45_scale);
	}
	return tmp;
}

Fortran

b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (b_m <= 6d-221) then
        tmp = (((-4.0d0) * (b_m * (a * (b_m * a)))) / (x_45scale * y_45scale)) / (x_45scale * y_45scale)
    else
        tmp = (b_m * (b_m * (a * ((-4.0d0) / (x_45scale * y_45scale))))) * ((a / x_45scale) / y_45scale)
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 6e-221) {
		tmp = ((-4.0 * (b_m * (a * (b_m * a)))) / (x_45_scale * y_45_scale)) / (x_45_scale * y_45_scale);
	} else {
		tmp = (b_m * (b_m * (a * (-4.0 / (x_45_scale * y_45_scale))))) * ((a / x_45_scale) / y_45_scale);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 6e-221:
		tmp = ((-4.0 * (b_m * (a * (b_m * a)))) / (x_45_scale * y_45_scale)) / (x_45_scale * y_45_scale)
	else:
		tmp = (b_m * (b_m * (a * (-4.0 / (x_45_scale * y_45_scale))))) * ((a / x_45_scale) / y_45_scale)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 6e-221)
		tmp = Float64(Float64(Float64(-4.0 * Float64(b_m * Float64(a * Float64(b_m * a)))) / Float64(x_45_scale * y_45_scale)) / Float64(x_45_scale * y_45_scale));
	else
		tmp = Float64(Float64(b_m * Float64(b_m * Float64(a * Float64(-4.0 / Float64(x_45_scale * y_45_scale))))) * Float64(Float64(a / x_45_scale) / y_45_scale));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 6e-221)
		tmp = ((-4.0 * (b_m * (a * (b_m * a)))) / (x_45_scale * y_45_scale)) / (x_45_scale * y_45_scale);
	else
		tmp = (b_m * (b_m * (a * (-4.0 / (x_45_scale * y_45_scale))))) * ((a / x_45_scale) / y_45_scale);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 6e-221], N[(N[(N[(-4.0 * N[(b$95$m * N[(a * N[(b$95$m * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$m * N[(b$95$m * N[(a * N[(-4.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(a / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.0000000000000003e-221

   1. Initial program 30.4%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 53.3

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 53.3%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 60.9%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot \color{blue}{\frac{a}{x-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 69.8%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a}{x-scale}}{y-scale}} \]
   10. Step-by-step derivation

   11. Applied rewrites 81.5%

       \[\leadsto \frac{\frac{-4 \cdot \left(b \cdot \left(a \cdot \left(a \cdot b\right)\right)\right)}{x-scale \cdot y-scale}}{\color{blue}{x-scale \cdot y-scale}} \]

   if 6.0000000000000003e-221 < b

   1. Initial program 22.2%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      9. unpow2 N/A

         \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

      16. lower-*.f64 57.6

          \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   5. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 71.2%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot \color{blue}{\frac{a}{x-scale}} \]
   8. Step-by-step derivation

   9. Applied rewrites 81.0%

      \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a}{x-scale}}{y-scale}} \]
   10. Step-by-step derivation

   11. Applied rewrites 89.5%

       \[\leadsto \left(\left(\left(a \cdot \frac{-4}{x-scale \cdot y-scale}\right) \cdot b\right) \cdot b\right) \cdot \frac{\color{blue}{\frac{a}{x-scale}}}{y-scale} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{-221}:\\ \;\;\;\;\frac{\frac{-4 \cdot \left(b \cdot \left(a \cdot \left(b \cdot a\right)\right)\right)}{x-scale \cdot y-scale}}{x-scale \cdot y-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot \left(b \cdot \left(a \cdot \frac{-4}{x-scale \cdot y-scale}\right)\right)\right) \cdot \frac{\frac{a}{x-scale}}{y-scale}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.3% accurate, 40.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ a \cdot \left(\left(b\_m \cdot \left(a \cdot -4\right)\right) \cdot \frac{b\_m}{x-scale \cdot \left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (*
  a
  (* (* b_m (* a -4.0)) (/ b_m (* x-scale (* y-scale (* x-scale y-scale)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return a * ((b_m * (a * -4.0)) * (b_m / (x_45_scale * (y_45_scale * (x_45_scale * y_45_scale)))));
}

Fortran

b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = a * ((b_m * (a * (-4.0d0))) * (b_m / (x_45scale * (y_45scale * (x_45scale * y_45scale)))))
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return a * ((b_m * (a * -4.0)) * (b_m / (x_45_scale * (y_45_scale * (x_45_scale * y_45_scale)))));
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	return a * ((b_m * (a * -4.0)) * (b_m / (x_45_scale * (y_45_scale * (x_45_scale * y_45_scale)))))

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	return Float64(a * Float64(Float64(b_m * Float64(a * -4.0)) * Float64(b_m / Float64(x_45_scale * Float64(y_45_scale * Float64(x_45_scale * y_45_scale))))))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = a * ((b_m * (a * -4.0)) * (b_m / (x_45_scale * (y_45_scale * (x_45_scale * y_45_scale)))));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(a * N[(N[(b$95$m * N[(a * -4.0), $MachinePrecision]), $MachinePrecision] * N[(b$95$m / N[(x$45$scale * N[(y$45$scale * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.3%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   3. *-commutative N/A

      \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right) \cdot {a}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-4 \cdot {b}^{2}\right)} \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   7. unpow2 N/A

      \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\left(-4 \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {a}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(x-scale \cdot x-scale\right)} \cdot {y-scale}^{2}} \]

   12. associate-*l* N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{\color{blue}{x-scale \cdot \left(x-scale \cdot {y-scale}^{2}\right)}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \color{blue}{\left(x-scale \cdot {y-scale}^{2}\right)}} \]

   15. unpow2 N/A

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

   16. lower-*.f64 54.9

       \[\leadsto \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale\right)}\right)} \]

5. Applied rewrites 54.9%

   \[\leadsto \color{blue}{\frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 64.8%

   \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(y-scale \cdot y-scale\right)} \cdot \color{blue}{\frac{a}{x-scale}} \]
8. Step-by-step derivation

9. Applied rewrites 74.1%

   \[\leadsto \frac{a \cdot \left(-4 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{a}{x-scale}}{y-scale}} \]

11. Applied rewrites 77.9%

    \[\leadsto a \cdot \color{blue}{\left(\left(\left(a \cdot -4\right) \cdot b\right) \cdot \frac{b}{x-scale \cdot \left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\right)} \]

12. Final simplification 77.9%

    \[\leadsto a \cdot \left(\left(b \cdot \left(a \cdot -4\right)\right) \cdot \frac{b}{x-scale \cdot \left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\right) \]
13. Add Preprocessing

Alternative 8: 35.6% accurate, 1905.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ 0 \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale) :precision binary64 0.0)

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 0.0;
}

Fortran

b_m = abs(b)
real(8) function code(a, b_m, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = 0.0d0
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 0.0;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	return 0.0

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	return 0.0
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := 0.0

Derivation

1. Initial program 27.3%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{1}{8100} \cdot \frac{{\mathsf{PI}\left(\right)}^{2} \cdot {\left({b}^{2} - {a}^{2}\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} - 4 \cdot \left(\frac{{a}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{b}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right) - 4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

5. Applied rewrites 16.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \mathsf{fma}\left(0.0001234567901234568, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \pi\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}, -4 \cdot \mathsf{fma}\left(b, b \cdot \frac{\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, a \cdot a, 3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}, \frac{\left(a \cdot a\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, b \cdot b, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}\right)\right), \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}\right)} \]

6. Taylor expanded in b around 0

   \[\leadsto {angle}^{2} \cdot \color{blue}{\left(\frac{-1}{8100} \cdot \frac{{a}^{4} \cdot {\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{1}{8100} \cdot \frac{{a}^{4} \cdot {\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 23.7%

   \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{0} \]

9. Taylor expanded in angle around 0

   \[\leadsto 0 \]
10. Step-by-step derivation

11. Applied rewrites 34.7%

    \[\leadsto 0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(FPCore (a b angle x-scale y-scale)
  :name "Simplification of discriminant from scale-rotated-ellipse"
  :precision binary64
  (- (* (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale) (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale)) (* (* 4.0 (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale)) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale))))