Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.6% → 92.5%

Time: 15.4s

Alternatives: 11

Speedup: 40.5×

• 66.2% 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: 24.6% 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: 92.5% accurate, 26.8× speedup

Math

\[\begin{array}{l} t_0 := a \cdot \left|b\right|\\ \mathbf{if}\;\left|b\right| \leq 7 \cdot 10^{-246}:\\ \;\;\;\;\frac{\frac{t\_0 \cdot -4}{y-scale} \cdot \frac{t\_0}{x-scale}}{y-scale \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left|b\right|}{y-scale \cdot x-scale} \cdot a\right) \cdot \left(\left(\frac{-4}{y-scale \cdot x-scale} \cdot a\right) \cdot \left|b\right|\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* a (fabs b))))
  (if (<= (fabs b) 7e-246)
    (/
     (* (/ (* t_0 -4.0) y-scale) (/ t_0 x-scale))
     (* y-scale x-scale))
    (*
     (* (/ (fabs b) (* y-scale x-scale)) a)
     (* (* (/ -4.0 (* y-scale x-scale)) a) (fabs b))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * fabs(b);
	double tmp;
	if (fabs(b) <= 7e-246) {
		tmp = (((t_0 * -4.0) / y_45_scale) * (t_0 / x_45_scale)) / (y_45_scale * x_45_scale);
	} else {
		tmp = ((fabs(b) / (y_45_scale * x_45_scale)) * a) * (((-4.0 / (y_45_scale * x_45_scale)) * a) * fabs(b));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * abs(b)
    if (abs(b) <= 7d-246) then
        tmp = (((t_0 * (-4.0d0)) / y_45scale) * (t_0 / x_45scale)) / (y_45scale * x_45scale)
    else
        tmp = ((abs(b) / (y_45scale * x_45scale)) * a) * ((((-4.0d0) / (y_45scale * x_45scale)) * a) * abs(b))
    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 * Math.abs(b);
	double tmp;
	if (Math.abs(b) <= 7e-246) {
		tmp = (((t_0 * -4.0) / y_45_scale) * (t_0 / x_45_scale)) / (y_45_scale * x_45_scale);
	} else {
		tmp = ((Math.abs(b) / (y_45_scale * x_45_scale)) * a) * (((-4.0 / (y_45_scale * x_45_scale)) * a) * Math.abs(b));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = a * math.fabs(b)
	tmp = 0
	if math.fabs(b) <= 7e-246:
		tmp = (((t_0 * -4.0) / y_45_scale) * (t_0 / x_45_scale)) / (y_45_scale * x_45_scale)
	else:
		tmp = ((math.fabs(b) / (y_45_scale * x_45_scale)) * a) * (((-4.0 / (y_45_scale * x_45_scale)) * a) * math.fabs(b))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a * abs(b))
	tmp = 0.0
	if (abs(b) <= 7e-246)
		tmp = Float64(Float64(Float64(Float64(t_0 * -4.0) / y_45_scale) * Float64(t_0 / x_45_scale)) / Float64(y_45_scale * x_45_scale));
	else
		tmp = Float64(Float64(Float64(abs(b) / Float64(y_45_scale * x_45_scale)) * a) * Float64(Float64(Float64(-4.0 / Float64(y_45_scale * x_45_scale)) * a) * abs(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = a * abs(b);
	tmp = 0.0;
	if (abs(b) <= 7e-246)
		tmp = (((t_0 * -4.0) / y_45_scale) * (t_0 / x_45_scale)) / (y_45_scale * x_45_scale);
	else
		tmp = ((abs(b) / (y_45_scale * x_45_scale)) * a) * (((-4.0 / (y_45_scale * x_45_scale)) * a) * abs(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 7e-246], N[(N[(N[(N[(t$95$0 * -4.0), $MachinePrecision] / y$45$scale), $MachinePrecision] * N[(t$95$0 / x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[b], $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * N[(N[(N[(-4.0 / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.0000000000000003e-246

   1. Initial program 24.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-prod-down N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.5%

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

   6. Applied rewrites 83.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 83.3%

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 79.9%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 79.9%

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

   8. Applied rewrites 79.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. times-frac N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 N/A

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

      22. lower-/.f64 89.1%

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

   10. Applied rewrites 89.1%

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

   if 7.0000000000000003e-246 < b

   1. Initial program 24.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-prod-down N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.5%

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

   6. Applied rewrites 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 89.6%

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

   8. Applied rewrites 89.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 92.5%

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

   10. Applied rewrites 92.5%

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

Alternative 2: 92.5% accurate, 35.9× speedup

Math

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

FPCore

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

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) * (((-4.0 / (y_45_scale * x_45_scale)) * a) * b);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = ((b / (y_45scale * x_45scale)) * a) * ((((-4.0d0) / (y_45scale * x_45scale)) * a) * b)
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) * (((-4.0 / (y_45_scale * x_45_scale)) * a) * b);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.6%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 47.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-prod-down N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 64.5%

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

6. Applied rewrites 83.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 89.6%

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

8. Applied rewrites 89.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 92.5%

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

10. Applied rewrites 92.5%

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

Alternative 3: 90.5% accurate, 35.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.6%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 47.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-prod-down N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 64.5%

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

6. Applied rewrites 83.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 89.6%

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

8. Applied rewrites 89.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 90.5%

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 90.5%

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

10. Applied rewrites 90.5%

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

Alternative 4: 89.5% accurate, 35.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.6%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 47.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-prod-down N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 64.5%

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

6. Applied rewrites 83.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 89.6%

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

8. Applied rewrites 89.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 89.5%

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

10. Applied rewrites 89.5%

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

Alternative 5: 82.0% accurate, 26.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{b}{y-scale \cdot x-scale}\\ t_1 := \left|a\right| \cdot b\\ \mathbf{if}\;\left|a\right| \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(4 \cdot \left|a\right|\right) \cdot \left(t\_1 \cdot b\right)}{\left(\left(-x-scale\right) \cdot y-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\\ \mathbf{elif}\;\left|a\right| \leq 1.55 \cdot 10^{+125}:\\ \;\;\;\;\left(\left(\left|a\right| \cdot \left|a\right|\right) \cdot \left(t\_0 \cdot t\_0\right)\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\left|a\right| \cdot \frac{t\_1 \cdot \frac{b}{\left(y-scale \cdot y-scale\right) \cdot x-scale}}{x-scale}\right) \cdot -4\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (/ b (* y-scale x-scale))) (t_1 (* (fabs a) b)))
  (if (<= (fabs a) 1.6e-162)
    (/
     (* (* 4.0 (fabs a)) (* t_1 b))
     (* (* (- x-scale) y-scale) (* y-scale x-scale)))
    (if (<= (fabs a) 1.55e+125)
      (* (* (* (fabs a) (fabs a)) (* t_0 t_0)) -4.0)
      (*
       (*
        (fabs a)
        (/ (* t_1 (/ b (* (* y-scale y-scale) x-scale))) x-scale))
       -4.0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (y_45_scale * x_45_scale);
	double t_1 = fabs(a) * b;
	double tmp;
	if (fabs(a) <= 1.6e-162) {
		tmp = ((4.0 * fabs(a)) * (t_1 * b)) / ((-x_45_scale * y_45_scale) * (y_45_scale * x_45_scale));
	} else if (fabs(a) <= 1.55e+125) {
		tmp = ((fabs(a) * fabs(a)) * (t_0 * t_0)) * -4.0;
	} else {
		tmp = (fabs(a) * ((t_1 * (b / ((y_45_scale * y_45_scale) * x_45_scale))) / x_45_scale)) * -4.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = b / (y_45scale * x_45scale)
    t_1 = abs(a) * b
    if (abs(a) <= 1.6d-162) then
        tmp = ((4.0d0 * abs(a)) * (t_1 * b)) / ((-x_45scale * y_45scale) * (y_45scale * x_45scale))
    else if (abs(a) <= 1.55d+125) then
        tmp = ((abs(a) * abs(a)) * (t_0 * t_0)) * (-4.0d0)
    else
        tmp = (abs(a) * ((t_1 * (b / ((y_45scale * y_45scale) * x_45scale))) / x_45scale)) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (y_45_scale * x_45_scale);
	double t_1 = Math.abs(a) * b;
	double tmp;
	if (Math.abs(a) <= 1.6e-162) {
		tmp = ((4.0 * Math.abs(a)) * (t_1 * b)) / ((-x_45_scale * y_45_scale) * (y_45_scale * x_45_scale));
	} else if (Math.abs(a) <= 1.55e+125) {
		tmp = ((Math.abs(a) * Math.abs(a)) * (t_0 * t_0)) * -4.0;
	} else {
		tmp = (Math.abs(a) * ((t_1 * (b / ((y_45_scale * y_45_scale) * x_45_scale))) / x_45_scale)) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = b / (y_45_scale * x_45_scale)
	t_1 = math.fabs(a) * b
	tmp = 0
	if math.fabs(a) <= 1.6e-162:
		tmp = ((4.0 * math.fabs(a)) * (t_1 * b)) / ((-x_45_scale * y_45_scale) * (y_45_scale * x_45_scale))
	elif math.fabs(a) <= 1.55e+125:
		tmp = ((math.fabs(a) * math.fabs(a)) * (t_0 * t_0)) * -4.0
	else:
		tmp = (math.fabs(a) * ((t_1 * (b / ((y_45_scale * y_45_scale) * x_45_scale))) / x_45_scale)) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(y_45_scale * x_45_scale))
	t_1 = Float64(abs(a) * b)
	tmp = 0.0
	if (abs(a) <= 1.6e-162)
		tmp = Float64(Float64(Float64(4.0 * abs(a)) * Float64(t_1 * b)) / Float64(Float64(Float64(-x_45_scale) * y_45_scale) * Float64(y_45_scale * x_45_scale)));
	elseif (abs(a) <= 1.55e+125)
		tmp = Float64(Float64(Float64(abs(a) * abs(a)) * Float64(t_0 * t_0)) * -4.0);
	else
		tmp = Float64(Float64(abs(a) * Float64(Float64(t_1 * Float64(b / Float64(Float64(y_45_scale * y_45_scale) * x_45_scale))) / x_45_scale)) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = b / (y_45_scale * x_45_scale);
	t_1 = abs(a) * b;
	tmp = 0.0;
	if (abs(a) <= 1.6e-162)
		tmp = ((4.0 * abs(a)) * (t_1 * b)) / ((-x_45_scale * y_45_scale) * (y_45_scale * x_45_scale));
	elseif (abs(a) <= 1.55e+125)
		tmp = ((abs(a) * abs(a)) * (t_0 * t_0)) * -4.0;
	else
		tmp = (abs(a) * ((t_1 * (b / ((y_45_scale * y_45_scale) * x_45_scale))) / x_45_scale)) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[a], $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 1.6e-162], N[(N[(N[(4.0 * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * b), $MachinePrecision]), $MachinePrecision] / N[(N[((-x$45$scale) * y$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[a], $MachinePrecision], 1.55e+125], N[(N[(N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(N[Abs[a], $MachinePrecision] * N[(N[(t$95$1 * N[(b / N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.5999999999999999e-162

   1. Initial program 24.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-prod-down N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.5%

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

   6. Applied rewrites 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 89.6%

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

   8. Applied rewrites 89.6%

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

   10. Applied rewrites 74.9%

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

   if 1.5999999999999999e-162 < a < 1.55e125

   1. Initial program 24.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.9%

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

   6. Applied rewrites 54.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. swap-sqr N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 75.7%

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

   8. Applied rewrites 75.7%

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

   if 1.55e125 < a

   1. Initial program 24.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.9%

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

   6. Applied rewrites 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 60.5%

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. swap-sqr N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 76.9%

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. swap-sqr N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*l* N/A

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

   8. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 72.9%

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

   10. Applied rewrites 72.9%

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

Alternative 6: 81.5% accurate, 26.1× speedup

Math

\[\begin{array}{l} t_0 := \left|y-scale\right| \cdot x-scale\\ t_1 := \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{t\_0 \cdot t\_0}\right)\right)\right) \cdot -4\\ \mathbf{if}\;\left|y-scale\right| \leq 10^{-161}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\left|y-scale\right| \leq 2.9 \cdot 10^{+134}:\\ \;\;\;\;\left(a \cdot \frac{\left(a \cdot b\right) \cdot \frac{b}{\left(\left|y-scale\right| \cdot \left|y-scale\right|\right) \cdot x-scale}}{x-scale}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (fabs y-scale) x-scale))
       (t_1 (* (* a (* a (* b (/ b (* t_0 t_0))))) -4.0)))
  (if (<= (fabs y-scale) 1e-161)
    t_1
    (if (<= (fabs y-scale) 2.9e+134)
      (*
       (*
        a
        (/
         (*
          (* a b)
          (/ b (* (* (fabs y-scale) (fabs y-scale)) x-scale)))
         x-scale))
       -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 * (a * (b * (b / (t_0 * t_0))))) * -4.0;
	double tmp;
	if (fabs(y_45_scale) <= 1e-161) {
		tmp = t_1;
	} else if (fabs(y_45_scale) <= 2.9e+134) {
		tmp = (a * (((a * b) * (b / ((fabs(y_45_scale) * fabs(y_45_scale)) * x_45_scale))) / x_45_scale)) * -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 * (a * (b * (b / (t_0 * t_0))))) * (-4.0d0)
    if (abs(y_45scale) <= 1d-161) then
        tmp = t_1
    else if (abs(y_45scale) <= 2.9d+134) then
        tmp = (a * (((a * b) * (b / ((abs(y_45scale) * abs(y_45scale)) * x_45scale))) / x_45scale)) * (-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 * (a * (b * (b / (t_0 * t_0))))) * -4.0;
	double tmp;
	if (Math.abs(y_45_scale) <= 1e-161) {
		tmp = t_1;
	} else if (Math.abs(y_45_scale) <= 2.9e+134) {
		tmp = (a * (((a * b) * (b / ((Math.abs(y_45_scale) * Math.abs(y_45_scale)) * x_45_scale))) / x_45_scale)) * -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 * (a * (b * (b / (t_0 * t_0))))) * -4.0
	tmp = 0
	if math.fabs(y_45_scale) <= 1e-161:
		tmp = t_1
	elif math.fabs(y_45_scale) <= 2.9e+134:
		tmp = (a * (((a * b) * (b / ((math.fabs(y_45_scale) * math.fabs(y_45_scale)) * x_45_scale))) / x_45_scale)) * -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(a * Float64(a * Float64(b * Float64(b / Float64(t_0 * t_0))))) * -4.0)
	tmp = 0.0
	if (abs(y_45_scale) <= 1e-161)
		tmp = t_1;
	elseif (abs(y_45_scale) <= 2.9e+134)
		tmp = Float64(Float64(a * Float64(Float64(Float64(a * b) * Float64(b / Float64(Float64(abs(y_45_scale) * abs(y_45_scale)) * x_45_scale))) / x_45_scale)) * -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 * (a * (b * (b / (t_0 * t_0))))) * -4.0;
	tmp = 0.0;
	if (abs(y_45_scale) <= 1e-161)
		tmp = t_1;
	elseif (abs(y_45_scale) <= 2.9e+134)
		tmp = (a * (((a * b) * (b / ((abs(y_45_scale) * abs(y_45_scale)) * x_45_scale))) / x_45_scale)) * -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[(a * N[(a * N[(b * N[(b / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 1e-161], t$95$1, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 2.9e+134], N[(N[(a * N[(N[(N[(a * b), $MachinePrecision] * N[(b / N[(N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1e-161 or 2.9000000000000001e134 < y-scale

   1. Initial program 24.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.9%

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

   6. Applied rewrites 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 60.5%

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. swap-sqr N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 76.9%

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. swap-sqr N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*l* N/A

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

   8. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 76.9%

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

   10. Applied rewrites 76.9%

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

   if 1e-161 < y-scale < 2.9000000000000001e134

   1. Initial program 24.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.9%

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

   6. Applied rewrites 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 60.5%

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. swap-sqr N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 76.9%

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. swap-sqr N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*l* N/A

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

   8. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 72.9%

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

   10. Applied rewrites 72.9%

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

Alternative 7: 78.3% accurate, 26.4× speedup

Math

\[\begin{array}{l} t_0 := \left|y-scale\right| \cdot x-scale\\ t_1 := \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{t\_0 \cdot t\_0}\right)\right)\right) \cdot -4\\ \mathbf{if}\;\left|y-scale\right| \leq 2.55 \cdot 10^{-160}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\left|y-scale\right| \leq 8 \cdot 10^{+117}:\\ \;\;\;\;\left(-4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot a\right)\right)\right) \cdot \frac{1}{\left(\left(\left|y-scale\right| \cdot \left|y-scale\right|\right) \cdot x-scale\right) \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (fabs y-scale) x-scale))
       (t_1 (* (* a (* a (* b (/ b (* t_0 t_0))))) -4.0)))
  (if (<= (fabs y-scale) 2.55e-160)
    t_1
    (if (<= (fabs y-scale) 8e+117)
      (*
       (* -4.0 (* (* b a) (* b a)))
       (/
        1.0
        (* (* (* (fabs y-scale) (fabs y-scale)) x-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 * (a * (b * (b / (t_0 * t_0))))) * -4.0;
	double tmp;
	if (fabs(y_45_scale) <= 2.55e-160) {
		tmp = t_1;
	} else if (fabs(y_45_scale) <= 8e+117) {
		tmp = (-4.0 * ((b * a) * (b * a))) * (1.0 / (((fabs(y_45_scale) * fabs(y_45_scale)) * x_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 * (a * (b * (b / (t_0 * t_0))))) * (-4.0d0)
    if (abs(y_45scale) <= 2.55d-160) then
        tmp = t_1
    else if (abs(y_45scale) <= 8d+117) then
        tmp = ((-4.0d0) * ((b * a) * (b * a))) * (1.0d0 / (((abs(y_45scale) * abs(y_45scale)) * x_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 * (a * (b * (b / (t_0 * t_0))))) * -4.0;
	double tmp;
	if (Math.abs(y_45_scale) <= 2.55e-160) {
		tmp = t_1;
	} else if (Math.abs(y_45_scale) <= 8e+117) {
		tmp = (-4.0 * ((b * a) * (b * a))) * (1.0 / (((Math.abs(y_45_scale) * Math.abs(y_45_scale)) * x_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 * (a * (b * (b / (t_0 * t_0))))) * -4.0
	tmp = 0
	if math.fabs(y_45_scale) <= 2.55e-160:
		tmp = t_1
	elif math.fabs(y_45_scale) <= 8e+117:
		tmp = (-4.0 * ((b * a) * (b * a))) * (1.0 / (((math.fabs(y_45_scale) * math.fabs(y_45_scale)) * x_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(a * Float64(a * Float64(b * Float64(b / Float64(t_0 * t_0))))) * -4.0)
	tmp = 0.0
	if (abs(y_45_scale) <= 2.55e-160)
		tmp = t_1;
	elseif (abs(y_45_scale) <= 8e+117)
		tmp = Float64(Float64(-4.0 * Float64(Float64(b * a) * Float64(b * a))) * Float64(1.0 / Float64(Float64(Float64(abs(y_45_scale) * abs(y_45_scale)) * x_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 * (a * (b * (b / (t_0 * t_0))))) * -4.0;
	tmp = 0.0;
	if (abs(y_45_scale) <= 2.55e-160)
		tmp = t_1;
	elseif (abs(y_45_scale) <= 8e+117)
		tmp = (-4.0 * ((b * a) * (b * a))) * (1.0 / (((abs(y_45_scale) * abs(y_45_scale)) * x_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[(a * N[(a * N[(b * N[(b / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 2.55e-160], t$95$1, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 8e+117], N[(N[(-4.0 * N[(N[(b * a), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.55e-160 or 8.0000000000000004e117 < y-scale

   1. Initial program 24.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.9%

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

   6. Applied rewrites 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 60.5%

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. swap-sqr N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 76.9%

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. swap-sqr N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*l* N/A

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

   8. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 76.9%

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

   10. Applied rewrites 76.9%

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

   if 2.55e-160 < y-scale < 8.0000000000000004e117

   1. Initial program 24.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-prod-down N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. mult-flip N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 67.1%

      \[\leadsto \left(-4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot a\right)\right)\right) \cdot \color{blue}{\frac{1}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 78.2% accurate, 28.4× speedup

Math

\[\begin{array}{l} t_0 := \left|y-scale\right| \cdot x-scale\\ t_1 := \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{t\_0 \cdot t\_0}\right)\right)\right) \cdot -4\\ \mathbf{if}\;\left|y-scale\right| \leq 2.55 \cdot 10^{-160}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\left|y-scale\right| \leq 8 \cdot 10^{+117}:\\ \;\;\;\;\frac{-4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot a\right)\right)}{\left(\left(\left|y-scale\right| \cdot \left|y-scale\right|\right) \cdot x-scale\right) \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (fabs y-scale) x-scale))
       (t_1 (* (* a (* a (* b (/ b (* t_0 t_0))))) -4.0)))
  (if (<= (fabs y-scale) 2.55e-160)
    t_1
    (if (<= (fabs y-scale) 8e+117)
      (/
       (* -4.0 (* (* b a) (* b a)))
       (* (* (* (fabs y-scale) (fabs y-scale)) x-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 * (a * (b * (b / (t_0 * t_0))))) * -4.0;
	double tmp;
	if (fabs(y_45_scale) <= 2.55e-160) {
		tmp = t_1;
	} else if (fabs(y_45_scale) <= 8e+117) {
		tmp = (-4.0 * ((b * a) * (b * a))) / (((fabs(y_45_scale) * fabs(y_45_scale)) * x_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 * (a * (b * (b / (t_0 * t_0))))) * (-4.0d0)
    if (abs(y_45scale) <= 2.55d-160) then
        tmp = t_1
    else if (abs(y_45scale) <= 8d+117) then
        tmp = ((-4.0d0) * ((b * a) * (b * a))) / (((abs(y_45scale) * abs(y_45scale)) * x_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 * (a * (b * (b / (t_0 * t_0))))) * -4.0;
	double tmp;
	if (Math.abs(y_45_scale) <= 2.55e-160) {
		tmp = t_1;
	} else if (Math.abs(y_45_scale) <= 8e+117) {
		tmp = (-4.0 * ((b * a) * (b * a))) / (((Math.abs(y_45_scale) * Math.abs(y_45_scale)) * x_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 * (a * (b * (b / (t_0 * t_0))))) * -4.0
	tmp = 0
	if math.fabs(y_45_scale) <= 2.55e-160:
		tmp = t_1
	elif math.fabs(y_45_scale) <= 8e+117:
		tmp = (-4.0 * ((b * a) * (b * a))) / (((math.fabs(y_45_scale) * math.fabs(y_45_scale)) * x_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(a * Float64(a * Float64(b * Float64(b / Float64(t_0 * t_0))))) * -4.0)
	tmp = 0.0
	if (abs(y_45_scale) <= 2.55e-160)
		tmp = t_1;
	elseif (abs(y_45_scale) <= 8e+117)
		tmp = Float64(Float64(-4.0 * Float64(Float64(b * a) * Float64(b * a))) / Float64(Float64(Float64(abs(y_45_scale) * abs(y_45_scale)) * x_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 * (a * (b * (b / (t_0 * t_0))))) * -4.0;
	tmp = 0.0;
	if (abs(y_45_scale) <= 2.55e-160)
		tmp = t_1;
	elseif (abs(y_45_scale) <= 8e+117)
		tmp = (-4.0 * ((b * a) * (b * a))) / (((abs(y_45_scale) * abs(y_45_scale)) * x_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[(a * N[(a * N[(b * N[(b / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 2.55e-160], t$95$1, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 8e+117], N[(N[(-4.0 * N[(N[(b * a), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.55e-160 or 8.0000000000000004e117 < y-scale

   1. Initial program 24.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.9%

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

   6. Applied rewrites 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 60.5%

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. swap-sqr N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 76.9%

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. swap-sqr N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*l* N/A

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

   8. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 76.9%

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

   10. Applied rewrites 76.9%

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

   if 2.55e-160 < y-scale < 8.0000000000000004e117

   1. Initial program 24.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-prod-down N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   6. Applied rewrites 67.3%

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

Alternative 9: 76.9% accurate, 32.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|a\right| \leq 4.5 \cdot 10^{+219}:\\ \;\;\;\;\left(\left|a\right| \cdot \left(\left|a\right| \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\left|a\right| \cdot \frac{\left(\left|a\right| \cdot b\right) \cdot b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot -4\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (if (<= (fabs a) 4.5e+219)
  (*
   (*
    (fabs a)
    (*
     (fabs a)
     (* b (/ b (* (* y-scale x-scale) (* y-scale x-scale))))))
   -4.0)
  (*
   (*
    (fabs a)
    (/
     (* (* (fabs a) b) b)
     (* (* (* y-scale y-scale) x-scale) x-scale)))
   -4.0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (fabs(a) <= 4.5e+219) {
		tmp = (fabs(a) * (fabs(a) * (b * (b / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)))))) * -4.0;
	} else {
		tmp = (fabs(a) * (((fabs(a) * b) * b) / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale))) * -4.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (abs(a) <= 4.5d+219) then
        tmp = (abs(a) * (abs(a) * (b * (b / ((y_45scale * x_45scale) * (y_45scale * x_45scale)))))) * (-4.0d0)
    else
        tmp = (abs(a) * (((abs(a) * b) * b) / (((y_45scale * y_45scale) * x_45scale) * x_45scale))) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (Math.abs(a) <= 4.5e+219) {
		tmp = (Math.abs(a) * (Math.abs(a) * (b * (b / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)))))) * -4.0;
	} else {
		tmp = (Math.abs(a) * (((Math.abs(a) * b) * b) / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale))) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if math.fabs(a) <= 4.5e+219:
		tmp = (math.fabs(a) * (math.fabs(a) * (b * (b / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)))))) * -4.0
	else:
		tmp = (math.fabs(a) * (((math.fabs(a) * b) * b) / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale))) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (abs(a) <= 4.5e+219)
		tmp = Float64(Float64(abs(a) * Float64(abs(a) * Float64(b * Float64(b / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale)))))) * -4.0);
	else
		tmp = Float64(Float64(abs(a) * Float64(Float64(Float64(abs(a) * b) * b) / Float64(Float64(Float64(y_45_scale * y_45_scale) * x_45_scale) * x_45_scale))) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (abs(a) <= 4.5e+219)
		tmp = (abs(a) * (abs(a) * (b * (b / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)))))) * -4.0;
	else
		tmp = (abs(a) * (((abs(a) * b) * b) / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale))) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[N[Abs[a], $MachinePrecision], 4.5e+219], N[(N[(N[Abs[a], $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] * N[(b * N[(b / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(N[Abs[a], $MachinePrecision] * N[(N[(N[(N[Abs[a], $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] / N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.5000000000000002e219

   1. Initial program 24.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.9%

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

   6. Applied rewrites 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 60.5%

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. swap-sqr N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 76.9%

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. swap-sqr N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*l* N/A

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

   8. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 76.9%

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

   10. Applied rewrites 76.9%

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

   if 4.5000000000000002e219 < a

   1. Initial program 24.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.9%

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

   6. Applied rewrites 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 60.5%

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. swap-sqr N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 76.9%

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. swap-sqr N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*l* N/A

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

   8. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 66.2%

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

   10. Applied rewrites 66.2%

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

Alternative 10: 76.3% accurate, 32.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|a\right| \leq 4.2 \cdot 10^{+219}:\\ \;\;\;\;\left(\left|a\right| \cdot \left(\left|a\right| \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\left|a\right| \cdot \left(\left(\left|a\right| \cdot \frac{b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot b\right)\right) \cdot -4\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (if (<= (fabs a) 4.2e+219)
  (*
   (*
    (fabs a)
    (*
     (fabs a)
     (* b (/ b (* (* y-scale x-scale) (* y-scale x-scale))))))
   -4.0)
  (*
   (*
    (fabs a)
    (*
     (* (fabs a) (/ b (* (* (* y-scale y-scale) x-scale) x-scale)))
     b))
   -4.0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (fabs(a) <= 4.2e+219) {
		tmp = (fabs(a) * (fabs(a) * (b * (b / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)))))) * -4.0;
	} else {
		tmp = (fabs(a) * ((fabs(a) * (b / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale))) * b)) * -4.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (abs(a) <= 4.2d+219) then
        tmp = (abs(a) * (abs(a) * (b * (b / ((y_45scale * x_45scale) * (y_45scale * x_45scale)))))) * (-4.0d0)
    else
        tmp = (abs(a) * ((abs(a) * (b / (((y_45scale * y_45scale) * x_45scale) * x_45scale))) * b)) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (Math.abs(a) <= 4.2e+219) {
		tmp = (Math.abs(a) * (Math.abs(a) * (b * (b / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)))))) * -4.0;
	} else {
		tmp = (Math.abs(a) * ((Math.abs(a) * (b / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale))) * b)) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if math.fabs(a) <= 4.2e+219:
		tmp = (math.fabs(a) * (math.fabs(a) * (b * (b / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)))))) * -4.0
	else:
		tmp = (math.fabs(a) * ((math.fabs(a) * (b / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale))) * b)) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (abs(a) <= 4.2e+219)
		tmp = Float64(Float64(abs(a) * Float64(abs(a) * Float64(b * Float64(b / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale)))))) * -4.0);
	else
		tmp = Float64(Float64(abs(a) * Float64(Float64(abs(a) * Float64(b / Float64(Float64(Float64(y_45_scale * y_45_scale) * x_45_scale) * x_45_scale))) * b)) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (abs(a) <= 4.2e+219)
		tmp = (abs(a) * (abs(a) * (b * (b / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)))))) * -4.0;
	else
		tmp = (abs(a) * ((abs(a) * (b / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale))) * b)) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[N[Abs[a], $MachinePrecision], 4.2e+219], N[(N[(N[Abs[a], $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] * N[(b * N[(b / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(N[Abs[a], $MachinePrecision] * N[(N[(N[Abs[a], $MachinePrecision] * N[(b / N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.1999999999999998e219

   1. Initial program 24.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.9%

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

   6. Applied rewrites 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 60.5%

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. swap-sqr N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 76.9%

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. swap-sqr N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*l* N/A

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

   8. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 76.9%

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

   10. Applied rewrites 76.9%

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

   if 4.1999999999999998e219 < a

   1. Initial program 24.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.9%

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

   6. Applied rewrites 54.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 60.5%

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. swap-sqr N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 76.9%

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. swap-sqr N/A

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

      23. lift-*.f64 N/A

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

      24. associate-*l* N/A

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

   8. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right)\right)\right) \cdot -4 \]

      2. lift-*.f64 N/A

         \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right)\right)\right) \cdot -4 \]

      3. *-commutative N/A

         \[\leadsto \left(a \cdot \left(a \cdot \left(\frac{b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale} \cdot b\right)\right)\right) \cdot -4 \]

      4. associate-*r* N/A

         \[\leadsto \left(a \cdot \left(\left(a \cdot \frac{b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot b\right)\right) \cdot -4 \]

      5. lower-*.f64 N/A

         \[\leadsto \left(a \cdot \left(\left(a \cdot \frac{b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot b\right)\right) \cdot -4 \]

      6. lower-*.f64 70.2%

         \[\leadsto \left(a \cdot \left(\left(a \cdot \frac{b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot b\right)\right) \cdot -4 \]

   10. Applied rewrites 70.2%

       \[\leadsto \left(a \cdot \left(\left(a \cdot \frac{b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot b\right)\right) \cdot -4 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 76.2% accurate, 40.5× speedup

Math

\[\left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4 \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (*
 (* a (* a (* b (/ b (* (* y-scale x-scale) (* y-scale x-scale))))))
 -4.0))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (a * (a * (b * (b / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)))))) * -4.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (a * (a * (b * (b / ((y_45scale * x_45scale) * (y_45scale * x_45scale)))))) * (-4.0d0)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (a * (a * (b * (b / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)))))) * -4.0;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (a * (a * (b * (b / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)))))) * -4.0

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(a * Float64(a * Float64(b * Float64(b / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale)))))) * -4.0)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = (a * (a * (b * (b / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)))))) * -4.0;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(a * N[(a * N[(b * N[(b / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]

Derivation

1. Initial program 24.6%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -4 \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   2. lower-/.f64 N/A

      \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   3. lower-*.f64 N/A

      \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{\color{blue}{{x-scale}^{2}} \cdot {y-scale}^{2}} \]

   4. lower-pow.f64 N/A

      \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{\color{blue}{x-scale}}^{2} \cdot {y-scale}^{2}} \]

   5. lower-pow.f64 N/A

      \[\leadsto -4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{\color{blue}{2}} \cdot {y-scale}^{2}} \]

4. Applied rewrites 47.9%

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -4 \cdot \color{blue}{\frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   2. *-commutative N/A

      \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{-4} \]

   3. lower-*.f64 47.9%

      \[\leadsto \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{-4} \]

6. Applied rewrites 54.1%

   \[\leadsto \left(\left(a \cdot a\right) \cdot \frac{b \cdot b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot \color{blue}{-4} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(a \cdot a\right) \cdot \frac{b \cdot b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot -4 \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(a \cdot a\right) \cdot \frac{b \cdot b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot -4 \]

   3. associate-*l* N/A

      \[\leadsto \left(a \cdot \left(a \cdot \frac{b \cdot b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right)\right) \cdot -4 \]

   4. lower-*.f64 N/A

      \[\leadsto \left(a \cdot \left(a \cdot \frac{b \cdot b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right)\right) \cdot -4 \]

   5. lower-*.f64 60.5%

      \[\leadsto \left(a \cdot \left(a \cdot \frac{b \cdot b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right)\right) \cdot -4 \]

   6. lift-/.f64 N/A

      \[\leadsto \left(a \cdot \left(a \cdot \frac{b \cdot b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right)\right) \cdot -4 \]

   7. lift-*.f64 N/A

      \[\leadsto \left(a \cdot \left(a \cdot \frac{b \cdot b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right)\right) \cdot -4 \]

   8. associate-/l* N/A

      \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right)\right)\right) \cdot -4 \]

   9. lower-*.f64 N/A

      \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right)\right)\right) \cdot -4 \]

   10. lift-*.f64 N/A

       \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right)\right)\right) \cdot -4 \]

   11. lift-*.f64 N/A

       \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right)\right)\right) \cdot -4 \]

   12. associate-*l* N/A

       \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4 \]

   13. lift-*.f64 N/A

       \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4 \]

   14. swap-sqr N/A

       \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4 \]

   15. lift-*.f64 N/A

       \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4 \]

   16. lift-*.f64 N/A

       \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4 \]

   17. lift-*.f64 N/A

       \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4 \]

   18. lower-/.f64 76.9%

       \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4 \]

   19. lift-*.f64 N/A

       \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4 \]

   20. lift-*.f64 N/A

       \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4 \]

   21. lift-*.f64 N/A

       \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4 \]

   22. swap-sqr N/A

       \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4 \]

   23. lift-*.f64 N/A

       \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4 \]

   24. associate-*l* N/A

       \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right)\right)\right) \cdot -4 \]

8. Applied rewrites 68.2%

   \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right)\right)\right) \cdot -4 \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right)\right)\right) \cdot -4 \]

   2. lift-*.f64 N/A

      \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right)\right)\right) \cdot -4 \]

   3. associate-*l* N/A

      \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4 \]

   4. lift-*.f64 N/A

      \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4 \]

   5. unswap-sqr N/A

      \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4 \]

   6. lift-*.f64 N/A

      \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4 \]

   7. lift-*.f64 N/A

      \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4 \]

   8. lower-*.f64 76.9%

      \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4 \]

10. Applied rewrites 76.9%

    \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\right)\right)\right) \cdot -4 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(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))))