Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.6% → 93.9%

Time: 12.5s

Alternatives: 6

Speedup: 20.4×

• 33.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.9% accurate, 20.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 48.1%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 47.7

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

7. Applied rewrites 47.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. unswap-sqr N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 60.8

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. unswap-sqr N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 78.3

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

9. Applied rewrites 78.3%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. times-frac N/A

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

    9. lower-*.f64 N/A

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

    10. lower-/.f64 N/A

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

    11. lift-*.f64 N/A

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

    12. *-commutative N/A

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

    13. lower-*.f64 N/A

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

    14. lower-/.f64 N/A

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

    15. lift-*.f64 N/A

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

    16. *-commutative N/A

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

    17. lower-*.f64 93.9

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

11. Applied rewrites 93.9%

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

Alternative 2: 79.3% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)} \cdot -4\\ \mathbf{if}\;x-scale \leq 9.5 \cdot 10^{-147}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x-scale \leq 7.8 \cdot 10^{+146}:\\ \;\;\;\;\left(\left(a \cdot b\right) \cdot \frac{a \cdot b}{\left(\left(x-scale \cdot x-scale\right) \cdot y-scale\right) \cdot y-scale}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (*
          (/ (* (* a b) (* a b)) (* (* x-scale y-scale) (* x-scale y-scale)))
          -4.0)))
   (if (<= x-scale 9.5e-147)
     t_0
     (if (<= x-scale 7.8e+146)
       (*
        (* (* a b) (/ (* a b) (* (* (* x-scale x-scale) y-scale) y-scale)))
        -4.0)
       t_0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((a * b) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	double tmp;
	if (x_45_scale <= 9.5e-147) {
		tmp = t_0;
	} else if (x_45_scale <= 7.8e+146) {
		tmp = ((a * b) * ((a * b) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale))) * -4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((a * b) * (a * b)) / ((x_45scale * y_45scale) * (x_45scale * y_45scale))) * (-4.0d0)
    if (x_45scale <= 9.5d-147) then
        tmp = t_0
    else if (x_45scale <= 7.8d+146) then
        tmp = ((a * b) * ((a * b) / (((x_45scale * x_45scale) * y_45scale) * y_45scale))) * (-4.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((a * b) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	double tmp;
	if (x_45_scale <= 9.5e-147) {
		tmp = t_0;
	} else if (x_45_scale <= 7.8e+146) {
		tmp = ((a * b) * ((a * b) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale))) * -4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (((a * b) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0
	tmp = 0
	if x_45_scale <= 9.5e-147:
		tmp = t_0
	elif x_45_scale <= 7.8e+146:
		tmp = ((a * b) * ((a * b) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale))) * -4.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(Float64(a * b) * Float64(a * b)) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale))) * -4.0)
	tmp = 0.0
	if (x_45_scale <= 9.5e-147)
		tmp = t_0;
	elseif (x_45_scale <= 7.8e+146)
		tmp = Float64(Float64(Float64(a * b) * Float64(Float64(a * b) / Float64(Float64(Float64(x_45_scale * x_45_scale) * y_45_scale) * y_45_scale))) * -4.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (((a * b) * (a * b)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	tmp = 0.0;
	if (x_45_scale <= 9.5e-147)
		tmp = t_0;
	elseif (x_45_scale <= 7.8e+146)
		tmp = ((a * b) * ((a * b) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale))) * -4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(N[(a * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]}, If[LessEqual[x$45$scale, 9.5e-147], t$95$0, If[LessEqual[x$45$scale, 7.8e+146], N[(N[(N[(a * b), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] / N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 9.49999999999999986e-147 or 7.8e146 < x-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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 48.1%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 47.7

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

   7. Applied rewrites 47.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unswap-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 60.8

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 78.3

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

   9. Applied rewrites 78.3%

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

   if 9.49999999999999986e-147 < x-scale < 7.8e146

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 48.1%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 47.7

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

   7. Applied rewrites 47.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unswap-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 60.8

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 78.3

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

   9. Applied rewrites 78.3%

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

   11. Applied rewrites 71.7%

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

Alternative 3: 77.3% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x-scale \leq 2.05 \cdot 10^{-147}:\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{x-scale \cdot \left(y-scale \cdot \left(y-scale \cdot x-scale\right)\right)} \cdot -4\\ \mathbf{elif}\;x-scale \leq 7.8 \cdot 10^{+146}:\\ \;\;\;\;\left(\left(a \cdot b\right) \cdot \frac{a \cdot b}{\left(\left(x-scale \cdot x-scale\right) \cdot y-scale\right) \cdot y-scale}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(b \cdot \left(a \cdot b\right)\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)} \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= x-scale 2.05e-147)
   (* (/ (* (* a b) (* a b)) (* x-scale (* y-scale (* y-scale x-scale)))) -4.0)
   (if (<= x-scale 7.8e+146)
     (*
      (* (* a b) (/ (* a b) (* (* (* x-scale x-scale) y-scale) y-scale)))
      -4.0)
     (*
      (/ (* a (* b (* a b))) (* (* x-scale y-scale) (* x-scale y-scale)))
      -4.0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (x_45_scale <= 2.05e-147) {
		tmp = (((a * b) * (a * b)) / (x_45_scale * (y_45_scale * (y_45_scale * x_45_scale)))) * -4.0;
	} else if (x_45_scale <= 7.8e+146) {
		tmp = ((a * b) * ((a * b) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale))) * -4.0;
	} else {
		tmp = ((a * (b * (a * b))) / ((x_45_scale * y_45_scale) * (x_45_scale * y_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 (x_45scale <= 2.05d-147) then
        tmp = (((a * b) * (a * b)) / (x_45scale * (y_45scale * (y_45scale * x_45scale)))) * (-4.0d0)
    else if (x_45scale <= 7.8d+146) then
        tmp = ((a * b) * ((a * b) / (((x_45scale * x_45scale) * y_45scale) * y_45scale))) * (-4.0d0)
    else
        tmp = ((a * (b * (a * b))) / ((x_45scale * y_45scale) * (x_45scale * y_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 (x_45_scale <= 2.05e-147) {
		tmp = (((a * b) * (a * b)) / (x_45_scale * (y_45_scale * (y_45_scale * x_45_scale)))) * -4.0;
	} else if (x_45_scale <= 7.8e+146) {
		tmp = ((a * b) * ((a * b) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale))) * -4.0;
	} else {
		tmp = ((a * (b * (a * b))) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if x_45_scale <= 2.05e-147:
		tmp = (((a * b) * (a * b)) / (x_45_scale * (y_45_scale * (y_45_scale * x_45_scale)))) * -4.0
	elif x_45_scale <= 7.8e+146:
		tmp = ((a * b) * ((a * b) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale))) * -4.0
	else:
		tmp = ((a * (b * (a * b))) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (x_45_scale <= 2.05e-147)
		tmp = Float64(Float64(Float64(Float64(a * b) * Float64(a * b)) / Float64(x_45_scale * Float64(y_45_scale * Float64(y_45_scale * x_45_scale)))) * -4.0);
	elseif (x_45_scale <= 7.8e+146)
		tmp = Float64(Float64(Float64(a * b) * Float64(Float64(a * b) / Float64(Float64(Float64(x_45_scale * x_45_scale) * y_45_scale) * y_45_scale))) * -4.0);
	else
		tmp = Float64(Float64(Float64(a * Float64(b * Float64(a * b))) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_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 (x_45_scale <= 2.05e-147)
		tmp = (((a * b) * (a * b)) / (x_45_scale * (y_45_scale * (y_45_scale * x_45_scale)))) * -4.0;
	elseif (x_45_scale <= 7.8e+146)
		tmp = ((a * b) * ((a * b) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale))) * -4.0;
	else
		tmp = ((a * (b * (a * b))) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[x$45$scale, 2.05e-147], N[(N[(N[(N[(a * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(y$45$scale * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], If[LessEqual[x$45$scale, 7.8e+146], N[(N[(N[(a * b), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] / N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(N[(a * N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < 2.05e-147

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 48.1%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 47.7

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

   7. Applied rewrites 47.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unswap-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 60.8

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 78.3

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

   9. Applied rewrites 78.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 75.6

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

   11. Applied rewrites 75.6%

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

   if 2.05e-147 < x-scale < 7.8e146

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 48.1%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 47.7

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

   7. Applied rewrites 47.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unswap-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 60.8

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 78.3

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

   9. Applied rewrites 78.3%

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

   11. Applied rewrites 71.7%

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

   if 7.8e146 < x-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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 48.1%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 47.7

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

   7. Applied rewrites 47.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unswap-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 60.8

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 78.3

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

   9. Applied rewrites 78.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 75.6

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

   11. Applied rewrites 75.6%

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

Alternative 4: 77.3% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot \left(b \cdot \left(a \cdot b\right)\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)} \cdot -4\\ \mathbf{if}\;x-scale \leq 9 \cdot 10^{-160}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x-scale \leq 7.8 \cdot 10^{+146}:\\ \;\;\;\;\left(\left(a \cdot b\right) \cdot \frac{a \cdot b}{\left(\left(x-scale \cdot x-scale\right) \cdot y-scale\right) \cdot y-scale}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (*
          (/ (* a (* b (* a b))) (* (* x-scale y-scale) (* x-scale y-scale)))
          -4.0)))
   (if (<= x-scale 9e-160)
     t_0
     (if (<= x-scale 7.8e+146)
       (*
        (* (* a b) (/ (* a b) (* (* (* x-scale x-scale) y-scale) y-scale)))
        -4.0)
       t_0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((a * (b * (a * b))) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	double tmp;
	if (x_45_scale <= 9e-160) {
		tmp = t_0;
	} else if (x_45_scale <= 7.8e+146) {
		tmp = ((a * b) * ((a * b) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale))) * -4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((a * (b * (a * b))) / ((x_45scale * y_45scale) * (x_45scale * y_45scale))) * (-4.0d0)
    if (x_45scale <= 9d-160) then
        tmp = t_0
    else if (x_45scale <= 7.8d+146) then
        tmp = ((a * b) * ((a * b) / (((x_45scale * x_45scale) * y_45scale) * y_45scale))) * (-4.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((a * (b * (a * b))) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	double tmp;
	if (x_45_scale <= 9e-160) {
		tmp = t_0;
	} else if (x_45_scale <= 7.8e+146) {
		tmp = ((a * b) * ((a * b) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale))) * -4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = ((a * (b * (a * b))) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0
	tmp = 0
	if x_45_scale <= 9e-160:
		tmp = t_0
	elif x_45_scale <= 7.8e+146:
		tmp = ((a * b) * ((a * b) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale))) * -4.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(a * Float64(b * Float64(a * b))) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale))) * -4.0)
	tmp = 0.0
	if (x_45_scale <= 9e-160)
		tmp = t_0;
	elseif (x_45_scale <= 7.8e+146)
		tmp = Float64(Float64(Float64(a * b) * Float64(Float64(a * b) / Float64(Float64(Float64(x_45_scale * x_45_scale) * y_45_scale) * y_45_scale))) * -4.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = ((a * (b * (a * b))) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * -4.0;
	tmp = 0.0;
	if (x_45_scale <= 9e-160)
		tmp = t_0;
	elseif (x_45_scale <= 7.8e+146)
		tmp = ((a * b) * ((a * b) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale))) * -4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(a * N[(b * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]}, If[LessEqual[x$45$scale, 9e-160], t$95$0, If[LessEqual[x$45$scale, 7.8e+146], N[(N[(N[(a * b), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] / N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 9.00000000000000053e-160 or 7.8e146 < x-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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 48.1%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 47.7

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

   7. Applied rewrites 47.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unswap-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 60.8

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 78.3

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

   9. Applied rewrites 78.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-*.f64 75.6

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

   11. Applied rewrites 75.6%

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

   if 9.00000000000000053e-160 < x-scale < 7.8e146

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 48.1%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 47.7

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

   7. Applied rewrites 47.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unswap-sqr N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 60.8

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 78.3

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

   9. Applied rewrites 78.3%

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

   11. Applied rewrites 71.7%

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

Alternative 5: 71.7% accurate, 20.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 48.1%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 47.7

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

7. Applied rewrites 47.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. unswap-sqr N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 60.8

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. unswap-sqr N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 78.3

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

9. Applied rewrites 78.3%

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

11. Applied rewrites 71.7%

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

Alternative 6: 60.4% accurate, 20.4× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (* a (* a (/ (* b b) (* (* (* x-scale x-scale) y-scale) y-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) / (((x_45_scale * x_45_scale) * y_45_scale) * y_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) / (((x_45scale * x_45scale) * y_45scale) * y_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) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)))) * -4.0;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 48.1%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 47.7

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

7. Applied rewrites 47.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. unswap-sqr N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 60.8

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. unswap-sqr N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 78.3

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

9. Applied rewrites 78.3%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. pow2 N/A

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

    6. pow-prod-down N/A

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

    7. lift-*.f64 N/A

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

    8. lift-*.f64 N/A

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

    9. *-commutative N/A

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

    10. lift-*.f64 N/A

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

    11. *-commutative N/A

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

    12. unswap-sqr N/A

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

    13. lift-*.f64 N/A

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

    14. lift-*.f64 N/A

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

    15. lift-*.f64 N/A

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

    16. associate-*r/ N/A

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

    17. pow2 N/A

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

    18. pow2 N/A

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

    19. lift-*.f64 N/A

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

    20. lift-*.f64 N/A

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

    21. lift-*.f64 N/A

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

    22. associate-*l* N/A

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

    23. lower-*.f64 N/A

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

11. Applied rewrites 60.4%

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

Reproduce

herbie shell --seed 2025138 
(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))))