Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.0% → 94.3%

Time: 15.7s

Alternatives: 10

Speedup: 20.4×

• 65.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) 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: 25.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) 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: 94.3% accurate, 15.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{b}{x-scale} \cdot \frac{a}{\left|y-scale\right|}\\ t_1 := \frac{b}{\left|y-scale\right|} \cdot \frac{a}{x-scale}\\ \mathbf{if}\;\left|y-scale\right| \leq 1.2 \cdot 10^{-182}:\\ \;\;\;\;\left(t\_1 \cdot t\_1\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot -4\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ b x-scale) (/ a (fabs y-scale))))
        (t_1 (* (/ b (fabs y-scale)) (/ a x-scale))))
   (if (<= (fabs y-scale) 1.2e-182)
     (* (* t_1 t_1) -4.0)
     (* (* t_0 t_0) -4.0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b / x_45_scale) * (a / fabs(y_45_scale));
	double t_1 = (b / fabs(y_45_scale)) * (a / x_45_scale);
	double tmp;
	if (fabs(y_45_scale) <= 1.2e-182) {
		tmp = (t_1 * t_1) * -4.0;
	} else {
		tmp = (t_0 * t_0) * -4.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b / x_45scale) * (a / abs(y_45scale))
    t_1 = (b / abs(y_45scale)) * (a / x_45scale)
    if (abs(y_45scale) <= 1.2d-182) then
        tmp = (t_1 * t_1) * (-4.0d0)
    else
        tmp = (t_0 * t_0) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b / x_45_scale) * (a / Math.abs(y_45_scale));
	double t_1 = (b / Math.abs(y_45_scale)) * (a / x_45_scale);
	double tmp;
	if (Math.abs(y_45_scale) <= 1.2e-182) {
		tmp = (t_1 * t_1) * -4.0;
	} else {
		tmp = (t_0 * t_0) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (b / x_45_scale) * (a / math.fabs(y_45_scale))
	t_1 = (b / math.fabs(y_45_scale)) * (a / x_45_scale)
	tmp = 0
	if math.fabs(y_45_scale) <= 1.2e-182:
		tmp = (t_1 * t_1) * -4.0
	else:
		tmp = (t_0 * t_0) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b / x_45_scale) * Float64(a / abs(y_45_scale)))
	t_1 = Float64(Float64(b / abs(y_45_scale)) * Float64(a / x_45_scale))
	tmp = 0.0
	if (abs(y_45_scale) <= 1.2e-182)
		tmp = Float64(Float64(t_1 * t_1) * -4.0);
	else
		tmp = Float64(Float64(t_0 * t_0) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (b / x_45_scale) * (a / abs(y_45_scale));
	t_1 = (b / abs(y_45_scale)) * (a / x_45_scale);
	tmp = 0.0;
	if (abs(y_45_scale) <= 1.2e-182)
		tmp = (t_1 * t_1) * -4.0;
	else
		tmp = (t_0 * t_0) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b / x$45$scale), $MachinePrecision] * N[(a / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 1.2e-182], N[(N[(t$95$1 * t$95$1), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * -4.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.1999999999999999e-182

   1. Initial program 25.0%

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

   2. 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 48.7%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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 48.7%

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

   6. Applied rewrites 60.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. swap-sqr N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 89.7%

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

   10. Applied rewrites 89.7%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. associate-*l/ N/A

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

       5. lift-*.f64 N/A

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

       6. times-frac N/A

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

       7. lift-/.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 94.3%

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

   12. Applied rewrites 94.3%

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

   if 1.1999999999999999e-182 < y-scale

   1. Initial program 25.0%

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

   2. 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 48.7%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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 48.7%

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

   6. Applied rewrites 60.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. swap-sqr N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 89.3%

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

   10. Applied rewrites 89.3%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-/.f64 93.9%

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

   12. Applied rewrites 93.9%

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

Alternative 2: 94.3% accurate, 15.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{a \cdot b}{y-scale \cdot \left|x-scale\right|}\\ t_1 := \frac{b}{\left|x-scale\right|} \cdot \frac{a}{y-scale}\\ \mathbf{if}\;\left|x-scale\right| \leq 1.4 \cdot 10^{+260}:\\ \;\;\;\;\left(t\_1 \cdot t\_1\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot -4\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (* a b) (* y-scale (fabs x-scale))))
        (t_1 (* (/ b (fabs x-scale)) (/ a y-scale))))
   (if (<= (fabs x-scale) 1.4e+260)
     (* (* t_1 t_1) -4.0)
     (* (* t_0 t_0) -4.0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a * b) / (y_45_scale * fabs(x_45_scale));
	double t_1 = (b / fabs(x_45_scale)) * (a / y_45_scale);
	double tmp;
	if (fabs(x_45_scale) <= 1.4e+260) {
		tmp = (t_1 * t_1) * -4.0;
	} else {
		tmp = (t_0 * t_0) * -4.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (a * b) / (y_45scale * abs(x_45scale))
    t_1 = (b / abs(x_45scale)) * (a / y_45scale)
    if (abs(x_45scale) <= 1.4d+260) then
        tmp = (t_1 * t_1) * (-4.0d0)
    else
        tmp = (t_0 * t_0) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (a * b) / (y_45_scale * Math.abs(x_45_scale));
	double t_1 = (b / Math.abs(x_45_scale)) * (a / y_45_scale);
	double tmp;
	if (Math.abs(x_45_scale) <= 1.4e+260) {
		tmp = (t_1 * t_1) * -4.0;
	} else {
		tmp = (t_0 * t_0) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (a * b) / (y_45_scale * math.fabs(x_45_scale))
	t_1 = (b / math.fabs(x_45_scale)) * (a / y_45_scale)
	tmp = 0
	if math.fabs(x_45_scale) <= 1.4e+260:
		tmp = (t_1 * t_1) * -4.0
	else:
		tmp = (t_0 * t_0) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(a * b) / Float64(y_45_scale * abs(x_45_scale)))
	t_1 = Float64(Float64(b / abs(x_45_scale)) * Float64(a / y_45_scale))
	tmp = 0.0
	if (abs(x_45_scale) <= 1.4e+260)
		tmp = Float64(Float64(t_1 * t_1) * -4.0);
	else
		tmp = Float64(Float64(t_0 * t_0) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (a * b) / (y_45_scale * abs(x_45_scale));
	t_1 = (b / abs(x_45_scale)) * (a / y_45_scale);
	tmp = 0.0;
	if (abs(x_45_scale) <= 1.4e+260)
		tmp = (t_1 * t_1) * -4.0;
	else
		tmp = (t_0 * t_0) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(a * b), $MachinePrecision] / N[(y$45$scale * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[(a / y$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 1.4e+260], N[(N[(t$95$1 * t$95$1), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * -4.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.3999999999999999e260

   1. Initial program 25.0%

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

   2. 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 48.7%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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 48.7%

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

   6. Applied rewrites 60.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. swap-sqr N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 89.3%

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

   10. Applied rewrites 89.3%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-/.f64 93.9%

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

   12. Applied rewrites 93.9%

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

   if 1.3999999999999999e260 < x-scale

   1. Initial program 25.0%

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

   2. 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 48.7%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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 48.7%

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

   6. Applied rewrites 60.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. swap-sqr N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 92.8%

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

   10. Applied rewrites 92.8%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lift-*.f64 N/A

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

       5. lower-/.f64 94.3%

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

   12. Applied rewrites 94.3%

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

Alternative 3: 94.2% accurate, 20.1× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (* a b) (* y-scale x-scale)))) (* (* t_0 t_0) -4.0)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(a * b), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 * t$95$0), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Initial program 25.0%

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

2. 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 48.7%

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
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 48.7%

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

6. Applied rewrites 60.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. swap-sqr N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. associate-/l* N/A

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

   18. lower-*.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. associate-/l* N/A

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

   23. lower-*.f64 N/A

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

8. Applied rewrites 94.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lower-/.f64 92.8%

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

10. Applied rewrites 92.8%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-*r/ N/A

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

    4. lift-*.f64 N/A

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

    5. lower-/.f64 94.3%

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

12. Applied rewrites 94.3%

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

Alternative 4: 94.1% accurate, 20.1× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* a (/ b (* y-scale x-scale))))) (* (* t_0 t_0) -4.0)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a * N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 * t$95$0), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Initial program 25.0%

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

2. 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 48.7%

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
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 48.7%

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

6. Applied rewrites 60.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. swap-sqr N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. associate-/l* N/A

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

   18. lower-*.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. associate-/l* N/A

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

   23. lower-*.f64 N/A

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

8. Applied rewrites 94.3%

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

Alternative 5: 85.8% accurate, 20.1× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ a (* y-scale x-scale)))) (* (* b (* b (* t_0 t_0))) -4.0)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, N[(N[(b * N[(b * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Initial program 25.0%

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

2. 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 48.7%

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
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 48.7%

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

6. Applied rewrites 60.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 67.3%

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. swap-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 55.6%

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

8. Applied rewrites 55.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. unswap-sqr N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. times-frac N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lower-*.f64 85.8%

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

10. Applied rewrites 85.8%

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

Alternative 6: 81.2% accurate, 13.5× speedup

Math

\[\begin{array}{l} t_0 := \left|y-scale\right| \cdot x-scale\\ \mathbf{if}\;\left|y-scale\right| \leq 10^{-148}:\\ \;\;\;\;-4 \cdot \frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{\left(t\_0 \cdot \left|y-scale\right|\right) \cdot x-scale}\\ \mathbf{elif}\;\left|y-scale\right| \leq 1.95 \cdot 10^{+111}:\\ \;\;\;\;\left(\left(b \cdot \left(a \cdot \frac{b}{\left(\left|y-scale\right| \cdot \left|y-scale\right|\right) \cdot x-scale}\right)\right) \cdot \frac{a}{x-scale}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot \left(a \cdot \left(\frac{a}{\left(t\_0 \cdot x-scale\right) \cdot \left|y-scale\right|} \cdot b\right)\right)\right) \cdot -4\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (fabs y-scale) x-scale)))
   (if (<= (fabs y-scale) 1e-148)
     (* -4.0 (/ (* (* b a) (* b a)) (* (* t_0 (fabs y-scale)) x-scale)))
     (if (<= (fabs y-scale) 1.95e+111)
       (*
        (*
         (* b (* a (/ b (* (* (fabs y-scale) (fabs y-scale)) x-scale))))
         (/ a x-scale))
        -4.0)
       (* (* b (* a (* (/ a (* (* t_0 x-scale) (fabs y-scale))) b))) -4.0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(y_45_scale) * x_45_scale;
	double tmp;
	if (fabs(y_45_scale) <= 1e-148) {
		tmp = -4.0 * (((b * a) * (b * a)) / ((t_0 * fabs(y_45_scale)) * x_45_scale));
	} else if (fabs(y_45_scale) <= 1.95e+111) {
		tmp = ((b * (a * (b / ((fabs(y_45_scale) * fabs(y_45_scale)) * x_45_scale)))) * (a / x_45_scale)) * -4.0;
	} else {
		tmp = (b * (a * ((a / ((t_0 * x_45_scale) * fabs(y_45_scale))) * b))) * -4.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(y_45scale) * x_45scale
    if (abs(y_45scale) <= 1d-148) then
        tmp = (-4.0d0) * (((b * a) * (b * a)) / ((t_0 * abs(y_45scale)) * x_45scale))
    else if (abs(y_45scale) <= 1.95d+111) then
        tmp = ((b * (a * (b / ((abs(y_45scale) * abs(y_45scale)) * x_45scale)))) * (a / x_45scale)) * (-4.0d0)
    else
        tmp = (b * (a * ((a / ((t_0 * x_45scale) * abs(y_45scale))) * b))) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.abs(y_45_scale) * x_45_scale;
	double tmp;
	if (Math.abs(y_45_scale) <= 1e-148) {
		tmp = -4.0 * (((b * a) * (b * a)) / ((t_0 * Math.abs(y_45_scale)) * x_45_scale));
	} else if (Math.abs(y_45_scale) <= 1.95e+111) {
		tmp = ((b * (a * (b / ((Math.abs(y_45_scale) * Math.abs(y_45_scale)) * x_45_scale)))) * (a / x_45_scale)) * -4.0;
	} else {
		tmp = (b * (a * ((a / ((t_0 * x_45_scale) * Math.abs(y_45_scale))) * b))) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.fabs(y_45_scale) * x_45_scale
	tmp = 0
	if math.fabs(y_45_scale) <= 1e-148:
		tmp = -4.0 * (((b * a) * (b * a)) / ((t_0 * math.fabs(y_45_scale)) * x_45_scale))
	elif math.fabs(y_45_scale) <= 1.95e+111:
		tmp = ((b * (a * (b / ((math.fabs(y_45_scale) * math.fabs(y_45_scale)) * x_45_scale)))) * (a / x_45_scale)) * -4.0
	else:
		tmp = (b * (a * ((a / ((t_0 * x_45_scale) * math.fabs(y_45_scale))) * b))) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(y_45_scale) * x_45_scale)
	tmp = 0.0
	if (abs(y_45_scale) <= 1e-148)
		tmp = Float64(-4.0 * Float64(Float64(Float64(b * a) * Float64(b * a)) / Float64(Float64(t_0 * abs(y_45_scale)) * x_45_scale)));
	elseif (abs(y_45_scale) <= 1.95e+111)
		tmp = Float64(Float64(Float64(b * Float64(a * Float64(b / Float64(Float64(abs(y_45_scale) * abs(y_45_scale)) * x_45_scale)))) * Float64(a / x_45_scale)) * -4.0);
	else
		tmp = Float64(Float64(b * Float64(a * Float64(Float64(a / Float64(Float64(t_0 * x_45_scale) * abs(y_45_scale))) * b))) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(y_45_scale) * x_45_scale;
	tmp = 0.0;
	if (abs(y_45_scale) <= 1e-148)
		tmp = -4.0 * (((b * a) * (b * a)) / ((t_0 * abs(y_45_scale)) * x_45_scale));
	elseif (abs(y_45_scale) <= 1.95e+111)
		tmp = ((b * (a * (b / ((abs(y_45_scale) * abs(y_45_scale)) * x_45_scale)))) * (a / x_45_scale)) * -4.0;
	else
		tmp = (b * (a * ((a / ((t_0 * x_45_scale) * abs(y_45_scale))) * b))) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[y$45$scale], $MachinePrecision] * x$45$scale), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 1e-148], N[(-4.0 * N[(N[(N[(b * a), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 1.95e+111], N[(N[(N[(b * N[(a * N[(b / N[(N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a / x$45$scale), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(b * N[(a * N[(N[(a / N[(N[(t$95$0 * x$45$scale), $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 9.9999999999999994e-149

   1. Initial program 25.0%

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

   2. 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 48.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 61.4%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-pow.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. unpow2 N/A

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

      21. swap-sqr N/A

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

      22. lift-*.f64 N/A

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

      23. associate-*r* N/A

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

      24. *-commutative N/A

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

      25. lift-*.f64 N/A

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

      26. lower-*.f64 75.9%

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

      27. lift-*.f64 N/A

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

      28. *-commutative N/A

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

      29. lower-*.f64 75.9%

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

   6. Applied rewrites 75.9%

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

   if 9.9999999999999994e-149 < y-scale < 1.9499999999999999e111

   1. Initial program 25.0%

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

   2. 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 48.7%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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 48.7%

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

   6. Applied rewrites 60.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   8. Applied rewrites 76.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-/r* N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 79.9%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 73.6%

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

   10. Applied rewrites 73.6%

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

   if 1.9499999999999999e111 < y-scale

   1. Initial program 25.0%

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

   2. 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 48.7%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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 48.7%

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

   6. Applied rewrites 60.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 67.3%

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. swap-sqr N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 55.6%

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

   8. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 64.3%

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. unswap-sqr N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 78.3%

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

   10. Applied rewrites 78.3%

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

Alternative 7: 78.4% accurate, 15.8× speedup

Math

\[\begin{array}{l} t_0 := b \cdot \left|a\right|\\ \mathbf{if}\;\left|a\right| \leq 9 \cdot 10^{-239}:\\ \;\;\;\;-4 \cdot \frac{t\_0 \cdot t\_0}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot \left(\left|a\right| \cdot \left(\frac{\left|a\right|}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot b\right)\right)\right) \cdot -4\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* b (fabs a))))
   (if (<= (fabs a) 9e-239)
     (* -4.0 (/ (* t_0 t_0) (* (* (* y-scale x-scale) y-scale) x-scale)))
     (*
      (*
       b
       (*
        (fabs a)
        (* (/ (fabs a) (* (* (* y-scale x-scale) x-scale) y-scale)) b)))
      -4.0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b * fabs(a);
	double tmp;
	if (fabs(a) <= 9e-239) {
		tmp = -4.0 * ((t_0 * t_0) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale));
	} else {
		tmp = (b * (fabs(a) * ((fabs(a) / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale)) * b))) * -4.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b * abs(a)
    if (abs(a) <= 9d-239) then
        tmp = (-4.0d0) * ((t_0 * t_0) / (((y_45scale * x_45scale) * y_45scale) * x_45scale))
    else
        tmp = (b * (abs(a) * ((abs(a) / (((y_45scale * x_45scale) * x_45scale) * y_45scale)) * b))) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b * Math.abs(a);
	double tmp;
	if (Math.abs(a) <= 9e-239) {
		tmp = -4.0 * ((t_0 * t_0) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale));
	} else {
		tmp = (b * (Math.abs(a) * ((Math.abs(a) / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale)) * b))) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = b * math.fabs(a)
	tmp = 0
	if math.fabs(a) <= 9e-239:
		tmp = -4.0 * ((t_0 * t_0) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))
	else:
		tmp = (b * (math.fabs(a) * ((math.fabs(a) / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale)) * b))) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b * abs(a))
	tmp = 0.0
	if (abs(a) <= 9e-239)
		tmp = Float64(-4.0 * Float64(Float64(t_0 * t_0) / Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)));
	else
		tmp = Float64(Float64(b * Float64(abs(a) * Float64(Float64(abs(a) / Float64(Float64(Float64(y_45_scale * x_45_scale) * x_45_scale) * y_45_scale)) * b))) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = b * abs(a);
	tmp = 0.0;
	if (abs(a) <= 9e-239)
		tmp = -4.0 * ((t_0 * t_0) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale));
	else
		tmp = (b * (abs(a) * ((abs(a) / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale)) * b))) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 9e-239], N[(-4.0 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(N[Abs[a], $MachinePrecision] * N[(N[(N[Abs[a], $MachinePrecision] / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 9.0000000000000003e-239

   1. Initial program 25.0%

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

   2. 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 48.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 61.4%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-pow.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. unpow2 N/A

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

      21. swap-sqr N/A

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

      22. lift-*.f64 N/A

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

      23. associate-*r* N/A

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

      24. *-commutative N/A

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

      25. lift-*.f64 N/A

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

      26. lower-*.f64 75.9%

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

      27. lift-*.f64 N/A

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

      28. *-commutative N/A

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

      29. lower-*.f64 75.9%

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

   6. Applied rewrites 75.9%

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

   if 9.0000000000000003e-239 < a

   1. Initial program 25.0%

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

   2. 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 48.7%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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 48.7%

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

   6. Applied rewrites 60.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 67.3%

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. swap-sqr N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 55.6%

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

   8. Applied rewrites 55.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 64.3%

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. unswap-sqr N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 78.3%

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

   10. Applied rewrites 78.3%

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

Alternative 8: 78.3% accurate, 20.4× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (* b (* a (* (/ a (* (* (* y-scale x-scale) x-scale) y-scale)) b))) -4.0))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.0%

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

2. 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 48.7%

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
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 48.7%

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

6. Applied rewrites 60.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 67.3%

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. swap-sqr N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 55.6%

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

8. Applied rewrites 55.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 64.3%

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. unswap-sqr N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*r* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 78.3%

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

10. Applied rewrites 78.3%

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

Alternative 9: 69.5% accurate, 15.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\\ \mathbf{if}\;\left|b\right| \leq 5.6 \cdot 10^{-158}:\\ \;\;\;\;\left(\left(\frac{a \cdot a}{t\_0} \cdot \left|b\right|\right) \cdot \left|b\right|\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\left|b\right| \cdot \left|b\right|}{t\_0} \cdot a\right) \cdot a\right) \cdot -4\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* (* y-scale x-scale) x-scale) y-scale)))
   (if (<= (fabs b) 5.6e-158)
     (* (* (* (/ (* a a) t_0) (fabs b)) (fabs b)) -4.0)
     (* (* (* (/ (* (fabs b) (fabs b)) t_0) a) a) -4.0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale;
	double tmp;
	if (fabs(b) <= 5.6e-158) {
		tmp = ((((a * a) / t_0) * fabs(b)) * fabs(b)) * -4.0;
	} else {
		tmp = ((((fabs(b) * fabs(b)) / t_0) * a) * a) * -4.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y_45scale * x_45scale) * x_45scale) * y_45scale
    if (abs(b) <= 5.6d-158) then
        tmp = ((((a * a) / t_0) * abs(b)) * abs(b)) * (-4.0d0)
    else
        tmp = ((((abs(b) * abs(b)) / t_0) * a) * a) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale;
	double tmp;
	if (Math.abs(b) <= 5.6e-158) {
		tmp = ((((a * a) / t_0) * Math.abs(b)) * Math.abs(b)) * -4.0;
	} else {
		tmp = ((((Math.abs(b) * Math.abs(b)) / t_0) * a) * a) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = ((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale
	tmp = 0
	if math.fabs(b) <= 5.6e-158:
		tmp = ((((a * a) / t_0) * math.fabs(b)) * math.fabs(b)) * -4.0
	else:
		tmp = ((((math.fabs(b) * math.fabs(b)) / t_0) * a) * a) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(y_45_scale * x_45_scale) * x_45_scale) * y_45_scale)
	tmp = 0.0
	if (abs(b) <= 5.6e-158)
		tmp = Float64(Float64(Float64(Float64(Float64(a * a) / t_0) * abs(b)) * abs(b)) * -4.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(abs(b) * abs(b)) / t_0) * a) * a) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = ((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale;
	tmp = 0.0;
	if (abs(b) <= 5.6e-158)
		tmp = ((((a * a) / t_0) * abs(b)) * abs(b)) * -4.0;
	else
		tmp = ((((abs(b) * abs(b)) / t_0) * a) * a) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 5.6e-158], N[(N[(N[(N[(N[(a * a), $MachinePrecision] / t$95$0), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(N[(N[(N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 5.6e-158

   1. Initial program 25.0%

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

   2. 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 48.7%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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 48.7%

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

   6. Applied rewrites 60.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. swap-sqr N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. frac-times N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. pow-prod-down N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. unswap-sqr N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lift-*.f64 N/A

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

   10. Applied rewrites 67.0%

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

   if 5.6e-158 < b

   1. Initial program 25.0%

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

   2. 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 48.7%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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 48.7%

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

   6. Applied rewrites 60.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. swap-sqr N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. associate-/l* N/A

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

      23. lower-*.f64 N/A

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

   8. Applied rewrites 94.3%

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

   10. Applied rewrites 67.8%

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

Alternative 10: 67.0% accurate, 20.4× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (* (* (/ (* a a) (* (* (* y-scale x-scale) x-scale) y-scale)) b) b) -4.0))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.0%

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

2. 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 48.7%

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
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 48.7%

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

6. Applied rewrites 60.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. swap-sqr N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. associate-/l* N/A

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

   18. lower-*.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. associate-/l* N/A

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

   23. lower-*.f64 N/A

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

8. Applied rewrites 94.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*r/ N/A

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

   9. lift-*.f64 N/A

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

   10. frac-times N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. pow-prod-down N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. unswap-sqr N/A

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

   20. lift-*.f64 N/A

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

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

   23. lift-*.f64 N/A

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

10. Applied rewrites 67.0%

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

Reproduce

herbie shell --seed 2025187 
(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))))