Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.9% → 92.7%

Time: 13.9s

Alternatives: 7

Speedup: 21.0×

• 67.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.7% accurate, 20.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 47.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-prod-down N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 64.4%

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

6. Applied rewrites 83.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. frac-times N/A

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. times-frac N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

8. Applied rewrites 91.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. lift-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 92.7%

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

10. Applied rewrites 92.7%

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

Alternative 2: 91.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \frac{b}{y-scale \cdot x-scale}\\ t_3 := \cos t\_0\\ t_4 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_3}{x-scale}}{y-scale}\\ \mathbf{if}\;t\_4 \cdot t\_4 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_3\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_3\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \leq 0:\\ \;\;\;\;-4 \cdot \left(\left(\left(\frac{a}{y-scale \cdot x-scale} \cdot a\right) \cdot b\right) \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \left(\left(a \cdot t\_2\right) \cdot \frac{-4}{y-scale \cdot x-scale}\right)\\ \end{array} \]

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 (/ b (* y-scale x-scale)))
       (t_3 (cos t_0))
       (t_4
        (/
         (/
          (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_3)
          x-scale)
         y-scale)))
  (if (<=
       (-
        (* t_4 t_4)
        (*
         (*
          4.0
          (/
           (/ (+ (pow (* a t_1) 2.0) (pow (* b t_3) 2.0)) x-scale)
           x-scale))
         (/
          (/ (+ (pow (* a t_3) 2.0) (pow (* b t_1) 2.0)) y-scale)
          y-scale)))
       0.0)
    (* -4.0 (* (* (* (/ a (* y-scale x-scale)) a) b) t_2))
    (* (* a b) (* (* a t_2) (/ -4.0 (* y-scale x-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 = b / (y_45_scale * x_45_scale);
	double t_3 = cos(t_0);
	double t_4 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_3) / x_45_scale) / y_45_scale;
	double tmp;
	if (((t_4 * t_4) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_3), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_3), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))) <= 0.0) {
		tmp = -4.0 * ((((a / (y_45_scale * x_45_scale)) * a) * b) * t_2);
	} else {
		tmp = (a * b) * ((a * t_2) * (-4.0 / (y_45_scale * x_45_scale)));
	}
	return tmp;
}

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 = b / (y_45_scale * x_45_scale);
	double t_3 = Math.cos(t_0);
	double t_4 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_3) / x_45_scale) / y_45_scale;
	double tmp;
	if (((t_4 * t_4) - ((4.0 * (((Math.pow((a * t_1), 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_1), 2.0)) / y_45_scale) / y_45_scale))) <= 0.0) {
		tmp = -4.0 * ((((a / (y_45_scale * x_45_scale)) * a) * b) * t_2);
	} else {
		tmp = (a * b) * ((a * t_2) * (-4.0 / (y_45_scale * x_45_scale)));
	}
	return tmp;
}

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 = b / (y_45_scale * x_45_scale)
	t_3 = math.cos(t_0)
	t_4 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_3) / x_45_scale) / y_45_scale
	tmp = 0
	if ((t_4 * t_4) - ((4.0 * (((math.pow((a * t_1), 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_1), 2.0)) / y_45_scale) / y_45_scale))) <= 0.0:
		tmp = -4.0 * ((((a / (y_45_scale * x_45_scale)) * a) * b) * t_2)
	else:
		tmp = (a * b) * ((a * t_2) * (-4.0 / (y_45_scale * x_45_scale)))
	return tmp

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 = Float64(b / Float64(y_45_scale * x_45_scale))
	t_3 = cos(t_0)
	t_4 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * 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_1) ^ 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_1) ^ 2.0)) / y_45_scale) / y_45_scale))) <= 0.0)
		tmp = Float64(-4.0 * Float64(Float64(Float64(Float64(a / Float64(y_45_scale * x_45_scale)) * a) * b) * t_2));
	else
		tmp = Float64(Float64(a * b) * Float64(Float64(a * t_2) * Float64(-4.0 / Float64(y_45_scale * x_45_scale))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = b / (y_45_scale * x_45_scale);
	t_3 = cos(t_0);
	t_4 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_3) / x_45_scale) / y_45_scale;
	tmp = 0.0;
	if (((t_4 * t_4) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_3) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_3) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale))) <= 0.0)
		tmp = -4.0 * ((((a / (y_45_scale * x_45_scale)) * a) * b) * t_2);
	else
		tmp = (a * b) * ((a * t_2) * (-4.0 / (y_45_scale * x_45_scale)));
	end
	tmp_2 = tmp;
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[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$0], $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$1), $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$1), $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$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(-4.0 * N[(N[(N[(N[(a / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(a * b), $MachinePrecision] * N[(N[(a * t$95$2), $MachinePrecision] * N[(-4.0 / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < 0.0

   1. Initial program 24.9%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-prod-down N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 60.9%

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. associate-*r* N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 58.9%

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

   6. Applied rewrites 58.9%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-pow.f64 58.9%

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 67.1%

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 67.1%

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

   8. Applied rewrites 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. frac-times N/A

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

      14. lift-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-/l* N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

   10. Applied rewrites 84.6%

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

   if 0.0 < (-.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 24.9%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-prod-down N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.4%

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

   6. Applied rewrites 83.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 90.5%

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

   8. Applied rewrites 90.5%

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

Alternative 3: 84.7% accurate, 14.2× speedup

Math

\[\begin{array}{l} t_0 := b \cdot \left|a\right|\\ t_1 := \frac{\left(t\_0 \cdot t\_0\right) \cdot -4}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\\ \mathbf{if}\;\left|a\right| \leq 7 \cdot 10^{-195}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\left|a\right| \leq 2.7 \cdot 10^{+160}:\\ \;\;\;\;-4 \cdot \left(\left(\left(\frac{\left|a\right|}{y-scale \cdot x-scale} \cdot \left|a\right|\right) \cdot b\right) \cdot \frac{b}{y-scale \cdot x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* b (fabs a)))
       (t_1
        (/
         (* (* t_0 t_0) -4.0)
         (* (* (* y-scale x-scale) y-scale) x-scale))))
  (if (<= (fabs a) 7e-195)
    t_1
    (if (<= (fabs a) 2.7e+160)
      (*
       -4.0
       (*
        (* (* (/ (fabs a) (* y-scale x-scale)) (fabs a)) b)
        (/ b (* y-scale x-scale))))
      t_1))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b * fabs(a);
	double t_1 = ((t_0 * t_0) * -4.0) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale);
	double tmp;
	if (fabs(a) <= 7e-195) {
		tmp = t_1;
	} else if (fabs(a) <= 2.7e+160) {
		tmp = -4.0 * ((((fabs(a) / (y_45_scale * x_45_scale)) * fabs(a)) * b) * (b / (y_45_scale * x_45_scale)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = b * abs(a)
    t_1 = ((t_0 * t_0) * (-4.0d0)) / (((y_45scale * x_45scale) * y_45scale) * x_45scale)
    if (abs(a) <= 7d-195) then
        tmp = t_1
    else if (abs(a) <= 2.7d+160) then
        tmp = (-4.0d0) * ((((abs(a) / (y_45scale * x_45scale)) * abs(a)) * b) * (b / (y_45scale * x_45scale)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b * Math.abs(a);
	double t_1 = ((t_0 * t_0) * -4.0) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale);
	double tmp;
	if (Math.abs(a) <= 7e-195) {
		tmp = t_1;
	} else if (Math.abs(a) <= 2.7e+160) {
		tmp = -4.0 * ((((Math.abs(a) / (y_45_scale * x_45_scale)) * Math.abs(a)) * b) * (b / (y_45_scale * x_45_scale)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = b * math.fabs(a)
	t_1 = ((t_0 * t_0) * -4.0) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)
	tmp = 0
	if math.fabs(a) <= 7e-195:
		tmp = t_1
	elif math.fabs(a) <= 2.7e+160:
		tmp = -4.0 * ((((math.fabs(a) / (y_45_scale * x_45_scale)) * math.fabs(a)) * b) * (b / (y_45_scale * x_45_scale)))
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b * abs(a))
	t_1 = Float64(Float64(Float64(t_0 * t_0) * -4.0) / Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))
	tmp = 0.0
	if (abs(a) <= 7e-195)
		tmp = t_1;
	elseif (abs(a) <= 2.7e+160)
		tmp = Float64(-4.0 * Float64(Float64(Float64(Float64(abs(a) / Float64(y_45_scale * x_45_scale)) * abs(a)) * b) * Float64(b / Float64(y_45_scale * x_45_scale))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = b * abs(a);
	t_1 = ((t_0 * t_0) * -4.0) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale);
	tmp = 0.0;
	if (abs(a) <= 7e-195)
		tmp = t_1;
	elseif (abs(a) <= 2.7e+160)
		tmp = -4.0 * ((((abs(a) / (y_45_scale * x_45_scale)) * abs(a)) * b) * (b / (y_45_scale * x_45_scale)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * -4.0), $MachinePrecision] / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 7e-195], t$95$1, If[LessEqual[N[Abs[a], $MachinePrecision], 2.7e+160], N[(-4.0 * N[(N[(N[(N[(N[Abs[a], $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < 7.0000000000000003e-195 or 2.7e160 < a

   1. Initial program 24.9%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-prod-down N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 74.6%

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

   if 7.0000000000000003e-195 < a < 2.7e160

   1. Initial program 24.9%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-prod-down N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 60.9%

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. associate-*r* N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 58.9%

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

   6. Applied rewrites 58.9%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-pow.f64 58.9%

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 67.1%

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 67.1%

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

   8. Applied rewrites 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. frac-times N/A

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

      14. lift-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-/l* N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

   10. Applied rewrites 84.6%

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

Alternative 4: 84.1% accurate, 14.2× speedup

Math

\[\begin{array}{l} t_0 := b \cdot \left|a\right|\\ t_1 := \frac{\left(t\_0 \cdot t\_0\right) \cdot -4}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\\ \mathbf{if}\;\left|a\right| \leq 7 \cdot 10^{-195}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\left|a\right| \leq 2.7 \cdot 10^{+160}:\\ \;\;\;\;-4 \cdot \left(b \cdot \left(\frac{b}{y-scale \cdot x-scale} \cdot \left(\frac{\left|a\right|}{y-scale \cdot x-scale} \cdot \left|a\right|\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* b (fabs a)))
       (t_1
        (/
         (* (* t_0 t_0) -4.0)
         (* (* (* y-scale x-scale) y-scale) x-scale))))
  (if (<= (fabs a) 7e-195)
    t_1
    (if (<= (fabs a) 2.7e+160)
      (*
       -4.0
       (*
        b
        (*
         (/ b (* y-scale x-scale))
         (* (/ (fabs a) (* y-scale x-scale)) (fabs a)))))
      t_1))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b * fabs(a);
	double t_1 = ((t_0 * t_0) * -4.0) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale);
	double tmp;
	if (fabs(a) <= 7e-195) {
		tmp = t_1;
	} else if (fabs(a) <= 2.7e+160) {
		tmp = -4.0 * (b * ((b / (y_45_scale * x_45_scale)) * ((fabs(a) / (y_45_scale * x_45_scale)) * fabs(a))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = b * abs(a)
    t_1 = ((t_0 * t_0) * (-4.0d0)) / (((y_45scale * x_45scale) * y_45scale) * x_45scale)
    if (abs(a) <= 7d-195) then
        tmp = t_1
    else if (abs(a) <= 2.7d+160) then
        tmp = (-4.0d0) * (b * ((b / (y_45scale * x_45scale)) * ((abs(a) / (y_45scale * x_45scale)) * abs(a))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b * Math.abs(a);
	double t_1 = ((t_0 * t_0) * -4.0) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale);
	double tmp;
	if (Math.abs(a) <= 7e-195) {
		tmp = t_1;
	} else if (Math.abs(a) <= 2.7e+160) {
		tmp = -4.0 * (b * ((b / (y_45_scale * x_45_scale)) * ((Math.abs(a) / (y_45_scale * x_45_scale)) * Math.abs(a))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = b * math.fabs(a)
	t_1 = ((t_0 * t_0) * -4.0) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)
	tmp = 0
	if math.fabs(a) <= 7e-195:
		tmp = t_1
	elif math.fabs(a) <= 2.7e+160:
		tmp = -4.0 * (b * ((b / (y_45_scale * x_45_scale)) * ((math.fabs(a) / (y_45_scale * x_45_scale)) * math.fabs(a))))
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b * abs(a))
	t_1 = Float64(Float64(Float64(t_0 * t_0) * -4.0) / Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))
	tmp = 0.0
	if (abs(a) <= 7e-195)
		tmp = t_1;
	elseif (abs(a) <= 2.7e+160)
		tmp = Float64(-4.0 * Float64(b * Float64(Float64(b / Float64(y_45_scale * x_45_scale)) * Float64(Float64(abs(a) / Float64(y_45_scale * x_45_scale)) * abs(a)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = b * abs(a);
	t_1 = ((t_0 * t_0) * -4.0) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale);
	tmp = 0.0;
	if (abs(a) <= 7e-195)
		tmp = t_1;
	elseif (abs(a) <= 2.7e+160)
		tmp = -4.0 * (b * ((b / (y_45_scale * x_45_scale)) * ((abs(a) / (y_45_scale * x_45_scale)) * abs(a))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * -4.0), $MachinePrecision] / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 7e-195], t$95$1, If[LessEqual[N[Abs[a], $MachinePrecision], 2.7e+160], N[(-4.0 * N[(b * N[(N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[a], $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < 7.0000000000000003e-195 or 2.7e160 < a

   1. Initial program 24.9%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-prod-down N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 74.6%

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

   if 7.0000000000000003e-195 < a < 2.7e160

   1. Initial program 24.9%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-prod-down N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 60.9%

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. associate-*r* N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 58.9%

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

   6. Applied rewrites 58.9%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-pow.f64 58.9%

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 67.1%

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 67.1%

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

   8. Applied rewrites 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower-*.f64 75.6%

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 84.5%

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

   10. Applied rewrites 84.5%

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

Alternative 5: 83.5% accurate, 20.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 47.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-pow.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. pow-prod-down N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. pow2 N/A

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

   18. lift-*.f64 N/A

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

   19. lower-/.f64 60.9%

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

   20. lift-*.f64 N/A

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

   21. lift-*.f64 N/A

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

   22. associate-*r* N/A

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

   23. lower-*.f64 N/A

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

   24. lower-*.f64 58.9%

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

6. Applied rewrites 58.9%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-pow.f64 58.9%

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 67.1%

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

   13. lift-pow.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 67.1%

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

8. Applied rewrites 67.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. frac-times N/A

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

   14. lift-*.f64 N/A

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

   15. associate-/l* N/A

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

   16. lift-/.f64 N/A

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

   17. associate-*l* N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 N/A

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

10. Applied rewrites 84.7%

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

Alternative 6: 78.0% accurate, 21.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 47.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-pow.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. pow-prod-down N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. pow2 N/A

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

   18. lift-*.f64 N/A

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

   19. lower-/.f64 60.9%

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

   20. lift-*.f64 N/A

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

   21. lift-*.f64 N/A

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

   22. associate-*r* N/A

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

   23. lower-*.f64 N/A

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

   24. lower-*.f64 58.9%

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

6. Applied rewrites 58.9%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-pow.f64 58.9%

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 67.1%

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

   13. lift-pow.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 67.1%

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

8. Applied rewrites 67.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 78.0%

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

10. Applied rewrites 78.0%

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

Alternative 7: 69.5% accurate, 21.0× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 47.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-pow.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. pow-prod-down N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. pow2 N/A

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

   18. lift-*.f64 N/A

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

   19. lower-/.f64 60.9%

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

   20. lift-*.f64 N/A

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

   21. lift-*.f64 N/A

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

   22. associate-*r* N/A

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

   23. lower-*.f64 N/A

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

   24. lower-*.f64 58.9%

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

6. Applied rewrites 58.9%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-pow.f64 58.9%

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 67.1%

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

   13. lift-pow.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 67.1%

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

8. Applied rewrites 67.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lower-*.f64 69.5%

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

10. Applied rewrites 69.5%

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

Reproduce

herbie shell --seed 2025212 
(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))))