Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.1% → 94.2%

Time: 15.1s

Alternatives: 12

Speedup: 20.4×

• 65.9% 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: 24.1% 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: 94.2% accurate, 15.2× speedup

Math

\[\begin{array}{l} t_0 := a \cdot \frac{b}{\left|y-scale\right| \cdot x-scale}\\ t_1 := \frac{\frac{b}{\left|y-scale\right|} \cdot a}{x-scale}\\ \mathbf{if}\;\left|y-scale\right| \leq 2 \cdot 10^{-66}:\\ \;\;\;\;-4 \cdot \left(t\_1 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t\_0 \cdot t\_0\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* a (/ b (* (fabs y-scale) x-scale))))
        (t_1 (/ (* (/ b (fabs y-scale)) a) x-scale)))
   (if (<= (fabs y-scale) 2e-66) (* -4.0 (* t_1 t_1)) (* -4.0 (* t_0 t_0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * (b / (fabs(y_45_scale) * x_45_scale));
	double t_1 = ((b / fabs(y_45_scale)) * a) / x_45_scale;
	double tmp;
	if (fabs(y_45_scale) <= 2e-66) {
		tmp = -4.0 * (t_1 * t_1);
	} else {
		tmp = -4.0 * (t_0 * t_0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a * (b / (abs(y_45scale) * x_45scale))
    t_1 = ((b / abs(y_45scale)) * a) / x_45scale
    if (abs(y_45scale) <= 2d-66) then
        tmp = (-4.0d0) * (t_1 * t_1)
    else
        tmp = (-4.0d0) * (t_0 * t_0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * (b / (Math.abs(y_45_scale) * x_45_scale));
	double t_1 = ((b / Math.abs(y_45_scale)) * a) / x_45_scale;
	double tmp;
	if (Math.abs(y_45_scale) <= 2e-66) {
		tmp = -4.0 * (t_1 * t_1);
	} else {
		tmp = -4.0 * (t_0 * t_0);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = a * (b / (math.fabs(y_45_scale) * x_45_scale))
	t_1 = ((b / math.fabs(y_45_scale)) * a) / x_45_scale
	tmp = 0
	if math.fabs(y_45_scale) <= 2e-66:
		tmp = -4.0 * (t_1 * t_1)
	else:
		tmp = -4.0 * (t_0 * t_0)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a * Float64(b / Float64(abs(y_45_scale) * x_45_scale)))
	t_1 = Float64(Float64(Float64(b / abs(y_45_scale)) * a) / x_45_scale)
	tmp = 0.0
	if (abs(y_45_scale) <= 2e-66)
		tmp = Float64(-4.0 * Float64(t_1 * t_1));
	else
		tmp = Float64(-4.0 * Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = a * (b / (abs(y_45_scale) * x_45_scale));
	t_1 = ((b / abs(y_45_scale)) * a) / x_45_scale;
	tmp = 0.0;
	if (abs(y_45_scale) <= 2e-66)
		tmp = -4.0 * (t_1 * t_1);
	else
		tmp = -4.0 * (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a * N[(b / N[(N[Abs[y$45$scale], $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] / x$45$scale), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 2e-66], N[(-4.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2e-66

   1. Initial program 24.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 64.2%

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. unswap-sqr N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 83.3%

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

   6. Applied rewrites 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 94.2%

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

   8. Applied rewrites 94.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 89.4%

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

   10. Applied rewrites 89.4%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-/r* N/A

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

       6. associate-*l/ N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 93.7%

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

   12. Applied rewrites 93.7%

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

   if 2e-66 < y-scale

   1. Initial program 24.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 64.2%

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. unswap-sqr N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 83.3%

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

   6. Applied rewrites 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 94.2%

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

   8. Applied rewrites 94.2%

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

Alternative 2: 93.9% accurate, 15.2× speedup

Math

\[\begin{array}{l} t_0 := a \cdot \frac{b}{\left|y-scale\right| \cdot x-scale}\\ t_1 := \frac{b}{\left|y-scale\right|} \cdot \frac{a}{x-scale}\\ \mathbf{if}\;\left|y-scale\right| \leq 2 \cdot 10^{-66}:\\ \;\;\;\;-4 \cdot \left(t\_1 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t\_0 \cdot t\_0\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* a (/ b (* (fabs y-scale) x-scale))))
        (t_1 (* (/ b (fabs y-scale)) (/ a x-scale))))
   (if (<= (fabs y-scale) 2e-66) (* -4.0 (* t_1 t_1)) (* -4.0 (* t_0 t_0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * (b / (fabs(y_45_scale) * x_45_scale));
	double t_1 = (b / fabs(y_45_scale)) * (a / x_45_scale);
	double tmp;
	if (fabs(y_45_scale) <= 2e-66) {
		tmp = -4.0 * (t_1 * t_1);
	} else {
		tmp = -4.0 * (t_0 * t_0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a * (b / (abs(y_45scale) * x_45scale))
    t_1 = (b / abs(y_45scale)) * (a / x_45scale)
    if (abs(y_45scale) <= 2d-66) then
        tmp = (-4.0d0) * (t_1 * t_1)
    else
        tmp = (-4.0d0) * (t_0 * t_0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * (b / (Math.abs(y_45_scale) * x_45_scale));
	double t_1 = (b / Math.abs(y_45_scale)) * (a / x_45_scale);
	double tmp;
	if (Math.abs(y_45_scale) <= 2e-66) {
		tmp = -4.0 * (t_1 * t_1);
	} else {
		tmp = -4.0 * (t_0 * t_0);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = a * (b / (math.fabs(y_45_scale) * x_45_scale))
	t_1 = (b / math.fabs(y_45_scale)) * (a / x_45_scale)
	tmp = 0
	if math.fabs(y_45_scale) <= 2e-66:
		tmp = -4.0 * (t_1 * t_1)
	else:
		tmp = -4.0 * (t_0 * t_0)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a * Float64(b / Float64(abs(y_45_scale) * x_45_scale)))
	t_1 = Float64(Float64(b / abs(y_45_scale)) * Float64(a / x_45_scale))
	tmp = 0.0
	if (abs(y_45_scale) <= 2e-66)
		tmp = Float64(-4.0 * Float64(t_1 * t_1));
	else
		tmp = Float64(-4.0 * Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = a * (b / (abs(y_45_scale) * x_45_scale));
	t_1 = (b / abs(y_45_scale)) * (a / x_45_scale);
	tmp = 0.0;
	if (abs(y_45_scale) <= 2e-66)
		tmp = -4.0 * (t_1 * t_1);
	else
		tmp = -4.0 * (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a * N[(b / N[(N[Abs[y$45$scale], $MachinePrecision] * x$45$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], 2e-66], N[(-4.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2e-66

   1. Initial program 24.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 64.2%

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. unswap-sqr N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 83.3%

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

   6. Applied rewrites 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 94.2%

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

   8. Applied rewrites 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 89.3%

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

   10. Applied rewrites 89.3%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. times-frac N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-/.f64 93.9%

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

   12. Applied rewrites 93.9%

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

   if 2e-66 < y-scale

   1. Initial program 24.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 64.2%

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. unswap-sqr N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 83.3%

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

   6. Applied rewrites 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 94.2%

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

   8. Applied rewrites 94.2%

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

Alternative 3: 93.8% accurate, 20.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.1%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 47.5%

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
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-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. pow-prod-down N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 64.2%

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

   12. lift-*.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. lift-pow.f64 N/A

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

   19. pow2 N/A

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

   20. unswap-sqr N/A

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

   21. lower-*.f64 N/A

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

   22. lower-*.f64 N/A

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

   23. lower-*.f64 83.3%

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

6. Applied rewrites 83.3%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 94.2%

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

8. Applied rewrites 94.2%

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

Alternative 4: 92.4% accurate, 20.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.1%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 47.5%

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
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-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. pow-prod-down N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 64.2%

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

   12. lift-*.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. lift-pow.f64 N/A

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

   19. pow2 N/A

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

   20. unswap-sqr N/A

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

   21. lower-*.f64 N/A

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

   22. lower-*.f64 N/A

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

   23. lower-*.f64 83.3%

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

6. Applied rewrites 83.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/l* N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

8. Applied rewrites 92.4%

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

Alternative 5: 88.5% accurate, 15.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|a\right| \leq 1.08 \cdot 10^{+157}:\\ \;\;\;\;\left(\frac{\left(\frac{b}{y-scale \cdot x-scale} \cdot \left|a\right|\right) \cdot \left|a\right|}{y-scale \cdot x-scale} \cdot b\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{y-scale \cdot x-scale} \cdot \left(\left(\left(\left|a\right| \cdot b\right) \cdot b\right) \cdot \frac{\left|a\right|}{y-scale \cdot x-scale}\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= (fabs a) 1.08e+157)
   (*
    (*
     (/
      (* (* (/ b (* y-scale x-scale)) (fabs a)) (fabs a))
      (* y-scale x-scale))
     b)
    -4.0)
   (*
    (/ -4.0 (* y-scale x-scale))
    (* (* (* (fabs a) b) b) (/ (fabs a) (* y-scale x-scale))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (fabs(a) <= 1.08e+157) {
		tmp = (((((b / (y_45_scale * x_45_scale)) * fabs(a)) * fabs(a)) / (y_45_scale * x_45_scale)) * b) * -4.0;
	} else {
		tmp = (-4.0 / (y_45_scale * x_45_scale)) * (((fabs(a) * b) * b) * (fabs(a) / (y_45_scale * x_45_scale)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (abs(a) <= 1.08d+157) then
        tmp = (((((b / (y_45scale * x_45scale)) * abs(a)) * abs(a)) / (y_45scale * x_45scale)) * b) * (-4.0d0)
    else
        tmp = ((-4.0d0) / (y_45scale * x_45scale)) * (((abs(a) * b) * b) * (abs(a) / (y_45scale * x_45scale)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (Math.abs(a) <= 1.08e+157) {
		tmp = (((((b / (y_45_scale * x_45_scale)) * Math.abs(a)) * Math.abs(a)) / (y_45_scale * x_45_scale)) * b) * -4.0;
	} else {
		tmp = (-4.0 / (y_45_scale * x_45_scale)) * (((Math.abs(a) * b) * b) * (Math.abs(a) / (y_45_scale * x_45_scale)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if math.fabs(a) <= 1.08e+157:
		tmp = (((((b / (y_45_scale * x_45_scale)) * math.fabs(a)) * math.fabs(a)) / (y_45_scale * x_45_scale)) * b) * -4.0
	else:
		tmp = (-4.0 / (y_45_scale * x_45_scale)) * (((math.fabs(a) * b) * b) * (math.fabs(a) / (y_45_scale * x_45_scale)))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (abs(a) <= 1.08e+157)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(b / Float64(y_45_scale * x_45_scale)) * abs(a)) * abs(a)) / Float64(y_45_scale * x_45_scale)) * b) * -4.0);
	else
		tmp = Float64(Float64(-4.0 / Float64(y_45_scale * x_45_scale)) * Float64(Float64(Float64(abs(a) * b) * b) * Float64(abs(a) / Float64(y_45_scale * x_45_scale))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (abs(a) <= 1.08e+157)
		tmp = (((((b / (y_45_scale * x_45_scale)) * abs(a)) * abs(a)) / (y_45_scale * x_45_scale)) * b) * -4.0;
	else
		tmp = (-4.0 / (y_45_scale * x_45_scale)) * (((abs(a) * b) * b) * (abs(a) / (y_45_scale * x_45_scale)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[N[Abs[a], $MachinePrecision], 1.08e+157], N[(N[(N[(N[(N[(N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(-4.0 / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Abs[a], $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.08e157

   1. Initial program 24.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 64.2%

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. unswap-sqr N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 83.3%

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

   6. Applied rewrites 83.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

   8. Applied rewrites 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. associate-/l/ N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

   10. Applied rewrites 88.0%

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

   if 1.08e157 < a

   1. Initial program 24.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 64.2%

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. unswap-sqr N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 83.3%

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

   6. Applied rewrites 83.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. lift-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 83.2%

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

   8. Applied rewrites 82.7%

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

Alternative 6: 88.0% accurate, 13.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|a\right|}{y-scale \cdot x-scale}\\ t_1 := \left(\left(\left(t\_0 \cdot t\_0\right) \cdot b\right) \cdot b\right) \cdot -4\\ \mathbf{if}\;\left|a\right| \leq 10^{-166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\left|a\right| \leq 3.7 \cdot 10^{+59}:\\ \;\;\;\;\left(\left(\left(\left|a\right| \cdot t\_0\right) \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot b\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (fabs a) (* y-scale x-scale)))
        (t_1 (* (* (* (* t_0 t_0) b) b) -4.0)))
   (if (<= (fabs a) 1e-166)
     t_1
     (if (<= (fabs a) 3.7e+59)
       (* (* (* (* (fabs a) t_0) (/ b (* y-scale x-scale))) b) -4.0)
       t_1))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(a) / (y_45_scale * x_45_scale);
	double t_1 = (((t_0 * t_0) * b) * b) * -4.0;
	double tmp;
	if (fabs(a) <= 1e-166) {
		tmp = t_1;
	} else if (fabs(a) <= 3.7e+59) {
		tmp = (((fabs(a) * t_0) * (b / (y_45_scale * x_45_scale))) * b) * -4.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs(a) / (y_45scale * x_45scale)
    t_1 = (((t_0 * t_0) * b) * b) * (-4.0d0)
    if (abs(a) <= 1d-166) then
        tmp = t_1
    else if (abs(a) <= 3.7d+59) then
        tmp = (((abs(a) * t_0) * (b / (y_45scale * x_45scale))) * b) * (-4.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.abs(a) / (y_45_scale * x_45_scale);
	double t_1 = (((t_0 * t_0) * b) * b) * -4.0;
	double tmp;
	if (Math.abs(a) <= 1e-166) {
		tmp = t_1;
	} else if (Math.abs(a) <= 3.7e+59) {
		tmp = (((Math.abs(a) * t_0) * (b / (y_45_scale * x_45_scale))) * b) * -4.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.fabs(a) / (y_45_scale * x_45_scale)
	t_1 = (((t_0 * t_0) * b) * b) * -4.0
	tmp = 0
	if math.fabs(a) <= 1e-166:
		tmp = t_1
	elif math.fabs(a) <= 3.7e+59:
		tmp = (((math.fabs(a) * t_0) * (b / (y_45_scale * x_45_scale))) * b) * -4.0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(a) / Float64(y_45_scale * x_45_scale))
	t_1 = Float64(Float64(Float64(Float64(t_0 * t_0) * b) * b) * -4.0)
	tmp = 0.0
	if (abs(a) <= 1e-166)
		tmp = t_1;
	elseif (abs(a) <= 3.7e+59)
		tmp = Float64(Float64(Float64(Float64(abs(a) * t_0) * Float64(b / Float64(y_45_scale * x_45_scale))) * b) * -4.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(a) / (y_45_scale * x_45_scale);
	t_1 = (((t_0 * t_0) * b) * b) * -4.0;
	tmp = 0.0;
	if (abs(a) <= 1e-166)
		tmp = t_1;
	elseif (abs(a) <= 3.7e+59)
		tmp = (((abs(a) * t_0) * (b / (y_45_scale * x_45_scale))) * b) * -4.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[a], $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * -4.0), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 1e-166], t$95$1, If[LessEqual[N[Abs[a], $MachinePrecision], 3.7e+59], N[(N[(N[(N[(N[Abs[a], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * -4.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.00000000000000004e-166 or 3.69999999999999997e59 < a

   1. Initial program 24.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 64.2%

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. unswap-sqr N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 83.3%

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

   6. Applied rewrites 83.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

   8. Applied rewrites 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 85.4%

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

   10. Applied rewrites 85.4%

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

   if 1.00000000000000004e-166 < a < 3.69999999999999997e59

   1. Initial program 24.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 64.2%

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. unswap-sqr N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 83.3%

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

   6. Applied rewrites 83.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

   8. Applied rewrites 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. associate-/l* N/A

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

      14. associate-/r* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lower-*.f64 N/A

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

   10. Applied rewrites 84.9%

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

Alternative 7: 87.5% accurate, 20.1× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (((((b / (y_45_scale * x_45_scale)) * a) * a) / (y_45_scale * x_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 / (y_45scale * x_45scale)) * a) * a) / (y_45scale * x_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 / (y_45_scale * x_45_scale)) * a) * a) / (y_45_scale * x_45_scale)) * b) * -4.0;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.1%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 47.5%

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
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-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. pow-prod-down N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 64.2%

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

   12. lift-*.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. lift-pow.f64 N/A

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

   19. pow2 N/A

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

   20. unswap-sqr N/A

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

   21. lower-*.f64 N/A

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

   22. lower-*.f64 N/A

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

   23. lower-*.f64 83.3%

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

6. Applied rewrites 83.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

8. Applied rewrites 74.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l/ N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l/ N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/r* N/A

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

   12. associate-/l/ N/A

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

   13. lift-*.f64 N/A

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

   14. lower-/.f64 N/A

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

10. Applied rewrites 88.0%

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

Alternative 8: 85.0% accurate, 17.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq 2.1 \cdot 10^{+200}:\\ \;\;\;\;\left(\left(\left(a \cdot \frac{a}{y-scale \cdot x-scale}\right) \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot b\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot b\right) \cdot -4\right) \cdot \frac{a \cdot b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= a 2.1e+200)
   (* (* (* (* a (/ a (* y-scale x-scale))) (/ b (* y-scale x-scale))) b) -4.0)
   (*
    (* (* a b) -4.0)
    (/ (* a b) (* (* (* y-scale x-scale) y-scale) x-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 2.1e+200) {
		tmp = (((a * (a / (y_45_scale * x_45_scale))) * (b / (y_45_scale * x_45_scale))) * b) * -4.0;
	} else {
		tmp = ((a * b) * -4.0) * ((a * b) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (a <= 2.1d+200) then
        tmp = (((a * (a / (y_45scale * x_45scale))) * (b / (y_45scale * x_45scale))) * b) * (-4.0d0)
    else
        tmp = ((a * b) * (-4.0d0)) * ((a * b) / (((y_45scale * x_45scale) * y_45scale) * x_45scale))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 2.1e+200) {
		tmp = (((a * (a / (y_45_scale * x_45_scale))) * (b / (y_45_scale * x_45_scale))) * b) * -4.0;
	} else {
		tmp = ((a * b) * -4.0) * ((a * b) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a <= 2.1e+200:
		tmp = (((a * (a / (y_45_scale * x_45_scale))) * (b / (y_45_scale * x_45_scale))) * b) * -4.0
	else:
		tmp = ((a * b) * -4.0) * ((a * b) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a <= 2.1e+200)
		tmp = Float64(Float64(Float64(Float64(a * Float64(a / Float64(y_45_scale * x_45_scale))) * Float64(b / Float64(y_45_scale * x_45_scale))) * b) * -4.0);
	else
		tmp = Float64(Float64(Float64(a * b) * -4.0) * Float64(Float64(a * b) / Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a <= 2.1e+200)
		tmp = (((a * (a / (y_45_scale * x_45_scale))) * (b / (y_45_scale * x_45_scale))) * b) * -4.0;
	else
		tmp = ((a * b) * -4.0) * ((a * b) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a, 2.1e+200], N[(N[(N[(N[(a * N[(a / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(N[(a * b), $MachinePrecision] * -4.0), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.09999999999999997e200

   1. Initial program 24.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 64.2%

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. unswap-sqr N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 83.3%

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

   6. Applied rewrites 83.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

   8. Applied rewrites 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. times-frac N/A

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

      13. associate-/l* N/A

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

      14. associate-/r* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. lower-*.f64 N/A

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

   10. Applied rewrites 84.9%

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

   if 2.09999999999999997e200 < a

   1. Initial program 24.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 64.2%

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. unswap-sqr N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 83.3%

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

   6. Applied rewrites 83.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 80.2%

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

Alternative 9: 80.7% accurate, 13.7× speedup

Math

\[\begin{array}{l} t_0 := \left|y-scale\right| \cdot x-scale\\ t_1 := \left(\left(\left(\frac{a}{t\_0 \cdot t\_0} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4\\ \mathbf{if}\;\left|y-scale\right| \leq 1.8 \cdot 10^{-80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\left|y-scale\right| \leq 5 \cdot 10^{+222}:\\ \;\;\;\;\left(\left(a \cdot b\right) \cdot -4\right) \cdot \frac{a \cdot b}{\left(t\_0 \cdot \left|y-scale\right|\right) \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (fabs y-scale) x-scale))
        (t_1 (* (* (* (* (/ a (* t_0 t_0)) a) b) b) -4.0)))
   (if (<= (fabs y-scale) 1.8e-80)
     t_1
     (if (<= (fabs y-scale) 5e+222)
       (* (* (* a b) -4.0) (/ (* a b) (* (* t_0 (fabs y-scale)) x-scale)))
       t_1))))

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 t_1 = ((((a / (t_0 * t_0)) * a) * b) * b) * -4.0;
	double tmp;
	if (fabs(y_45_scale) <= 1.8e-80) {
		tmp = t_1;
	} else if (fabs(y_45_scale) <= 5e+222) {
		tmp = ((a * b) * -4.0) * ((a * b) / ((t_0 * fabs(y_45_scale)) * x_45_scale));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs(y_45scale) * x_45scale
    t_1 = ((((a / (t_0 * t_0)) * a) * b) * b) * (-4.0d0)
    if (abs(y_45scale) <= 1.8d-80) then
        tmp = t_1
    else if (abs(y_45scale) <= 5d+222) then
        tmp = ((a * b) * (-4.0d0)) * ((a * b) / ((t_0 * abs(y_45scale)) * x_45scale))
    else
        tmp = t_1
    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 t_1 = ((((a / (t_0 * t_0)) * a) * b) * b) * -4.0;
	double tmp;
	if (Math.abs(y_45_scale) <= 1.8e-80) {
		tmp = t_1;
	} else if (Math.abs(y_45_scale) <= 5e+222) {
		tmp = ((a * b) * -4.0) * ((a * b) / ((t_0 * Math.abs(y_45_scale)) * x_45_scale));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.fabs(y_45_scale) * x_45_scale
	t_1 = ((((a / (t_0 * t_0)) * a) * b) * b) * -4.0
	tmp = 0
	if math.fabs(y_45_scale) <= 1.8e-80:
		tmp = t_1
	elif math.fabs(y_45_scale) <= 5e+222:
		tmp = ((a * b) * -4.0) * ((a * b) / ((t_0 * math.fabs(y_45_scale)) * x_45_scale))
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(y_45_scale) * x_45_scale)
	t_1 = Float64(Float64(Float64(Float64(Float64(a / Float64(t_0 * t_0)) * a) * b) * b) * -4.0)
	tmp = 0.0
	if (abs(y_45_scale) <= 1.8e-80)
		tmp = t_1;
	elseif (abs(y_45_scale) <= 5e+222)
		tmp = Float64(Float64(Float64(a * b) * -4.0) * Float64(Float64(a * b) / Float64(Float64(t_0 * abs(y_45_scale)) * x_45_scale)));
	else
		tmp = t_1;
	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;
	t_1 = ((((a / (t_0 * t_0)) * a) * b) * b) * -4.0;
	tmp = 0.0;
	if (abs(y_45_scale) <= 1.8e-80)
		tmp = t_1;
	elseif (abs(y_45_scale) <= 5e+222)
		tmp = ((a * b) * -4.0) * ((a * b) / ((t_0 * abs(y_45_scale)) * x_45_scale));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(N[(N[(a / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * -4.0), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 1.8e-80], t$95$1, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 5e+222], N[(N[(N[(a * b), $MachinePrecision] * -4.0), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] / N[(N[(t$95$0 * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.8e-80 or 5.00000000000000023e222 < y-scale

   1. Initial program 24.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 64.2%

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. unswap-sqr N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 83.3%

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

   6. Applied rewrites 83.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

   8. Applied rewrites 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 77.3%

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

   10. Applied rewrites 77.3%

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

   if 1.8e-80 < y-scale < 5.00000000000000023e222

   1. Initial program 24.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 64.2%

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. unswap-sqr N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 83.3%

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

   6. Applied rewrites 83.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 80.2%

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

Alternative 10: 78.8% accurate, 13.7× speedup

Math

\[\begin{array}{l} t_0 := \left|y-scale\right| \cdot x-scale\\ t_1 := \left(\left(\left(\frac{a}{t\_0 \cdot t\_0} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4\\ \mathbf{if}\;\left|y-scale\right| \leq 4.3 \cdot 10^{-140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\left|y-scale\right| \leq 3.45 \cdot 10^{+99}:\\ \;\;\;\;a \cdot \left(\left(-4 \cdot b\right) \cdot \left(\frac{b}{\left(t\_0 \cdot \left|y-scale\right|\right) \cdot x-scale} \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (fabs y-scale) x-scale))
        (t_1 (* (* (* (* (/ a (* t_0 t_0)) a) b) b) -4.0)))
   (if (<= (fabs y-scale) 4.3e-140)
     t_1
     (if (<= (fabs y-scale) 3.45e+99)
       (* a (* (* -4.0 b) (* (/ b (* (* t_0 (fabs y-scale)) x-scale)) a)))
       t_1))))

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 t_1 = ((((a / (t_0 * t_0)) * a) * b) * b) * -4.0;
	double tmp;
	if (fabs(y_45_scale) <= 4.3e-140) {
		tmp = t_1;
	} else if (fabs(y_45_scale) <= 3.45e+99) {
		tmp = a * ((-4.0 * b) * ((b / ((t_0 * fabs(y_45_scale)) * x_45_scale)) * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs(y_45scale) * x_45scale
    t_1 = ((((a / (t_0 * t_0)) * a) * b) * b) * (-4.0d0)
    if (abs(y_45scale) <= 4.3d-140) then
        tmp = t_1
    else if (abs(y_45scale) <= 3.45d+99) then
        tmp = a * (((-4.0d0) * b) * ((b / ((t_0 * abs(y_45scale)) * x_45scale)) * a))
    else
        tmp = t_1
    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 t_1 = ((((a / (t_0 * t_0)) * a) * b) * b) * -4.0;
	double tmp;
	if (Math.abs(y_45_scale) <= 4.3e-140) {
		tmp = t_1;
	} else if (Math.abs(y_45_scale) <= 3.45e+99) {
		tmp = a * ((-4.0 * b) * ((b / ((t_0 * Math.abs(y_45_scale)) * x_45_scale)) * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.fabs(y_45_scale) * x_45_scale
	t_1 = ((((a / (t_0 * t_0)) * a) * b) * b) * -4.0
	tmp = 0
	if math.fabs(y_45_scale) <= 4.3e-140:
		tmp = t_1
	elif math.fabs(y_45_scale) <= 3.45e+99:
		tmp = a * ((-4.0 * b) * ((b / ((t_0 * math.fabs(y_45_scale)) * x_45_scale)) * a))
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(y_45_scale) * x_45_scale)
	t_1 = Float64(Float64(Float64(Float64(Float64(a / Float64(t_0 * t_0)) * a) * b) * b) * -4.0)
	tmp = 0.0
	if (abs(y_45_scale) <= 4.3e-140)
		tmp = t_1;
	elseif (abs(y_45_scale) <= 3.45e+99)
		tmp = Float64(a * Float64(Float64(-4.0 * b) * Float64(Float64(b / Float64(Float64(t_0 * abs(y_45_scale)) * x_45_scale)) * a)));
	else
		tmp = t_1;
	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;
	t_1 = ((((a / (t_0 * t_0)) * a) * b) * b) * -4.0;
	tmp = 0.0;
	if (abs(y_45_scale) <= 4.3e-140)
		tmp = t_1;
	elseif (abs(y_45_scale) <= 3.45e+99)
		tmp = a * ((-4.0 * b) * ((b / ((t_0 * abs(y_45_scale)) * x_45_scale)) * a));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(N[(N[(a / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * -4.0), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 4.3e-140], t$95$1, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 3.45e+99], N[(a * N[(N[(-4.0 * b), $MachinePrecision] * N[(N[(b / N[(N[(t$95$0 * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 4.29999999999999962e-140 or 3.45e99 < y-scale

   1. Initial program 24.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-prod-down N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 64.2%

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

      12. lift-*.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. unswap-sqr N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 83.3%

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

   6. Applied rewrites 83.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

   8. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\left(\left(\left(\frac{a}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{a}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{a}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4 \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(\left(\frac{a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4 \]

      5. lower-*.f64 77.3%

         \[\leadsto \left(\left(\left(\frac{a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4 \]

   10. Applied rewrites 77.3%

       \[\leadsto \left(\left(\left(\frac{a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4 \]

   if 4.29999999999999962e-140 < y-scale < 3.45e99

   1. Initial program 24.1%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{x-scale}^{2}} \cdot {y-scale}^{2}} \]

      4. lower-pow.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{\color{blue}{x-scale}}^{2} \cdot {y-scale}^{2}} \]

      5. lower-pow.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{\color{blue}{2}} \cdot {y-scale}^{2}} \]

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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-*.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot \color{blue}{{y-scale}^{2}}} \]

      3. lift-pow.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {\color{blue}{y-scale}}^{2}} \]

      4. lift-pow.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{\color{blue}{2}}} \]

      5. pow-prod-down N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{\left(x-scale \cdot y-scale\right)}^{\color{blue}{2}}} \]

      6. *-commutative N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{\left(y-scale \cdot x-scale\right)}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{\left(y-scale \cdot x-scale\right)}^{2}} \]

      8. pow2 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\left(y-scale \cdot x-scale\right) \cdot \color{blue}{\left(y-scale \cdot x-scale\right)}} \]

      9. associate-/r* N/A

         \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}}{\color{blue}{y-scale \cdot x-scale}} \]

      10. lower-/.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}}{\color{blue}{y-scale \cdot x-scale}} \]

      11. lower-/.f64 64.2%

          \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}}{\color{blue}{y-scale} \cdot x-scale} \]

      12. lift-*.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      13. lift-pow.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      14. pow2 N/A

          \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot \left(b \cdot b\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      15. lift-*.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot \left(b \cdot b\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      16. *-commutative N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot b\right) \cdot {a}^{2}}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      17. lift-*.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot b\right) \cdot {a}^{2}}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      18. lift-pow.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot b\right) \cdot {a}^{2}}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      19. pow2 N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      20. unswap-sqr N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      21. lower-*.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      22. lower-*.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      23. lower-*.f64 83.3%

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

   6. Applied rewrites 83.3%

      \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{\color{blue}{y-scale \cdot x-scale}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{\color{blue}{y-scale \cdot x-scale}} \]

      2. lift-/.f64 N/A

         \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{\color{blue}{y-scale} \cdot x-scale} \]

      3. associate-/l/ N/A

         \[\leadsto -4 \cdot \frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto -4 \cdot \frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{\color{blue}{\left(y-scale \cdot x-scale\right)} \cdot \left(y-scale \cdot x-scale\right)} \]

      5. times-frac N/A

         \[\leadsto -4 \cdot \left(\frac{b \cdot a}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{b \cdot a}{y-scale \cdot x-scale}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -4 \cdot \left(\frac{b \cdot a}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{b \cdot a}{y-scale \cdot x-scale}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto -4 \cdot \left(\frac{b \cdot a}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{b} \cdot a}{y-scale \cdot x-scale}\right) \]

      8. *-commutative N/A

         \[\leadsto -4 \cdot \left(\frac{a \cdot b}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{b} \cdot a}{y-scale \cdot x-scale}\right) \]

      9. associate-/l* N/A

         \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \frac{\color{blue}{b \cdot a}}{y-scale \cdot x-scale}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \frac{\color{blue}{b \cdot a}}{y-scale \cdot x-scale}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \frac{b \cdot \color{blue}{a}}{y-scale \cdot x-scale}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \frac{b \cdot a}{\color{blue}{y-scale} \cdot x-scale}\right) \]

      13. *-commutative N/A

          \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \frac{a \cdot b}{\color{blue}{y-scale} \cdot x-scale}\right) \]

      14. associate-/l* N/A

          \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(a \cdot \color{blue}{\frac{b}{y-scale \cdot x-scale}}\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(a \cdot \color{blue}{\frac{b}{y-scale \cdot x-scale}}\right)\right) \]

      16. lower-/.f64 94.2%

          \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(a \cdot \frac{b}{\color{blue}{y-scale \cdot x-scale}}\right)\right) \]

   8. Applied rewrites 94.2%

      \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \color{blue}{\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right)}\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(a \cdot \frac{b}{y-scale \cdot x-scale}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \color{blue}{\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(\color{blue}{a} \cdot \frac{b}{y-scale \cdot x-scale}\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(a \cdot \frac{b}{y-scale \cdot x-scale}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto -4 \cdot \left(\frac{a \cdot b}{y-scale \cdot x-scale} \cdot \left(\color{blue}{a} \cdot \frac{b}{y-scale \cdot x-scale}\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto -4 \cdot \left(\frac{a \cdot b}{y-scale \cdot x-scale} \cdot \left(a \cdot \frac{b}{y-scale \cdot x-scale}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto -4 \cdot \frac{\left(a \cdot b\right) \cdot \left(a \cdot \frac{b}{y-scale \cdot x-scale}\right)}{\color{blue}{y-scale \cdot x-scale}} \]

      8. associate-*r/ N/A

         \[\leadsto -4 \cdot \left(\left(a \cdot b\right) \cdot \color{blue}{\frac{a \cdot \frac{b}{y-scale \cdot x-scale}}{y-scale \cdot x-scale}}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto -4 \cdot \left(\left(a \cdot b\right) \cdot \frac{a \cdot \frac{b}{y-scale \cdot x-scale}}{\color{blue}{y-scale \cdot x-scale}}\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(-4 \cdot \left(a \cdot b\right)\right) \cdot \color{blue}{\frac{a \cdot \frac{b}{y-scale \cdot x-scale}}{y-scale \cdot x-scale}} \]

      11. *-commutative N/A

          \[\leadsto \left(\left(a \cdot b\right) \cdot -4\right) \cdot \frac{\color{blue}{a \cdot \frac{b}{y-scale \cdot x-scale}}}{y-scale \cdot x-scale} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(a \cdot b\right) \cdot -4\right) \cdot \frac{\color{blue}{a} \cdot \frac{b}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      13. associate-*l* N/A

          \[\leadsto \left(a \cdot \left(b \cdot -4\right)\right) \cdot \frac{\color{blue}{a \cdot \frac{b}{y-scale \cdot x-scale}}}{y-scale \cdot x-scale} \]

      14. associate-*l* N/A

          \[\leadsto a \cdot \color{blue}{\left(\left(b \cdot -4\right) \cdot \frac{a \cdot \frac{b}{y-scale \cdot x-scale}}{y-scale \cdot x-scale}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(\left(b \cdot -4\right) \cdot \frac{a \cdot \frac{b}{y-scale \cdot x-scale}}{y-scale \cdot x-scale}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto a \cdot \left(\left(b \cdot -4\right) \cdot \color{blue}{\frac{a \cdot \frac{b}{y-scale \cdot x-scale}}{y-scale \cdot x-scale}}\right) \]

   10. Applied rewrites 77.7%

       \[\leadsto a \cdot \color{blue}{\left(\left(-4 \cdot b\right) \cdot \left(\frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot a\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 78.7% accurate, 13.7× speedup

Math

\[\begin{array}{l} t_0 := \left|y-scale\right| \cdot x-scale\\ t_1 := \left(\left(\left(\frac{a}{t\_0 \cdot t\_0} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4\\ \mathbf{if}\;\left|y-scale\right| \leq 9.5 \cdot 10^{-81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\left|y-scale\right| \leq 10^{+108}:\\ \;\;\;\;\left(\frac{b}{\left(t\_0 \cdot \left|y-scale\right|\right) \cdot x-scale} \cdot a\right) \cdot \left(\left(a \cdot b\right) \cdot -4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (fabs y-scale) x-scale))
        (t_1 (* (* (* (* (/ a (* t_0 t_0)) a) b) b) -4.0)))
   (if (<= (fabs y-scale) 9.5e-81)
     t_1
     (if (<= (fabs y-scale) 1e+108)
       (* (* (/ b (* (* t_0 (fabs y-scale)) x-scale)) a) (* (* a b) -4.0))
       t_1))))

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 t_1 = ((((a / (t_0 * t_0)) * a) * b) * b) * -4.0;
	double tmp;
	if (fabs(y_45_scale) <= 9.5e-81) {
		tmp = t_1;
	} else if (fabs(y_45_scale) <= 1e+108) {
		tmp = ((b / ((t_0 * fabs(y_45_scale)) * x_45_scale)) * a) * ((a * b) * -4.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs(y_45scale) * x_45scale
    t_1 = ((((a / (t_0 * t_0)) * a) * b) * b) * (-4.0d0)
    if (abs(y_45scale) <= 9.5d-81) then
        tmp = t_1
    else if (abs(y_45scale) <= 1d+108) then
        tmp = ((b / ((t_0 * abs(y_45scale)) * x_45scale)) * a) * ((a * b) * (-4.0d0))
    else
        tmp = t_1
    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 t_1 = ((((a / (t_0 * t_0)) * a) * b) * b) * -4.0;
	double tmp;
	if (Math.abs(y_45_scale) <= 9.5e-81) {
		tmp = t_1;
	} else if (Math.abs(y_45_scale) <= 1e+108) {
		tmp = ((b / ((t_0 * Math.abs(y_45_scale)) * x_45_scale)) * a) * ((a * b) * -4.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.fabs(y_45_scale) * x_45_scale
	t_1 = ((((a / (t_0 * t_0)) * a) * b) * b) * -4.0
	tmp = 0
	if math.fabs(y_45_scale) <= 9.5e-81:
		tmp = t_1
	elif math.fabs(y_45_scale) <= 1e+108:
		tmp = ((b / ((t_0 * math.fabs(y_45_scale)) * x_45_scale)) * a) * ((a * b) * -4.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(y_45_scale) * x_45_scale)
	t_1 = Float64(Float64(Float64(Float64(Float64(a / Float64(t_0 * t_0)) * a) * b) * b) * -4.0)
	tmp = 0.0
	if (abs(y_45_scale) <= 9.5e-81)
		tmp = t_1;
	elseif (abs(y_45_scale) <= 1e+108)
		tmp = Float64(Float64(Float64(b / Float64(Float64(t_0 * abs(y_45_scale)) * x_45_scale)) * a) * Float64(Float64(a * b) * -4.0));
	else
		tmp = t_1;
	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;
	t_1 = ((((a / (t_0 * t_0)) * a) * b) * b) * -4.0;
	tmp = 0.0;
	if (abs(y_45_scale) <= 9.5e-81)
		tmp = t_1;
	elseif (abs(y_45_scale) <= 1e+108)
		tmp = ((b / ((t_0 * abs(y_45_scale)) * x_45_scale)) * a) * ((a * b) * -4.0);
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(N[(N[(a / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * -4.0), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 9.5e-81], t$95$1, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 1e+108], N[(N[(N[(b / N[(N[(t$95$0 * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 9.49999999999999917e-81 or 1e108 < y-scale

   1. Initial program 24.1%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{x-scale}^{2}} \cdot {y-scale}^{2}} \]

      4. lower-pow.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{\color{blue}{x-scale}}^{2} \cdot {y-scale}^{2}} \]

      5. lower-pow.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{\color{blue}{2}} \cdot {y-scale}^{2}} \]

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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-*.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot \color{blue}{{y-scale}^{2}}} \]

      3. lift-pow.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {\color{blue}{y-scale}}^{2}} \]

      4. lift-pow.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{\color{blue}{2}}} \]

      5. pow-prod-down N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{\left(x-scale \cdot y-scale\right)}^{\color{blue}{2}}} \]

      6. *-commutative N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{\left(y-scale \cdot x-scale\right)}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{\left(y-scale \cdot x-scale\right)}^{2}} \]

      8. pow2 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\left(y-scale \cdot x-scale\right) \cdot \color{blue}{\left(y-scale \cdot x-scale\right)}} \]

      9. associate-/r* N/A

         \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}}{\color{blue}{y-scale \cdot x-scale}} \]

      10. lower-/.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}}{\color{blue}{y-scale \cdot x-scale}} \]

      11. lower-/.f64 64.2%

          \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}}{\color{blue}{y-scale} \cdot x-scale} \]

      12. lift-*.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      13. lift-pow.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      14. pow2 N/A

          \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot \left(b \cdot b\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      15. lift-*.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot \left(b \cdot b\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      16. *-commutative N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot b\right) \cdot {a}^{2}}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      17. lift-*.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot b\right) \cdot {a}^{2}}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      18. lift-pow.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot b\right) \cdot {a}^{2}}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      19. pow2 N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      20. unswap-sqr N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      21. lower-*.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      22. lower-*.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      23. lower-*.f64 83.3%

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

   6. Applied rewrites 83.3%

      \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{\color{blue}{y-scale \cdot x-scale}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \cdot \color{blue}{-4} \]

   8. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\left(\left(\left(\frac{a}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{a}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{a}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4 \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(\left(\frac{a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4 \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\frac{a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4 \]

      5. lower-*.f64 77.3%

         \[\leadsto \left(\left(\left(\frac{a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4 \]

   10. Applied rewrites 77.3%

       \[\leadsto \left(\left(\left(\frac{a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4 \]

   if 9.49999999999999917e-81 < y-scale < 1e108

   1. Initial program 24.1%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{x-scale}^{2}} \cdot {y-scale}^{2}} \]

      4. lower-pow.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{\color{blue}{x-scale}}^{2} \cdot {y-scale}^{2}} \]

      5. lower-pow.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{\color{blue}{2}} \cdot {y-scale}^{2}} \]

   4. Applied rewrites 47.5%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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-*.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot \color{blue}{{y-scale}^{2}}} \]

      3. lift-pow.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {\color{blue}{y-scale}}^{2}} \]

      4. lift-pow.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{\color{blue}{2}}} \]

      5. pow-prod-down N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{\left(x-scale \cdot y-scale\right)}^{\color{blue}{2}}} \]

      6. *-commutative N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{\left(y-scale \cdot x-scale\right)}^{2}} \]

      7. lift-*.f64 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{\left(y-scale \cdot x-scale\right)}^{2}} \]

      8. pow2 N/A

         \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\left(y-scale \cdot x-scale\right) \cdot \color{blue}{\left(y-scale \cdot x-scale\right)}} \]

      9. associate-/r* N/A

         \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}}{\color{blue}{y-scale \cdot x-scale}} \]

      10. lower-/.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}}{\color{blue}{y-scale \cdot x-scale}} \]

      11. lower-/.f64 64.2%

          \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}}{\color{blue}{y-scale} \cdot x-scale} \]

      12. lift-*.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      13. lift-pow.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      14. pow2 N/A

          \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot \left(b \cdot b\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      15. lift-*.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot \left(b \cdot b\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      16. *-commutative N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot b\right) \cdot {a}^{2}}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      17. lift-*.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot b\right) \cdot {a}^{2}}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      18. lift-pow.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot b\right) \cdot {a}^{2}}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      19. pow2 N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      20. unswap-sqr N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      21. lower-*.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      22. lower-*.f64 N/A

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

      23. lower-*.f64 83.3%

          \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

   6. Applied rewrites 83.3%

      \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{\color{blue}{y-scale \cdot x-scale}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{\color{blue}{y-scale \cdot x-scale}} \]

      2. lift-/.f64 N/A

         \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{\color{blue}{y-scale} \cdot x-scale} \]

      3. associate-/l/ N/A

         \[\leadsto -4 \cdot \frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto -4 \cdot \frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{\color{blue}{\left(y-scale \cdot x-scale\right)} \cdot \left(y-scale \cdot x-scale\right)} \]

      5. times-frac N/A

         \[\leadsto -4 \cdot \left(\frac{b \cdot a}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{b \cdot a}{y-scale \cdot x-scale}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -4 \cdot \left(\frac{b \cdot a}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{b \cdot a}{y-scale \cdot x-scale}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto -4 \cdot \left(\frac{b \cdot a}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{b} \cdot a}{y-scale \cdot x-scale}\right) \]

      8. *-commutative N/A

         \[\leadsto -4 \cdot \left(\frac{a \cdot b}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{b} \cdot a}{y-scale \cdot x-scale}\right) \]

      9. associate-/l* N/A

         \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \frac{\color{blue}{b \cdot a}}{y-scale \cdot x-scale}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \frac{\color{blue}{b \cdot a}}{y-scale \cdot x-scale}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \frac{b \cdot \color{blue}{a}}{y-scale \cdot x-scale}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \frac{b \cdot a}{\color{blue}{y-scale} \cdot x-scale}\right) \]

      13. *-commutative N/A

          \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \frac{a \cdot b}{\color{blue}{y-scale} \cdot x-scale}\right) \]

      14. associate-/l* N/A

          \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(a \cdot \color{blue}{\frac{b}{y-scale \cdot x-scale}}\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(a \cdot \color{blue}{\frac{b}{y-scale \cdot x-scale}}\right)\right) \]

      16. lower-/.f64 94.2%

          \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(a \cdot \frac{b}{\color{blue}{y-scale \cdot x-scale}}\right)\right) \]

   8. Applied rewrites 94.2%

      \[\leadsto -4 \cdot \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \color{blue}{\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right)}\right) \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -4 \cdot \color{blue}{\left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(a \cdot \frac{b}{y-scale \cdot x-scale}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(a \cdot \frac{b}{y-scale \cdot x-scale}\right)\right) \cdot \color{blue}{-4} \]

      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. lift-*.f64 N/A

         \[\leadsto \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(a \cdot \frac{b}{y-scale \cdot x-scale}\right)\right) \cdot -4 \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(a \cdot \frac{b}{y-scale \cdot x-scale}\right)\right) \cdot -4 \]

      6. associate-*r/ N/A

         \[\leadsto \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \frac{a \cdot b}{y-scale \cdot x-scale}\right) \cdot -4 \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \frac{a \cdot b}{y-scale \cdot x-scale}\right) \cdot -4 \]

      8. associate-*r/ N/A

         \[\leadsto \frac{\left(a \cdot \frac{b}{y-scale \cdot x-scale}\right) \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot -4 \]

      9. associate-*l/ N/A

         \[\leadsto \left(\frac{a \cdot \frac{b}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \cdot \left(a \cdot b\right)\right) \cdot -4 \]

      10. lift-/.f64 N/A

          \[\leadsto \left(\frac{a \cdot \frac{b}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \cdot \left(a \cdot b\right)\right) \cdot -4 \]

      11. associate-*l* N/A

          \[\leadsto \frac{a \cdot \frac{b}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \cdot \color{blue}{\left(\left(a \cdot b\right) \cdot -4\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{a \cdot \frac{b}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \cdot \left(\left(a \cdot b\right) \cdot \color{blue}{-4}\right) \]

      13. lower-*.f64 90.4%

          \[\leadsto \frac{a \cdot \frac{b}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \cdot \color{blue}{\left(\left(a \cdot b\right) \cdot -4\right)} \]

   10. Applied rewrites 78.1%

       \[\leadsto \left(\frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot a\right) \cdot \color{blue}{\left(\left(a \cdot b\right) \cdot -4\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 77.3% accurate, 20.4× speedup

Math

\[\left(\left(\left(\frac{a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4 \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (* (* (* (/ a (* (* y-scale x-scale) (* y-scale x-scale))) a) b) b) -4.0))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((((a / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * a) * 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 / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * a) * 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 / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * a) * b) * b) * -4.0;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return ((((a / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * a) * b) * b) * -4.0

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(Float64(Float64(Float64(a / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * a) * b) * b) * -4.0)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = ((((a / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * a) * b) * b) * -4.0;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(N[(N[(a / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * -4.0), $MachinePrecision]

Derivation

1. Initial program 24.1%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -4 \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   2. lower-/.f64 N/A

      \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   3. lower-*.f64 N/A

      \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{x-scale}^{2}} \cdot {y-scale}^{2}} \]

   4. lower-pow.f64 N/A

      \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{\color{blue}{x-scale}}^{2} \cdot {y-scale}^{2}} \]

   5. lower-pow.f64 N/A

      \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{\color{blue}{2}} \cdot {y-scale}^{2}} \]

4. Applied rewrites 47.5%

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
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-*.f64 N/A

      \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot \color{blue}{{y-scale}^{2}}} \]

   3. lift-pow.f64 N/A

      \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {\color{blue}{y-scale}}^{2}} \]

   4. lift-pow.f64 N/A

      \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{\color{blue}{2}}} \]

   5. pow-prod-down N/A

      \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{\left(x-scale \cdot y-scale\right)}^{\color{blue}{2}}} \]

   6. *-commutative N/A

      \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{\left(y-scale \cdot x-scale\right)}^{2}} \]

   7. lift-*.f64 N/A

      \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{\left(y-scale \cdot x-scale\right)}^{2}} \]

   8. pow2 N/A

      \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\left(y-scale \cdot x-scale\right) \cdot \color{blue}{\left(y-scale \cdot x-scale\right)}} \]

   9. associate-/r* N/A

      \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}}{\color{blue}{y-scale \cdot x-scale}} \]

   10. lower-/.f64 N/A

       \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}}{\color{blue}{y-scale \cdot x-scale}} \]

   11. lower-/.f64 64.2%

       \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}}{\color{blue}{y-scale} \cdot x-scale} \]

   12. lift-*.f64 N/A

       \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

   13. lift-pow.f64 N/A

       \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot {b}^{2}}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

   14. pow2 N/A

       \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot \left(b \cdot b\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

   15. lift-*.f64 N/A

       \[\leadsto -4 \cdot \frac{\frac{{a}^{2} \cdot \left(b \cdot b\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

   16. *-commutative N/A

       \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot b\right) \cdot {a}^{2}}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

   17. lift-*.f64 N/A

       \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot b\right) \cdot {a}^{2}}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

   18. lift-pow.f64 N/A

       \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot b\right) \cdot {a}^{2}}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

   19. pow2 N/A

       \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot b\right) \cdot \left(a \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

   20. unswap-sqr N/A

       \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

   21. lower-*.f64 N/A

       \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

   22. lower-*.f64 N/A

       \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

   23. lower-*.f64 83.3%

       \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \]

6. Applied rewrites 83.3%

   \[\leadsto -4 \cdot \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{\color{blue}{y-scale \cdot x-scale}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -4 \cdot \color{blue}{\frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \cdot \color{blue}{-4} \]

8. Applied rewrites 74.6%

   \[\leadsto \color{blue}{\left(\left(\left(\frac{a}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(\frac{a}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4 \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(\frac{a}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4 \]

   3. associate-*l* N/A

      \[\leadsto \left(\left(\left(\frac{a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4 \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(\frac{a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4 \]

   5. lower-*.f64 77.3%

      \[\leadsto \left(\left(\left(\frac{a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4 \]

10. Applied rewrites 77.3%

    \[\leadsto \left(\left(\left(\frac{a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(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))))