Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.1% → 91.8%

Time: 14.4s

Alternatives: 9

Speedup: 20.4×

• 33.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.8% accurate, 15.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{a}{y-scale \cdot x-scale}\\ t_1 := \frac{-4}{y-scale \cdot x-scale}\\ \mathbf{if}\;b\_m \leq 1.85 \cdot 10^{-166}:\\ \;\;\;\;\left(t\_1 \cdot \left(\left(t\_0 \cdot b\_m\right) \cdot a\right)\right) \cdot b\_m\\ \mathbf{elif}\;b\_m \leq 10^{+139}:\\ \;\;\;\;\left(\left(\left(b\_m \cdot b\_m\right) \cdot t\_0\right) \cdot t\_0\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(b\_m \cdot \left(a \cdot \frac{a \cdot b\_m}{y-scale \cdot x-scale}\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ a (* y-scale x-scale))) (t_1 (/ -4.0 (* y-scale x-scale))))
   (if (<= b_m 1.85e-166)
     (* (* t_1 (* (* t_0 b_m) a)) b_m)
     (if (<= b_m 1e+139)
       (* (* (* (* b_m b_m) t_0) t_0) -4.0)
       (* t_1 (* b_m (* a (/ (* a b_m) (* y-scale x-scale)))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a / (y_45_scale * x_45_scale);
	double t_1 = -4.0 / (y_45_scale * x_45_scale);
	double tmp;
	if (b_m <= 1.85e-166) {
		tmp = (t_1 * ((t_0 * b_m) * a)) * b_m;
	} else if (b_m <= 1e+139) {
		tmp = (((b_m * b_m) * t_0) * t_0) * -4.0;
	} else {
		tmp = t_1 * (b_m * (a * ((a * b_m) / (y_45_scale * x_45_scale))));
	}
	return tmp;
}

Fortran

b_m =     private
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_m, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a / (y_45scale * x_45scale)
    t_1 = (-4.0d0) / (y_45scale * x_45scale)
    if (b_m <= 1.85d-166) then
        tmp = (t_1 * ((t_0 * b_m) * a)) * b_m
    else if (b_m <= 1d+139) then
        tmp = (((b_m * b_m) * t_0) * t_0) * (-4.0d0)
    else
        tmp = t_1 * (b_m * (a * ((a * b_m) / (y_45scale * x_45scale))))
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a / (y_45_scale * x_45_scale);
	double t_1 = -4.0 / (y_45_scale * x_45_scale);
	double tmp;
	if (b_m <= 1.85e-166) {
		tmp = (t_1 * ((t_0 * b_m) * a)) * b_m;
	} else if (b_m <= 1e+139) {
		tmp = (((b_m * b_m) * t_0) * t_0) * -4.0;
	} else {
		tmp = t_1 * (b_m * (a * ((a * b_m) / (y_45_scale * x_45_scale))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = a / (y_45_scale * x_45_scale)
	t_1 = -4.0 / (y_45_scale * x_45_scale)
	tmp = 0
	if b_m <= 1.85e-166:
		tmp = (t_1 * ((t_0 * b_m) * a)) * b_m
	elif b_m <= 1e+139:
		tmp = (((b_m * b_m) * t_0) * t_0) * -4.0
	else:
		tmp = t_1 * (b_m * (a * ((a * b_m) / (y_45_scale * x_45_scale))))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a / Float64(y_45_scale * x_45_scale))
	t_1 = Float64(-4.0 / Float64(y_45_scale * x_45_scale))
	tmp = 0.0
	if (b_m <= 1.85e-166)
		tmp = Float64(Float64(t_1 * Float64(Float64(t_0 * b_m) * a)) * b_m);
	elseif (b_m <= 1e+139)
		tmp = Float64(Float64(Float64(Float64(b_m * b_m) * t_0) * t_0) * -4.0);
	else
		tmp = Float64(t_1 * Float64(b_m * Float64(a * Float64(Float64(a * b_m) / Float64(y_45_scale * x_45_scale)))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = a / (y_45_scale * x_45_scale);
	t_1 = -4.0 / (y_45_scale * x_45_scale);
	tmp = 0.0;
	if (b_m <= 1.85e-166)
		tmp = (t_1 * ((t_0 * b_m) * a)) * b_m;
	elseif (b_m <= 1e+139)
		tmp = (((b_m * b_m) * t_0) * t_0) * -4.0;
	else
		tmp = t_1 * (b_m * (a * ((a * b_m) / (y_45_scale * x_45_scale))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 1.85e-166], N[(N[(t$95$1 * N[(N[(t$95$0 * b$95$m), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision], If[LessEqual[b$95$m, 1e+139], N[(N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] * -4.0), $MachinePrecision], N[(t$95$1 * N[(b$95$m * N[(a * N[(N[(a * b$95$m), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.8500000000000001e-166

   1. Initial program 25.1%

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

   2. 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 48.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.6

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

   6. Applied rewrites 71.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 88.6

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

   8. Applied rewrites 88.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 88.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 89.8

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

   10. Applied rewrites 89.8%

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

   if 1.8500000000000001e-166 < b < 1.00000000000000003e139

   1. Initial program 25.1%

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

   2. 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 48.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.6

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

   6. Applied rewrites 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

      17. lift-*.f64 N/A

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

   8. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. times-frac N/A

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

      17. lift-/.f64 N/A

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

      18. lower-*.f64 N/A

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

   10. Applied rewrites 78.8%

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

   if 1.00000000000000003e139 < b

   1. Initial program 25.1%

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

   2. 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 48.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.6

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

   6. Applied rewrites 71.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 88.6

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

   8. Applied rewrites 88.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 89.3

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 89.3

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

   10. Applied rewrites 89.3%

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

Alternative 2: 89.8% accurate, 15.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{a}{y-scale \cdot x-scale}\\ t_1 := \left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\\ \mathbf{if}\;b\_m \leq 1.85 \cdot 10^{-166}:\\ \;\;\;\;\left(\left(a \cdot b\_m\right) \cdot \left(\frac{b\_m}{t\_1} \cdot a\right)\right) \cdot -4\\ \mathbf{elif}\;b\_m \leq 7.5 \cdot 10^{+151}:\\ \;\;\;\;\left(\left(\left(b\_m \cdot b\_m\right) \cdot t\_0\right) \cdot t\_0\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m \cdot \frac{\left(a \cdot b\_m\right) \cdot a}{t\_1}\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ a (* y-scale x-scale)))
        (t_1 (* (* (* y-scale x-scale) x-scale) y-scale)))
   (if (<= b_m 1.85e-166)
     (* (* (* a b_m) (* (/ b_m t_1) a)) -4.0)
     (if (<= b_m 7.5e+151)
       (* (* (* (* b_m b_m) t_0) t_0) -4.0)
       (* (* b_m (/ (* (* a b_m) a) t_1)) -4.0)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a / (y_45_scale * x_45_scale);
	double t_1 = ((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale;
	double tmp;
	if (b_m <= 1.85e-166) {
		tmp = ((a * b_m) * ((b_m / t_1) * a)) * -4.0;
	} else if (b_m <= 7.5e+151) {
		tmp = (((b_m * b_m) * t_0) * t_0) * -4.0;
	} else {
		tmp = (b_m * (((a * b_m) * a) / t_1)) * -4.0;
	}
	return tmp;
}

Fortran

b_m =     private
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_m, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a / (y_45scale * x_45scale)
    t_1 = ((y_45scale * x_45scale) * x_45scale) * y_45scale
    if (b_m <= 1.85d-166) then
        tmp = ((a * b_m) * ((b_m / t_1) * a)) * (-4.0d0)
    else if (b_m <= 7.5d+151) then
        tmp = (((b_m * b_m) * t_0) * t_0) * (-4.0d0)
    else
        tmp = (b_m * (((a * b_m) * a) / t_1)) * (-4.0d0)
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a / (y_45_scale * x_45_scale);
	double t_1 = ((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale;
	double tmp;
	if (b_m <= 1.85e-166) {
		tmp = ((a * b_m) * ((b_m / t_1) * a)) * -4.0;
	} else if (b_m <= 7.5e+151) {
		tmp = (((b_m * b_m) * t_0) * t_0) * -4.0;
	} else {
		tmp = (b_m * (((a * b_m) * a) / t_1)) * -4.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = a / (y_45_scale * x_45_scale)
	t_1 = ((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale
	tmp = 0
	if b_m <= 1.85e-166:
		tmp = ((a * b_m) * ((b_m / t_1) * a)) * -4.0
	elif b_m <= 7.5e+151:
		tmp = (((b_m * b_m) * t_0) * t_0) * -4.0
	else:
		tmp = (b_m * (((a * b_m) * a) / t_1)) * -4.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a / Float64(y_45_scale * x_45_scale))
	t_1 = Float64(Float64(Float64(y_45_scale * x_45_scale) * x_45_scale) * y_45_scale)
	tmp = 0.0
	if (b_m <= 1.85e-166)
		tmp = Float64(Float64(Float64(a * b_m) * Float64(Float64(b_m / t_1) * a)) * -4.0);
	elseif (b_m <= 7.5e+151)
		tmp = Float64(Float64(Float64(Float64(b_m * b_m) * t_0) * t_0) * -4.0);
	else
		tmp = Float64(Float64(b_m * Float64(Float64(Float64(a * b_m) * a) / t_1)) * -4.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = a / (y_45_scale * x_45_scale);
	t_1 = ((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale;
	tmp = 0.0;
	if (b_m <= 1.85e-166)
		tmp = ((a * b_m) * ((b_m / t_1) * a)) * -4.0;
	elseif (b_m <= 7.5e+151)
		tmp = (((b_m * b_m) * t_0) * t_0) * -4.0;
	else
		tmp = (b_m * (((a * b_m) * a) / t_1)) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]}, If[LessEqual[b$95$m, 1.85e-166], N[(N[(N[(a * b$95$m), $MachinePrecision] * N[(N[(b$95$m / t$95$1), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], If[LessEqual[b$95$m, 7.5e+151], N[(N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(b$95$m * N[(N[(N[(a * b$95$m), $MachinePrecision] * a), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.8500000000000001e-166

   1. Initial program 25.1%

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

   2. 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 48.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.6

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

   6. Applied rewrites 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

      17. lift-*.f64 N/A

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

   8. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 78.7

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

   10. Applied rewrites 78.7%

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

   if 1.8500000000000001e-166 < b < 7.49999999999999977e151

   1. Initial program 25.1%

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

   2. 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 48.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.6

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

   6. Applied rewrites 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

      17. lift-*.f64 N/A

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

   8. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. times-frac N/A

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

      17. lift-/.f64 N/A

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

      18. lower-*.f64 N/A

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

   10. Applied rewrites 78.8%

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

   if 7.49999999999999977e151 < b

   1. Initial program 25.1%

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

   2. 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 48.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.6

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

   6. Applied rewrites 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

      17. lift-*.f64 N/A

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

   8. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 74.4

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

   10. Applied rewrites 74.4%

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

Alternative 3: 86.2% accurate, 20.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.1%

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

2. 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 48.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. pow-prod-down N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 64.6

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

6. Applied rewrites 71.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 88.6

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

8. Applied rewrites 88.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 88.7

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 89.8

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

10. Applied rewrites 89.8%

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

Alternative 4: 78.9% accurate, 15.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(y-scale \cdot x-scale\right) \cdot x-scale\\ t_1 := t\_0 \cdot y-scale\\ \mathbf{if}\;b\_m \leq 1.55 \cdot 10^{-100}:\\ \;\;\;\;\left(\left(a \cdot b\_m\right) \cdot \left(\frac{b\_m}{t\_1} \cdot a\right)\right) \cdot -4\\ \mathbf{elif}\;b\_m \leq 3.85 \cdot 10^{+114}:\\ \;\;\;\;\left(\left(b\_m \cdot \left(b\_m \cdot \frac{a}{t\_0}\right)\right) \cdot \frac{a}{y-scale}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m \cdot \frac{\left(a \cdot b\_m\right) \cdot a}{t\_1}\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* y-scale x-scale) x-scale)) (t_1 (* t_0 y-scale)))
   (if (<= b_m 1.55e-100)
     (* (* (* a b_m) (* (/ b_m t_1) a)) -4.0)
     (if (<= b_m 3.85e+114)
       (* (* (* b_m (* b_m (/ a t_0))) (/ a y-scale)) -4.0)
       (* (* b_m (/ (* (* a b_m) a) t_1)) -4.0)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (y_45_scale * x_45_scale) * x_45_scale;
	double t_1 = t_0 * y_45_scale;
	double tmp;
	if (b_m <= 1.55e-100) {
		tmp = ((a * b_m) * ((b_m / t_1) * a)) * -4.0;
	} else if (b_m <= 3.85e+114) {
		tmp = ((b_m * (b_m * (a / t_0))) * (a / y_45_scale)) * -4.0;
	} else {
		tmp = (b_m * (((a * b_m) * a) / t_1)) * -4.0;
	}
	return tmp;
}

Fortran

b_m =     private
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_m, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y_45scale * x_45scale) * x_45scale
    t_1 = t_0 * y_45scale
    if (b_m <= 1.55d-100) then
        tmp = ((a * b_m) * ((b_m / t_1) * a)) * (-4.0d0)
    else if (b_m <= 3.85d+114) then
        tmp = ((b_m * (b_m * (a / t_0))) * (a / y_45scale)) * (-4.0d0)
    else
        tmp = (b_m * (((a * b_m) * a) / t_1)) * (-4.0d0)
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (y_45_scale * x_45_scale) * x_45_scale;
	double t_1 = t_0 * y_45_scale;
	double tmp;
	if (b_m <= 1.55e-100) {
		tmp = ((a * b_m) * ((b_m / t_1) * a)) * -4.0;
	} else if (b_m <= 3.85e+114) {
		tmp = ((b_m * (b_m * (a / t_0))) * (a / y_45_scale)) * -4.0;
	} else {
		tmp = (b_m * (((a * b_m) * a) / t_1)) * -4.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = (y_45_scale * x_45_scale) * x_45_scale
	t_1 = t_0 * y_45_scale
	tmp = 0
	if b_m <= 1.55e-100:
		tmp = ((a * b_m) * ((b_m / t_1) * a)) * -4.0
	elif b_m <= 3.85e+114:
		tmp = ((b_m * (b_m * (a / t_0))) * (a / y_45_scale)) * -4.0
	else:
		tmp = (b_m * (((a * b_m) * a) / t_1)) * -4.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(y_45_scale * x_45_scale) * x_45_scale)
	t_1 = Float64(t_0 * y_45_scale)
	tmp = 0.0
	if (b_m <= 1.55e-100)
		tmp = Float64(Float64(Float64(a * b_m) * Float64(Float64(b_m / t_1) * a)) * -4.0);
	elseif (b_m <= 3.85e+114)
		tmp = Float64(Float64(Float64(b_m * Float64(b_m * Float64(a / t_0))) * Float64(a / y_45_scale)) * -4.0);
	else
		tmp = Float64(Float64(b_m * Float64(Float64(Float64(a * b_m) * a) / t_1)) * -4.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = (y_45_scale * x_45_scale) * x_45_scale;
	t_1 = t_0 * y_45_scale;
	tmp = 0.0;
	if (b_m <= 1.55e-100)
		tmp = ((a * b_m) * ((b_m / t_1) * a)) * -4.0;
	elseif (b_m <= 3.85e+114)
		tmp = ((b_m * (b_m * (a / t_0))) * (a / y_45_scale)) * -4.0;
	else
		tmp = (b_m * (((a * b_m) * a) / t_1)) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * y$45$scale), $MachinePrecision]}, If[LessEqual[b$95$m, 1.55e-100], N[(N[(N[(a * b$95$m), $MachinePrecision] * N[(N[(b$95$m / t$95$1), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], If[LessEqual[b$95$m, 3.85e+114], N[(N[(N[(b$95$m * N[(b$95$m * N[(a / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(b$95$m * N[(N[(N[(a * b$95$m), $MachinePrecision] * a), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.5499999999999999e-100

   1. Initial program 25.1%

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

   2. 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 48.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.6

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

   6. Applied rewrites 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

      17. lift-*.f64 N/A

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

   8. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 78.7

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

   10. Applied rewrites 78.7%

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

   if 1.5499999999999999e-100 < b < 3.8500000000000001e114

   1. Initial program 25.1%

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

   2. 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 48.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 48.0

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

   6. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 68.8

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

   8. Applied rewrites 68.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 77.5

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

   10. Applied rewrites 77.5%

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

   if 3.8500000000000001e114 < b

   1. Initial program 25.1%

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

   2. 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 48.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.6

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

   6. Applied rewrites 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

      17. lift-*.f64 N/A

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

   8. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 74.4

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

   10. Applied rewrites 74.4%

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

Alternative 5: 78.8% accurate, 15.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(y-scale \cdot x-scale\right) \cdot x-scale\\ t_1 := t\_0 \cdot y-scale\\ \mathbf{if}\;b\_m \leq 2 \cdot 10^{-51}:\\ \;\;\;\;\left(\left(a \cdot b\_m\right) \cdot \left(\frac{b\_m}{t\_1} \cdot a\right)\right) \cdot -4\\ \mathbf{elif}\;b\_m \leq 2.6 \cdot 10^{+150}:\\ \;\;\;\;\left(\left(a \cdot \frac{b\_m \cdot b\_m}{t\_0}\right) \cdot \frac{a}{y-scale}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m \cdot \frac{\left(a \cdot b\_m\right) \cdot a}{t\_1}\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* y-scale x-scale) x-scale)) (t_1 (* t_0 y-scale)))
   (if (<= b_m 2e-51)
     (* (* (* a b_m) (* (/ b_m t_1) a)) -4.0)
     (if (<= b_m 2.6e+150)
       (* (* (* a (/ (* b_m b_m) t_0)) (/ a y-scale)) -4.0)
       (* (* b_m (/ (* (* a b_m) a) t_1)) -4.0)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (y_45_scale * x_45_scale) * x_45_scale;
	double t_1 = t_0 * y_45_scale;
	double tmp;
	if (b_m <= 2e-51) {
		tmp = ((a * b_m) * ((b_m / t_1) * a)) * -4.0;
	} else if (b_m <= 2.6e+150) {
		tmp = ((a * ((b_m * b_m) / t_0)) * (a / y_45_scale)) * -4.0;
	} else {
		tmp = (b_m * (((a * b_m) * a) / t_1)) * -4.0;
	}
	return tmp;
}

Fortran

b_m =     private
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_m, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y_45scale * x_45scale) * x_45scale
    t_1 = t_0 * y_45scale
    if (b_m <= 2d-51) then
        tmp = ((a * b_m) * ((b_m / t_1) * a)) * (-4.0d0)
    else if (b_m <= 2.6d+150) then
        tmp = ((a * ((b_m * b_m) / t_0)) * (a / y_45scale)) * (-4.0d0)
    else
        tmp = (b_m * (((a * b_m) * a) / t_1)) * (-4.0d0)
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (y_45_scale * x_45_scale) * x_45_scale;
	double t_1 = t_0 * y_45_scale;
	double tmp;
	if (b_m <= 2e-51) {
		tmp = ((a * b_m) * ((b_m / t_1) * a)) * -4.0;
	} else if (b_m <= 2.6e+150) {
		tmp = ((a * ((b_m * b_m) / t_0)) * (a / y_45_scale)) * -4.0;
	} else {
		tmp = (b_m * (((a * b_m) * a) / t_1)) * -4.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = (y_45_scale * x_45_scale) * x_45_scale
	t_1 = t_0 * y_45_scale
	tmp = 0
	if b_m <= 2e-51:
		tmp = ((a * b_m) * ((b_m / t_1) * a)) * -4.0
	elif b_m <= 2.6e+150:
		tmp = ((a * ((b_m * b_m) / t_0)) * (a / y_45_scale)) * -4.0
	else:
		tmp = (b_m * (((a * b_m) * a) / t_1)) * -4.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(y_45_scale * x_45_scale) * x_45_scale)
	t_1 = Float64(t_0 * y_45_scale)
	tmp = 0.0
	if (b_m <= 2e-51)
		tmp = Float64(Float64(Float64(a * b_m) * Float64(Float64(b_m / t_1) * a)) * -4.0);
	elseif (b_m <= 2.6e+150)
		tmp = Float64(Float64(Float64(a * Float64(Float64(b_m * b_m) / t_0)) * Float64(a / y_45_scale)) * -4.0);
	else
		tmp = Float64(Float64(b_m * Float64(Float64(Float64(a * b_m) * a) / t_1)) * -4.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = (y_45_scale * x_45_scale) * x_45_scale;
	t_1 = t_0 * y_45_scale;
	tmp = 0.0;
	if (b_m <= 2e-51)
		tmp = ((a * b_m) * ((b_m / t_1) * a)) * -4.0;
	elseif (b_m <= 2.6e+150)
		tmp = ((a * ((b_m * b_m) / t_0)) * (a / y_45_scale)) * -4.0;
	else
		tmp = (b_m * (((a * b_m) * a) / t_1)) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * y$45$scale), $MachinePrecision]}, If[LessEqual[b$95$m, 2e-51], N[(N[(N[(a * b$95$m), $MachinePrecision] * N[(N[(b$95$m / t$95$1), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], If[LessEqual[b$95$m, 2.6e+150], N[(N[(N[(a * N[(N[(b$95$m * b$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(b$95$m * N[(N[(N[(a * b$95$m), $MachinePrecision] * a), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2e-51

   1. Initial program 25.1%

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

   2. 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 48.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.6

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

   6. Applied rewrites 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

      17. lift-*.f64 N/A

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

   8. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 78.7

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

   10. Applied rewrites 78.7%

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

   if 2e-51 < b < 2.60000000000000006e150

   1. Initial program 25.1%

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

   2. 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 48.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 48.0

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

   6. Applied rewrites 59.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 68.8

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

   8. Applied rewrites 68.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 68.8

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

   10. Applied rewrites 68.8%

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

   if 2.60000000000000006e150 < b

   1. Initial program 25.1%

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

   2. 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 48.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.6

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

   6. Applied rewrites 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

      17. lift-*.f64 N/A

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

   8. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 74.4

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

   10. Applied rewrites 74.4%

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

Alternative 6: 78.7% accurate, 17.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\\ \mathbf{if}\;b\_m \leq 2.4 \cdot 10^{+150}:\\ \;\;\;\;\left(\left(a \cdot b\_m\right) \cdot \left(\frac{b\_m}{t\_0} \cdot a\right)\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(b\_m \cdot \frac{\left(a \cdot b\_m\right) \cdot a}{t\_0}\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* (* y-scale x-scale) x-scale) y-scale)))
   (if (<= b_m 2.4e+150)
     (* (* (* a b_m) (* (/ b_m t_0) a)) -4.0)
     (* (* b_m (/ (* (* a b_m) a) t_0)) -4.0))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale;
	double tmp;
	if (b_m <= 2.4e+150) {
		tmp = ((a * b_m) * ((b_m / t_0) * a)) * -4.0;
	} else {
		tmp = (b_m * (((a * b_m) * a) / t_0)) * -4.0;
	}
	return tmp;
}

Fortran

b_m =     private
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_m, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y_45scale * x_45scale) * x_45scale) * y_45scale
    if (b_m <= 2.4d+150) then
        tmp = ((a * b_m) * ((b_m / t_0) * a)) * (-4.0d0)
    else
        tmp = (b_m * (((a * b_m) * a) / t_0)) * (-4.0d0)
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale;
	double tmp;
	if (b_m <= 2.4e+150) {
		tmp = ((a * b_m) * ((b_m / t_0) * a)) * -4.0;
	} else {
		tmp = (b_m * (((a * b_m) * a) / t_0)) * -4.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = ((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale
	tmp = 0
	if b_m <= 2.4e+150:
		tmp = ((a * b_m) * ((b_m / t_0) * a)) * -4.0
	else:
		tmp = (b_m * (((a * b_m) * a) / t_0)) * -4.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(y_45_scale * x_45_scale) * x_45_scale) * y_45_scale)
	tmp = 0.0
	if (b_m <= 2.4e+150)
		tmp = Float64(Float64(Float64(a * b_m) * Float64(Float64(b_m / t_0) * a)) * -4.0);
	else
		tmp = Float64(Float64(b_m * Float64(Float64(Float64(a * b_m) * a) / t_0)) * -4.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = ((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale;
	tmp = 0.0;
	if (b_m <= 2.4e+150)
		tmp = ((a * b_m) * ((b_m / t_0) * a)) * -4.0;
	else
		tmp = (b_m * (((a * b_m) * a) / t_0)) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]}, If[LessEqual[b$95$m, 2.4e+150], N[(N[(N[(a * b$95$m), $MachinePrecision] * N[(N[(b$95$m / t$95$0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(b$95$m * N[(N[(N[(a * b$95$m), $MachinePrecision] * a), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.40000000000000003e150

   1. Initial program 25.1%

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

   2. 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 48.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.6

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

   6. Applied rewrites 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

      17. lift-*.f64 N/A

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

   8. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 78.7

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

   10. Applied rewrites 78.7%

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

   if 2.40000000000000003e150 < b

   1. Initial program 25.1%

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

   2. 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 48.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-prod-down N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.6

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

   6. Applied rewrites 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

      17. lift-*.f64 N/A

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

   8. Applied rewrites 75.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 74.4

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

   10. Applied rewrites 74.4%

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

Alternative 7: 78.4% accurate, 20.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.1%

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

2. 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 48.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. pow-prod-down N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 64.6

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

6. Applied rewrites 71.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l* N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*l/ N/A

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

   16. lift-/.f64 N/A

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

   17. lift-*.f64 N/A

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

8. Applied rewrites 75.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 78.7

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

10. Applied rewrites 78.7%

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

Alternative 8: 78.2% accurate, 20.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.1%

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

2. 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 48.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. pow-prod-down N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 64.6

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

6. Applied rewrites 71.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

      \[\leadsto \frac{-4 \cdot \left(\left(\left(b \cdot b\right) \cdot a\right) \cdot a\right)}{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{-4 \cdot \left(\left(\left(b \cdot b\right) \cdot a\right) \cdot a\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot \color{blue}{x-scale}\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{-4 \cdot \left(\left(\left(b \cdot b\right) \cdot a\right) \cdot a\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(x-scale \cdot \color{blue}{y-scale}\right)} \]

   7. associate-*l* N/A

      \[\leadsto \frac{-4 \cdot \left(\left(\left(b \cdot b\right) \cdot a\right) \cdot a\right)}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot \color{blue}{y-scale}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{-4 \cdot \left(\left(\left(b \cdot b\right) \cdot a\right) \cdot a\right)}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{-4 \cdot \left(\left(\left(b \cdot b\right) \cdot a\right) \cdot a\right)}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot \color{blue}{y-scale}} \]

   10. associate-/l* N/A

       \[\leadsto -4 \cdot \color{blue}{\frac{\left(\left(b \cdot b\right) \cdot a\right) \cdot a}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}} \]

   11. lift-*.f64 N/A

       \[\leadsto -4 \cdot \frac{\left(\left(b \cdot b\right) \cdot a\right) \cdot a}{\color{blue}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right)} \cdot y-scale} \]

   12. lift-*.f64 N/A

       \[\leadsto -4 \cdot \frac{\left(\left(b \cdot b\right) \cdot a\right) \cdot a}{\left(\color{blue}{\left(y-scale \cdot x-scale\right)} \cdot x-scale\right) \cdot y-scale} \]

   13. associate-*l* N/A

       \[\leadsto -4 \cdot \frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right)} \cdot y-scale} \]

   14. lift-*.f64 N/A

       \[\leadsto -4 \cdot \frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{\left(\left(y-scale \cdot x-scale\right) \cdot \color{blue}{x-scale}\right) \cdot y-scale} \]

   15. associate-*l/ N/A

       \[\leadsto -4 \cdot \left(\frac{b \cdot b}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   16. lift-/.f64 N/A

       \[\leadsto -4 \cdot \left(\frac{b \cdot b}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot \left(\color{blue}{a} \cdot a\right)\right) \]

   17. lift-*.f64 N/A

       \[\leadsto -4 \cdot \left(\frac{b \cdot b}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

8. Applied rewrites 75.0%

   \[\leadsto \left(\left(\left(b \cdot \frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}\right) \cdot a\right) \cdot a\right) \cdot \color{blue}{-4} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(b \cdot \frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}\right) \cdot a\right) \cdot a\right) \cdot -4 \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(b \cdot \frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}\right) \cdot a\right) \cdot a\right) \cdot -4 \]

   3. associate-*l* N/A

      \[\leadsto \left(\left(b \cdot \left(\frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot a\right)\right) \cdot a\right) \cdot -4 \]

   4. *-commutative N/A

      \[\leadsto \left(\left(\left(\frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot a\right) \cdot b\right) \cdot a\right) \cdot -4 \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\left(\left(\frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot a\right) \cdot b\right) \cdot a\right) \cdot -4 \]

   6. lower-*.f64 78.2

      \[\leadsto \left(\left(\left(\frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot a\right) \cdot b\right) \cdot a\right) \cdot -4 \]

10. Applied rewrites 78.2%

    \[\leadsto \left(\left(\left(\frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot a\right) \cdot b\right) \cdot a\right) \cdot -4 \]
11. Add Preprocessing

Alternative 9: 75.0% accurate, 20.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \left(\left(\left(b\_m \cdot \frac{b\_m}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}\right) \cdot a\right) \cdot a\right) \cdot -4 \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (*
  (* (* (* b_m (/ b_m (* (* (* y-scale x-scale) x-scale) y-scale))) a) a)
  -4.0))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return (((b_m * (b_m / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))) * a) * a) * -4.0;
}

Fortran

b_m =     private
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_m, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (((b_m * (b_m / (((y_45scale * x_45scale) * x_45scale) * y_45scale))) * a) * a) * (-4.0d0)
end function

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return (((b_m * (b_m / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))) * a) * a) * -4.0;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	return (((b_m * (b_m / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))) * a) * a) * -4.0

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	return Float64(Float64(Float64(Float64(b_m * Float64(b_m / Float64(Float64(Float64(y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))) * a) * a) * -4.0)
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = (((b_m * (b_m / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))) * a) * a) * -4.0;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(N[(b$95$m * N[(b$95$m / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * -4.0), $MachinePrecision]

Derivation

1. Initial program 25.1%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

2. 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 48.0%

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -4 \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   2. lift-/.f64 N/A

      \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot \color{blue}{{y-scale}^{2}}} \]

   5. *-commutative N/A

      \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot \color{blue}{{x-scale}^{2}}} \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot {\color{blue}{x-scale}}^{2}} \]

   7. lift-pow.f64 N/A

      \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{y-scale}^{2} \cdot {x-scale}^{\color{blue}{2}}} \]

   8. pow-prod-down N/A

      \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{\left(y-scale \cdot x-scale\right)}^{\color{blue}{2}}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{\left(y-scale \cdot x-scale\right)}^{2}} \]

   10. pow2 N/A

       \[\leadsto \frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{\left(y-scale \cdot x-scale\right) \cdot \color{blue}{\left(y-scale \cdot x-scale\right)}} \]

   11. times-frac N/A

       \[\leadsto \frac{-4}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{-4}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}} \]

   13. lower-/.f64 N/A

       \[\leadsto \frac{-4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{{a}^{2} \cdot {b}^{2}}}{y-scale \cdot x-scale} \]

   14. lower-/.f64 64.6

       \[\leadsto \frac{-4}{y-scale \cdot x-scale} \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{y-scale \cdot x-scale}} \]

6. Applied rewrites 71.8%

   \[\leadsto \frac{-4}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{\left(\left(b \cdot b\right) \cdot a\right) \cdot a}{y-scale \cdot x-scale}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{-4}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{\left(\left(b \cdot b\right) \cdot a\right) \cdot a}{y-scale \cdot x-scale}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{-4}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(\left(b \cdot b\right) \cdot a\right) \cdot a}}{y-scale \cdot x-scale} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{-4}{y-scale \cdot x-scale} \cdot \frac{\left(\left(b \cdot b\right) \cdot a\right) \cdot a}{\color{blue}{y-scale \cdot x-scale}} \]

   4. frac-times N/A

      \[\leadsto \frac{-4 \cdot \left(\left(\left(b \cdot b\right) \cdot a\right) \cdot a\right)}{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{-4 \cdot \left(\left(\left(b \cdot b\right) \cdot a\right) \cdot a\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot \color{blue}{x-scale}\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{-4 \cdot \left(\left(\left(b \cdot b\right) \cdot a\right) \cdot a\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(x-scale \cdot \color{blue}{y-scale}\right)} \]

   7. associate-*l* N/A

      \[\leadsto \frac{-4 \cdot \left(\left(\left(b \cdot b\right) \cdot a\right) \cdot a\right)}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot \color{blue}{y-scale}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{-4 \cdot \left(\left(\left(b \cdot b\right) \cdot a\right) \cdot a\right)}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{-4 \cdot \left(\left(\left(b \cdot b\right) \cdot a\right) \cdot a\right)}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot \color{blue}{y-scale}} \]

   10. associate-/l* N/A

       \[\leadsto -4 \cdot \color{blue}{\frac{\left(\left(b \cdot b\right) \cdot a\right) \cdot a}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}} \]

   11. lift-*.f64 N/A

       \[\leadsto -4 \cdot \frac{\left(\left(b \cdot b\right) \cdot a\right) \cdot a}{\color{blue}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right)} \cdot y-scale} \]

   12. lift-*.f64 N/A

       \[\leadsto -4 \cdot \frac{\left(\left(b \cdot b\right) \cdot a\right) \cdot a}{\left(\color{blue}{\left(y-scale \cdot x-scale\right)} \cdot x-scale\right) \cdot y-scale} \]

   13. associate-*l* N/A

       \[\leadsto -4 \cdot \frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{\color{blue}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right)} \cdot y-scale} \]

   14. lift-*.f64 N/A

       \[\leadsto -4 \cdot \frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{\left(\left(y-scale \cdot x-scale\right) \cdot \color{blue}{x-scale}\right) \cdot y-scale} \]

   15. associate-*l/ N/A

       \[\leadsto -4 \cdot \left(\frac{b \cdot b}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   16. lift-/.f64 N/A

       \[\leadsto -4 \cdot \left(\frac{b \cdot b}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot \left(\color{blue}{a} \cdot a\right)\right) \]

   17. lift-*.f64 N/A

       \[\leadsto -4 \cdot \left(\frac{b \cdot b}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

8. Applied rewrites 75.0%

   \[\leadsto \left(\left(\left(b \cdot \frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}\right) \cdot a\right) \cdot a\right) \cdot \color{blue}{-4} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (a b angle x-scale y-scale)
  :name "Simplification of discriminant from scale-rotated-ellipse"
  :precision binary64
  (- (* (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale) (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale)) (* (* 4.0 (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale)) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale))))