Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.1% → 93.8%

Time: 17.2s

Alternatives: 4

Speedup: 40.5×

• 34.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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)))))

Initial Program: 24.1% accurate, 1.0× speedup

Math

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

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)))))

Alternative 1: 93.8% 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 21.9%

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

3. Taylor expanded in angle around 0

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

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

5. Applied rewrites 77.4%

   \[\leadsto \color{blue}{\frac{{\left(a \cdot b\right)}^{2}}{{\left(y-scale \cdot x-scale\right)}^{2}} \cdot -4} \]
6. 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 77.4

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

7. Applied rewrites 77.4%

   \[\leadsto \frac{{\left(a \cdot b\right)}^{2}}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot -4 \]
8. 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 77.4

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

9. Applied rewrites 77.4%

   \[\leadsto \frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot -4 \]
10. 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 94.8

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

11. Applied rewrites 94.8%

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

Alternative 2: 80.0% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{x-scale \cdot y-scale}\\ \mathbf{if}\;a \leq 2.5 \cdot 10^{-127} \lor \neg \left(a \leq 1.02 \cdot 10^{+129}\right):\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot \left(t\_0 \cdot t\_0\right)\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* x-scale y-scale))))
   (if (or (<= a 2.5e-127) (not (<= a 1.02e+129)))
     (*
      (/ (* (* a b) (* a b)) (* (* y-scale x-scale) (* y-scale x-scale)))
      -4.0)
     (* (* -4.0 (* t_0 t_0)) (* a a)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (x_45_scale * y_45_scale);
	double tmp;
	if ((a <= 2.5e-127) || !(a <= 1.02e+129)) {
		tmp = (((a * b) * (a * b)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0;
	} else {
		tmp = (-4.0 * (t_0 * t_0)) * (a * a);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b / (x_45scale * y_45scale)
    if ((a <= 2.5d-127) .or. (.not. (a <= 1.02d+129))) then
        tmp = (((a * b) * (a * b)) / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * (-4.0d0)
    else
        tmp = ((-4.0d0) * (t_0 * t_0)) * (a * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (x_45_scale * y_45_scale);
	double tmp;
	if ((a <= 2.5e-127) || !(a <= 1.02e+129)) {
		tmp = (((a * b) * (a * b)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0;
	} else {
		tmp = (-4.0 * (t_0 * t_0)) * (a * a);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = b / (x_45_scale * y_45_scale)
	tmp = 0
	if (a <= 2.5e-127) or not (a <= 1.02e+129):
		tmp = (((a * b) * (a * b)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0
	else:
		tmp = (-4.0 * (t_0 * t_0)) * (a * a)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(x_45_scale * y_45_scale))
	tmp = 0.0
	if ((a <= 2.5e-127) || !(a <= 1.02e+129))
		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);
	else
		tmp = Float64(Float64(-4.0 * Float64(t_0 * t_0)) * Float64(a * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = b / (x_45_scale * y_45_scale);
	tmp = 0.0;
	if ((a <= 2.5e-127) || ~((a <= 1.02e+129)))
		tmp = (((a * b) * (a * b)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0;
	else
		tmp = (-4.0 * (t_0 * t_0)) * (a * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[a, 2.5e-127], N[Not[LessEqual[a, 1.02e+129]], $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], N[(N[(-4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.4999999999999999e-127 or 1.01999999999999996e129 < a

   1. Initial program 20.6%

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

   3. Taylor expanded in angle around 0

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

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

   5. Applied rewrites 76.9%

      \[\leadsto \color{blue}{\frac{{\left(a \cdot b\right)}^{2}}{{\left(y-scale \cdot x-scale\right)}^{2}} \cdot -4} \]
   6. 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 76.9

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

   7. Applied rewrites 76.9%

      \[\leadsto \frac{{\left(a \cdot b\right)}^{2}}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot -4 \]
   8. 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 76.9

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

   9. Applied rewrites 76.9%

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

   if 2.4999999999999999e-127 < a < 1.01999999999999996e129

   1. Initial program 27.5%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 77.3

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

   8. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 77.3

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

   10. Applied rewrites 77.3%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. times-frac N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lift-*.f64 95.8

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

   12. Applied rewrites 95.8%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.5 \cdot 10^{-127} \lor \neg \left(a \leq 1.02 \cdot 10^{+129}\right):\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(-4 \cdot \left(\frac{b}{x-scale \cdot y-scale} \cdot \frac{b}{x-scale \cdot y-scale}\right)\right) \cdot \left(a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.3% 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 21.9%

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

3. Taylor expanded in angle around 0

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

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

5. Applied rewrites 77.4%

   \[\leadsto \color{blue}{\frac{{\left(a \cdot b\right)}^{2}}{{\left(y-scale \cdot x-scale\right)}^{2}} \cdot -4} \]
6. 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 77.4

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

7. Applied rewrites 77.4%

   \[\leadsto \frac{{\left(a \cdot b\right)}^{2}}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot -4 \]
8. 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 77.4

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

9. Applied rewrites 77.4%

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

Alternative 4: 61.3% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.9%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 49.3%

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

6. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. pow-prod-down N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-*.f64 56.4

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

8. Applied rewrites 56.4%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 56.4

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

10. Applied rewrites 56.4%

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

Reproduce

herbie shell --seed 2025051 
(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))))