Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.2% → 93.8%

Time: 24.1s

Alternatives: 6

Speedup: 40.5×

• 66.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (/ angle 180) PI))
       (t_1 (sin t_0))
       (t_2 (cos t_0))
       (t_3
        (/
         (/ (* (* (* 2 (- (pow b 2) (pow a 2))) t_1) t_2) x-scale)
         y-scale)))
  (-
   (* t_3 t_3)
   (*
    (*
     4
     (/ (/ (+ (pow (* a t_1) 2) (pow (* b t_2) 2)) x-scale) x-scale))
    (/ (/ (+ (pow (* a t_2) 2) (pow (* b t_1) 2)) 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), $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 * N[(N[Power[b, 2], $MachinePrecision] - N[Power[a, 2], $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 * N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Initial Program: 25.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (/ angle 180) PI))
       (t_1 (sin t_0))
       (t_2 (cos t_0))
       (t_3
        (/
         (/ (* (* (* 2 (- (pow b 2) (pow a 2))) t_1) t_2) x-scale)
         y-scale)))
  (-
   (* t_3 t_3)
   (*
    (*
     4
     (/ (/ (+ (pow (* a t_1) 2) (pow (* b t_2) 2)) x-scale) x-scale))
    (/ (/ (+ (pow (* a t_2) 2) (pow (* b t_1) 2)) 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), $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 * N[(N[Power[b, 2], $MachinePrecision] - N[Power[a, 2], $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 * N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Alternative 1: 93.8% accurate, 24.7× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| \cdot \frac{b}{y-scale \cdot x-scale}\\ t_1 := \frac{\frac{b}{y-scale} \cdot \left|a\right|}{x-scale}\\ \mathbf{if}\;\left|a\right| \leq 10000000000000000:\\ \;\;\;\;-4 \cdot \left(t\_1 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t\_0 \cdot t\_0\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (fabs a) (/ b (* y-scale x-scale))))
       (t_1 (/ (* (/ b y-scale) (fabs a)) x-scale)))
  (if (<= (fabs a) 10000000000000000)
    (* -4 (* t_1 t_1))
    (* -4 (* t_0 t_0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(a) * (b / (y_45_scale * x_45_scale));
	double t_1 = ((b / y_45_scale) * fabs(a)) / x_45_scale;
	double tmp;
	if (fabs(a) <= 1e+16) {
		tmp = -4.0 * (t_1 * t_1);
	} else {
		tmp = -4.0 * (t_0 * t_0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs(a) * (b / (y_45scale * x_45scale))
    t_1 = ((b / y_45scale) * abs(a)) / x_45scale
    if (abs(a) <= 1d+16) then
        tmp = (-4.0d0) * (t_1 * t_1)
    else
        tmp = (-4.0d0) * (t_0 * t_0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.abs(a) * (b / (y_45_scale * x_45_scale));
	double t_1 = ((b / y_45_scale) * Math.abs(a)) / x_45_scale;
	double tmp;
	if (Math.abs(a) <= 1e+16) {
		tmp = -4.0 * (t_1 * t_1);
	} else {
		tmp = -4.0 * (t_0 * t_0);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.fabs(a) * (b / (y_45_scale * x_45_scale))
	t_1 = ((b / y_45_scale) * math.fabs(a)) / x_45_scale
	tmp = 0
	if math.fabs(a) <= 1e+16:
		tmp = -4.0 * (t_1 * t_1)
	else:
		tmp = -4.0 * (t_0 * t_0)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(a) * Float64(b / Float64(y_45_scale * x_45_scale)))
	t_1 = Float64(Float64(Float64(b / y_45_scale) * abs(a)) / x_45_scale)
	tmp = 0.0
	if (abs(a) <= 1e+16)
		tmp = Float64(-4.0 * Float64(t_1 * t_1));
	else
		tmp = Float64(-4.0 * Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(a) * (b / (y_45_scale * x_45_scale));
	t_1 = ((b / y_45_scale) * abs(a)) / x_45_scale;
	tmp = 0.0;
	if (abs(a) <= 1e+16)
		tmp = -4.0 * (t_1 * t_1);
	else
		tmp = -4.0 * (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[a], $MachinePrecision] * N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b / y$45$scale), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 10000000000000000], N[(-4 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(-4 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1e16

   1. Initial program 25.2%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 60.6%

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. swap-sqr N/A

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

      18. unpow2 N/A

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

      19. lift-pow.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. *-commutative N/A

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

      23. *-commutative N/A

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

      24. lift-pow.f64 N/A

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

      25. unpow2 N/A

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

      26. lift-*.f64 N/A

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

   6. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 89.1%

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

   10. Applied rewrites 89.1%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-/r* N/A

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

       6. lift-/.f64 N/A

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

       7. associate-*l/ N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 93.6%

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

   12. Applied rewrites 93.6%

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

   if 1e16 < a

   1. Initial program 25.2%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 60.6%

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. swap-sqr N/A

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

      18. unpow2 N/A

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

      19. lift-pow.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. *-commutative N/A

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

      23. *-commutative N/A

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

      24. lift-pow.f64 N/A

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

      25. unpow2 N/A

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

      26. lift-*.f64 N/A

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

   6. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 93.8%

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

Alternative 2: 93.7% accurate, 35.9× speedup

Math

\[\begin{array}{l} t_0 := a \cdot \frac{b}{y-scale \cdot x-scale}\\ -4 \cdot \left(t\_0 \cdot t\_0\right) \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* a (/ b (* y-scale x-scale))))) (* -4 (* t_0 t_0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * (b / (y_45_scale * x_45_scale));
	return -4.0 * (t_0 * t_0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    t_0 = a * (b / (y_45scale * x_45scale))
    code = (-4.0d0) * (t_0 * t_0)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * (b / (y_45_scale * x_45_scale));
	return -4.0 * (t_0 * t_0);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = a * (b / (y_45_scale * x_45_scale))
	return -4.0 * (t_0 * t_0)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a * Float64(b / Float64(y_45_scale * x_45_scale)))
	return Float64(-4.0 * Float64(t_0 * t_0))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = a * (b / (y_45_scale * x_45_scale));
	tmp = -4.0 * (t_0 * t_0);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a * N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(-4 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 25.2%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 47.9%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-prod-down N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 60.6%

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. swap-sqr N/A

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

   18. unpow2 N/A

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

   19. lift-pow.f64 N/A

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. *-commutative N/A

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

   23. *-commutative N/A

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

   24. lift-pow.f64 N/A

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

   25. unpow2 N/A

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

   26. lift-*.f64 N/A

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

6. Applied rewrites 59.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lift-*.f64 N/A

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

   6. swap-sqr N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. associate-/l* N/A

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

   18. lower-*.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. associate-/l* N/A

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

   23. lower-*.f64 N/A

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

8. Applied rewrites 93.8%

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

Alternative 3: 80.1% accurate, 40.5× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (*
 -4
 (* (* a b) (/ (* a b) (* (* (* y-scale x-scale) y-scale) x-scale)))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return -4.0 * ((a * b) * ((a * b) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)))

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(-4 * N[(N[(a * b), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.2%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 47.9%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-prod-down N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 60.6%

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. swap-sqr N/A

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

   18. unpow2 N/A

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

   19. lift-pow.f64 N/A

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. *-commutative N/A

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

   23. *-commutative N/A

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

   24. lift-pow.f64 N/A

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

   25. unpow2 N/A

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

   26. lift-*.f64 N/A

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

6. Applied rewrites 59.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lift-*.f64 N/A

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

   6. swap-sqr N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 80.1%

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 80.1%

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

8. Applied rewrites 80.1%

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

Alternative 4: 76.9% accurate, 32.3× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale\\ \mathbf{if}\;\left|b\right| \leq 390000000000000022808896731591674565280901202904021504371261300347429246460755656065414657728987710303451007148907348751825844443255411334236814010413770216644787797229568:\\ \;\;\;\;-4 \cdot \left(a \cdot \left(\left|b\right| \cdot \left(\frac{a}{t\_0} \cdot \left|b\right|\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left|b\right| \cdot \frac{a \cdot \left(a \cdot \left|b\right|\right)}{t\_0}\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (* (* y-scale x-scale) y-scale) x-scale)))
  (if (<=
       (fabs b)
       390000000000000022808896731591674565280901202904021504371261300347429246460755656065414657728987710303451007148907348751825844443255411334236814010413770216644787797229568)
    (* -4 (* a (* (fabs b) (* (/ a t_0) (fabs b)))))
    (* -4 (* (fabs b) (/ (* a (* a (fabs b))) t_0))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale;
	double tmp;
	if (fabs(b) <= 3.9e+170) {
		tmp = -4.0 * (a * (fabs(b) * ((a / t_0) * fabs(b))));
	} else {
		tmp = -4.0 * (fabs(b) * ((a * (a * fabs(b))) / t_0));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    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 = ((y_45scale * x_45scale) * y_45scale) * x_45scale
    if (abs(b) <= 3.9d+170) then
        tmp = (-4.0d0) * (a * (abs(b) * ((a / t_0) * abs(b))))
    else
        tmp = (-4.0d0) * (abs(b) * ((a * (a * abs(b))) / t_0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale;
	double tmp;
	if (Math.abs(b) <= 3.9e+170) {
		tmp = -4.0 * (a * (Math.abs(b) * ((a / t_0) * Math.abs(b))));
	} else {
		tmp = -4.0 * (Math.abs(b) * ((a * (a * Math.abs(b))) / t_0));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = ((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale
	tmp = 0
	if math.fabs(b) <= 3.9e+170:
		tmp = -4.0 * (a * (math.fabs(b) * ((a / t_0) * math.fabs(b))))
	else:
		tmp = -4.0 * (math.fabs(b) * ((a * (a * math.fabs(b))) / t_0))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)
	tmp = 0.0
	if (abs(b) <= 3.9e+170)
		tmp = Float64(-4.0 * Float64(a * Float64(abs(b) * Float64(Float64(a / t_0) * abs(b)))));
	else
		tmp = Float64(-4.0 * Float64(abs(b) * Float64(Float64(a * Float64(a * abs(b))) / t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = ((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale;
	tmp = 0.0;
	if (abs(b) <= 3.9e+170)
		tmp = -4.0 * (a * (abs(b) * ((a / t_0) * abs(b))));
	else
		tmp = -4.0 * (abs(b) * ((a * (a * abs(b))) / t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 390000000000000022808896731591674565280901202904021504371261300347429246460755656065414657728987710303451007148907348751825844443255411334236814010413770216644787797229568], N[(-4 * N[(a * N[(N[Abs[b], $MachinePrecision] * N[(N[(a / t$95$0), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4 * N[(N[Abs[b], $MachinePrecision] * N[(N[(a * N[(a * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.9000000000000002e170

   1. Initial program 25.2%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 60.6%

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. swap-sqr N/A

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

      18. unpow2 N/A

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

      19. lift-pow.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. *-commutative N/A

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

      23. *-commutative N/A

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

      24. lift-pow.f64 N/A

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

      25. unpow2 N/A

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

      26. lift-*.f64 N/A

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

   6. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r/ N/A

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

   10. Applied rewrites 76.4%

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

   if 3.9000000000000002e170 < b

   1. Initial program 25.2%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 60.6%

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. swap-sqr N/A

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

      18. unpow2 N/A

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

      19. lift-pow.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. *-commutative N/A

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

      23. *-commutative N/A

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

      24. lift-pow.f64 N/A

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

      25. unpow2 N/A

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

      26. lift-*.f64 N/A

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

   6. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 73.3%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 73.3%

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

   8. Applied rewrites 73.3%

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

Alternative 5: 76.5% accurate, 32.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 15000000000000001222640764792521223979384301462143890135988560884996241183905177530591823112499256611932191444551219896571654622986967420155985920:\\ \;\;\;\;-4 \cdot \left(a \cdot \left(\left|b\right| \cdot \left(\frac{a}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot \left|b\right|\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(\left|b\right| \cdot \left(\left|b\right| \cdot \left(a \cdot \frac{a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (if (<=
     (fabs b)
     15000000000000001222640764792521223979384301462143890135988560884996241183905177530591823112499256611932191444551219896571654622986967420155985920)
  (*
   -4
   (*
    a
    (*
     (fabs b)
     (* (/ a (* (* (* y-scale x-scale) y-scale) x-scale)) (fabs b)))))
  (*
   -4
   (*
    (fabs b)
    (*
     (fabs b)
     (* a (/ a (* (* y-scale x-scale) (* y-scale x-scale)))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (fabs(b) <= 1.5e+145) {
		tmp = -4.0 * (a * (fabs(b) * ((a / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * fabs(b))));
	} else {
		tmp = -4.0 * (fabs(b) * (fabs(b) * (a * (a / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (abs(b) <= 1.5d+145) then
        tmp = (-4.0d0) * (a * (abs(b) * ((a / (((y_45scale * x_45scale) * y_45scale) * x_45scale)) * abs(b))))
    else
        tmp = (-4.0d0) * (abs(b) * (abs(b) * (a * (a / ((y_45scale * x_45scale) * (y_45scale * x_45scale))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (Math.abs(b) <= 1.5e+145) {
		tmp = -4.0 * (a * (Math.abs(b) * ((a / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * Math.abs(b))));
	} else {
		tmp = -4.0 * (Math.abs(b) * (Math.abs(b) * (a * (a / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))))));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if math.fabs(b) <= 1.5e+145:
		tmp = -4.0 * (a * (math.fabs(b) * ((a / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * math.fabs(b))))
	else:
		tmp = -4.0 * (math.fabs(b) * (math.fabs(b) * (a * (a / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))))))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (abs(b) <= 1.5e+145)
		tmp = Float64(-4.0 * Float64(a * Float64(abs(b) * Float64(Float64(a / Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * abs(b)))));
	else
		tmp = Float64(-4.0 * Float64(abs(b) * Float64(abs(b) * Float64(a * Float64(a / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (abs(b) <= 1.5e+145)
		tmp = -4.0 * (a * (abs(b) * ((a / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * abs(b))));
	else
		tmp = -4.0 * (abs(b) * (abs(b) * (a * (a / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[N[Abs[b], $MachinePrecision], 15000000000000001222640764792521223979384301462143890135988560884996241183905177530591823112499256611932191444551219896571654622986967420155985920], N[(-4 * N[(a * N[(N[Abs[b], $MachinePrecision] * N[(N[(a / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4 * N[(N[Abs[b], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[(a * N[(a / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5000000000000001e145

   1. Initial program 25.2%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 60.6%

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. swap-sqr N/A

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

      18. unpow2 N/A

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

      19. lift-pow.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. *-commutative N/A

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

      23. *-commutative N/A

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

      24. lift-pow.f64 N/A

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

      25. unpow2 N/A

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

      26. lift-*.f64 N/A

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

   6. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 93.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. frac-times N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r/ N/A

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

   10. Applied rewrites 76.4%

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

   if 1.5000000000000001e145 < b

   1. Initial program 25.2%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 60.6%

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. swap-sqr N/A

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

      18. unpow2 N/A

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

      19. lift-pow.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. *-commutative N/A

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

      23. *-commutative N/A

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

      24. lift-pow.f64 N/A

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

      25. unpow2 N/A

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

      26. lift-*.f64 N/A

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

   6. Applied rewrites 59.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 66.7%

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 74.2%

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

   8. Applied rewrites 74.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 76.6%

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

   10. Applied rewrites 76.6%

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

Alternative 6: 76.4% accurate, 40.5× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (*
 -4
 (* a (* b (* (/ a (* (* (* y-scale x-scale) y-scale) x-scale)) b)))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return -4.0 * (a * (b * ((a / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * b)))

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(-4 * N[(a * N[(b * N[(N[(a / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.2%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 47.9%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-prod-down N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 60.6%

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. swap-sqr N/A

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

   18. unpow2 N/A

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

   19. lift-pow.f64 N/A

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. *-commutative N/A

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

   23. *-commutative N/A

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

   24. lift-pow.f64 N/A

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

   25. unpow2 N/A

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

   26. lift-*.f64 N/A

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

6. Applied rewrites 59.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lift-*.f64 N/A

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

   6. swap-sqr N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. associate-/l* N/A

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

   18. lower-*.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. associate-/l* N/A

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

   23. lower-*.f64 N/A

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

8. Applied rewrites 93.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*r/ N/A

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

   9. lift-*.f64 N/A

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

   10. frac-times N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. associate-*r/ N/A

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

10. Applied rewrites 76.4%

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

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (a b angle x-scale y-scale)
  :name "Simplification of discriminant from scale-rotated-ellipse"
  :precision binary64
  (- (* (/ (/ (* (* (* 2 (- (pow b 2) (pow a 2))) (sin (* (/ angle 180) PI))) (cos (* (/ angle 180) PI))) x-scale) y-scale) (/ (/ (* (* (* 2 (- (pow b 2) (pow a 2))) (sin (* (/ angle 180) PI))) (cos (* (/ angle 180) PI))) x-scale) y-scale)) (* (* 4 (/ (/ (+ (pow (* a (sin (* (/ angle 180) PI))) 2) (pow (* b (cos (* (/ angle 180) PI))) 2)) x-scale) x-scale)) (/ (/ (+ (pow (* a (cos (* (/ angle 180) PI))) 2) (pow (* b (sin (* (/ angle 180) PI))) 2)) y-scale) y-scale))))