Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.5% → 93.5%

Time: 14.3s

Alternatives: 4

Speedup: 40.5×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.5% 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: 93.5% accurate, 35.9× 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 25.5%

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

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 77.8%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 77.8

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

6. Applied rewrites 77.8%

   \[\leadsto \frac{{\left(a \cdot b\right)}^{2}}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot -4 \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 77.8

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

8. Applied rewrites 77.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-*.f64 93.5

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

10. Applied rewrites 93.5%

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

Alternative 2: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{x-scale \cdot y-scale}\\ t_1 := \frac{angle}{180} \cdot \pi\\ t_2 := \sin t\_1\\ t_3 := \cos t\_1\\ t_4 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_2\right) \cdot t\_3}{x-scale}}{y-scale}\\ \mathbf{if}\;t\_4 \cdot t\_4 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_3\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_3\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{y-scale}}{y-scale} \leq \infty:\\ \;\;\;\;\left(-4 \cdot \left(t\_0 \cdot t\_0\right)\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ a (* x-scale y-scale)))
        (t_1 (* (/ angle 180.0) PI))
        (t_2 (sin t_1))
        (t_3 (cos t_1))
        (t_4
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_2) t_3) x-scale)
          y-scale)))
   (if (<=
        (-
         (* t_4 t_4)
         (*
          (*
           4.0
           (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_3) 2.0)) x-scale) x-scale))
          (/ (/ (+ (pow (* a t_3) 2.0) (pow (* b t_2) 2.0)) y-scale) y-scale)))
        INFINITY)
     (* (* -4.0 (* t_0 t_0)) (* b b))
     (*
      (/ (* (* a b) (* a b)) (* (* y-scale x-scale) (* y-scale x-scale)))
      -4.0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a / (x_45_scale * y_45_scale);
	double t_1 = (angle / 180.0) * ((double) M_PI);
	double t_2 = sin(t_1);
	double t_3 = cos(t_1);
	double t_4 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_2) * t_3) / x_45_scale) / y_45_scale;
	double tmp;
	if (((t_4 * t_4) - ((4.0 * (((pow((a * t_2), 2.0) + pow((b * t_3), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_3), 2.0) + pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale))) <= ((double) INFINITY)) {
		tmp = (-4.0 * (t_0 * t_0)) * (b * b);
	} else {
		tmp = (((a * b) * (a * b)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a / (x_45_scale * y_45_scale);
	double t_1 = (angle / 180.0) * Math.PI;
	double t_2 = Math.sin(t_1);
	double t_3 = Math.cos(t_1);
	double t_4 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_2) * t_3) / x_45_scale) / y_45_scale;
	double tmp;
	if (((t_4 * t_4) - ((4.0 * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_3), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_3), 2.0) + Math.pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale))) <= Double.POSITIVE_INFINITY) {
		tmp = (-4.0 * (t_0 * t_0)) * (b * b);
	} else {
		tmp = (((a * b) * (a * b)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = a / (x_45_scale * y_45_scale)
	t_1 = (angle / 180.0) * math.pi
	t_2 = math.sin(t_1)
	t_3 = math.cos(t_1)
	t_4 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_2) * t_3) / x_45_scale) / y_45_scale
	tmp = 0
	if ((t_4 * t_4) - ((4.0 * (((math.pow((a * t_2), 2.0) + math.pow((b * t_3), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_3), 2.0) + math.pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale))) <= math.inf:
		tmp = (-4.0 * (t_0 * t_0)) * (b * b)
	else:
		tmp = (((a * b) * (a * b)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a / Float64(x_45_scale * y_45_scale))
	t_1 = Float64(Float64(angle / 180.0) * pi)
	t_2 = sin(t_1)
	t_3 = cos(t_1)
	t_4 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_2) * t_3) / x_45_scale) / y_45_scale)
	tmp = 0.0
	if (Float64(Float64(t_4 * t_4) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_3) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_3) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / y_45_scale) / y_45_scale))) <= Inf)
		tmp = Float64(Float64(-4.0 * Float64(t_0 * t_0)) * Float64(b * b));
	else
		tmp = Float64(Float64(Float64(Float64(a * b) * Float64(a * b)) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = a / (x_45_scale * y_45_scale);
	t_1 = (angle / 180.0) * pi;
	t_2 = sin(t_1);
	t_3 = cos(t_1);
	t_4 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_2) * t_3) / x_45_scale) / y_45_scale;
	tmp = 0.0;
	if (((t_4 * t_4) - ((4.0 * (((((a * t_2) ^ 2.0) + ((b * t_3) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_3) ^ 2.0) + ((b * t_2) ^ 2.0)) / y_45_scale) / y_45_scale))) <= Inf)
		tmp = (-4.0 * (t_0 * t_0)) * (b * b);
	else
		tmp = (((a * b) * (a * b)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$4 * t$95$4), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$3), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale)) (*.f64 (*.f64 #s(literal 4 binary64) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale))) < +inf.0

   1. Initial program 68.2%

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

   2. Taylor expanded in b around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(-8 \cdot \frac{{a}^{2} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} - 4 \cdot \left(\frac{{a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)} \]

   4. Applied rewrites 78.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2}}{{\left(y-scale \cdot x-scale\right)}^{2}}, -8, -4 \cdot \frac{\mathsf{fma}\left({\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}, a \cdot a, {\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4} \cdot \left(a \cdot a\right)\right)}{{\left(y-scale \cdot x-scale\right)}^{2}}\right) \cdot \left(b \cdot b\right)} \]

   5. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 80.0

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

   7. Applied rewrites 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 80.0

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

   9. Applied rewrites 80.0%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. times-frac N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lift-*.f64 88.2

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

   11. Applied rewrites 88.2%

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

   if +inf.0 < (-.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale)) (*.f64 (*.f64 #s(literal 4 binary64) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)))

   1. Initial program 0.0%

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

   2. Taylor expanded in angle around 0

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

      1. *-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 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 76.8

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

   6. Applied rewrites 76.8%

      \[\leadsto \frac{{\left(a \cdot b\right)}^{2}}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot -4 \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 76.8

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

   8. Applied rewrites 76.8%

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

Alternative 3: 77.8% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.5%

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

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 77.8%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 77.8

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

6. Applied rewrites 77.8%

   \[\leadsto \frac{{\left(a \cdot b\right)}^{2}}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot -4 \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 77.8

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

8. Applied rewrites 77.8%

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

Alternative 4: 61.4% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.5%

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

2. Taylor expanded in b around 0

   \[\leadsto \color{blue}{{b}^{2} \cdot \left(-8 \cdot \frac{{a}^{2} \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} - 4 \cdot \left(\frac{{a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right)} \]

4. Applied rewrites 54.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}^{2}}{{\left(y-scale \cdot x-scale\right)}^{2}}, -8, -4 \cdot \frac{\mathsf{fma}\left({\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}, a \cdot a, {\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4} \cdot \left(a \cdot a\right)\right)}{{\left(y-scale \cdot x-scale\right)}^{2}}\right) \cdot \left(b \cdot b\right)} \]

5. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. pow-prod-down N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-*.f64 61.4

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

7. Applied rewrites 61.4%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 61.4

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

9. Applied rewrites 61.4%

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

Reproduce

herbie shell --seed 2025095 
(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))))