Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.0% → 93.8%

Time: 13.8s

Alternatives: 6

Speedup: 20.4×

• 33.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.8% accurate, 20.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.0%

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

2. Taylor expanded in angle around 0

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

   1. 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 48.3%

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
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 65.0

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-pow.f64 N/A

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

   18. pow2 N/A

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

   19. lift-pow.f64 N/A

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

   20. pow2 N/A

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

   21. unswap-sqr N/A

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

   22. lower-*.f64 N/A

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

   23. lower-*.f64 N/A

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

   24. lower-*.f64 83.6

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

6. Applied rewrites 83.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 91.4

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 91.4

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

8. Applied rewrites 91.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l/ N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. lift-/.f64 N/A

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

   14. unswap-sqr N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 N/A

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

10. Applied rewrites 93.8%

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

Alternative 2: 88.7% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{y-scale \cdot x-scale}\\ \mathbf{if}\;b \leq 1.32 \cdot 10^{+197}:\\ \;\;\;\;\left(\frac{-4}{y-scale \cdot x-scale} \cdot a\right) \cdot \left(\left(a \cdot b\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(t\_0 \cdot t\_0\right)\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* y-scale x-scale))))
   (if (<= b 1.32e+197)
     (* (* (/ -4.0 (* y-scale x-scale)) a) (* (* a b) t_0))
     (* (* a (* a (* t_0 t_0))) -4.0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (y_45_scale * x_45_scale);
	double tmp;
	if (b <= 1.32e+197) {
		tmp = ((-4.0 / (y_45_scale * x_45_scale)) * a) * ((a * b) * t_0);
	} else {
		tmp = (a * (a * (t_0 * t_0))) * -4.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b / (y_45scale * x_45scale)
    if (b <= 1.32d+197) then
        tmp = (((-4.0d0) / (y_45scale * x_45scale)) * a) * ((a * b) * t_0)
    else
        tmp = (a * (a * (t_0 * t_0))) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (y_45_scale * x_45_scale);
	double tmp;
	if (b <= 1.32e+197) {
		tmp = ((-4.0 / (y_45_scale * x_45_scale)) * a) * ((a * b) * t_0);
	} else {
		tmp = (a * (a * (t_0 * t_0))) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = b / (y_45_scale * x_45_scale)
	tmp = 0
	if b <= 1.32e+197:
		tmp = ((-4.0 / (y_45_scale * x_45_scale)) * a) * ((a * b) * t_0)
	else:
		tmp = (a * (a * (t_0 * t_0))) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(y_45_scale * x_45_scale))
	tmp = 0.0
	if (b <= 1.32e+197)
		tmp = Float64(Float64(Float64(-4.0 / Float64(y_45_scale * x_45_scale)) * a) * Float64(Float64(a * b) * t_0));
	else
		tmp = Float64(Float64(a * Float64(a * Float64(t_0 * t_0))) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = b / (y_45_scale * x_45_scale);
	tmp = 0.0;
	if (b <= 1.32e+197)
		tmp = ((-4.0 / (y_45_scale * x_45_scale)) * a) * ((a * b) * t_0);
	else
		tmp = (a * (a * (t_0 * t_0))) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.32e+197], N[(N[(N[(-4.0 / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(a * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.3200000000000001e197

   1. Initial program 25.0%

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

   2. Taylor expanded in angle around 0

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

      1. 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 48.3%

      \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
   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 65.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-pow.f64 N/A

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

      18. pow2 N/A

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

      19. lift-pow.f64 N/A

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

      20. pow2 N/A

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

      21. unswap-sqr N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 83.6

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

   6. Applied rewrites 83.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 91.4

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 91.4

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

   8. Applied rewrites 91.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*r/ N/A

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

      15. lift-*.f64 N/A

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

      16. associate-/l* N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*r* N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

   10. Applied rewrites 88.5%

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

   if 1.3200000000000001e197 < b

   1. Initial program 25.0%

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

   2. Taylor expanded in angle around 0

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

      1. 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 48.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 48.3

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

   6. Applied rewrites 60.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 66.9

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 74.3

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 84.8

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

   10. Applied rewrites 84.8%

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

Alternative 3: 84.8% accurate, 20.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.0%

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

2. Taylor expanded in angle around 0

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

   1. 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 48.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 48.3

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

6. Applied rewrites 60.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 66.9

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 74.3

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

8. Applied rewrites 74.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. frac-times N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 84.8

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

10. Applied rewrites 84.8%

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

Alternative 4: 78.5% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y-scale \leq 1.72 \cdot 10^{+189}:\\ \;\;\;\;\left(\left(a \cdot b\right) \cdot \left(\frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale} \cdot a\right)\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}\right)\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= y-scale 1.72e+189)
   (* (* (* a b) (* (/ b (* (* (* y-scale x-scale) y-scale) x-scale)) a)) -4.0)
   (*
    (* a (* a (* b (/ b (* (* (* y-scale x-scale) x-scale) y-scale)))))
    -4.0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 1.72e+189) {
		tmp = ((a * b) * ((b / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * a)) * -4.0;
	} else {
		tmp = (a * (a * (b * (b / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))))) * -4.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (y_45scale <= 1.72d+189) then
        tmp = ((a * b) * ((b / (((y_45scale * x_45scale) * y_45scale) * x_45scale)) * a)) * (-4.0d0)
    else
        tmp = (a * (a * (b * (b / (((y_45scale * x_45scale) * x_45scale) * y_45scale))))) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 1.72e+189) {
		tmp = ((a * b) * ((b / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * a)) * -4.0;
	} else {
		tmp = (a * (a * (b * (b / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))))) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if y_45_scale <= 1.72e+189:
		tmp = ((a * b) * ((b / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * a)) * -4.0
	else:
		tmp = (a * (a * (b * (b / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))))) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (y_45_scale <= 1.72e+189)
		tmp = Float64(Float64(Float64(a * b) * Float64(Float64(b / Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * a)) * -4.0);
	else
		tmp = Float64(Float64(a * Float64(a * Float64(b * Float64(b / Float64(Float64(Float64(y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))))) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (y_45_scale <= 1.72e+189)
		tmp = ((a * b) * ((b / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * a)) * -4.0;
	else
		tmp = (a * (a * (b * (b / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))))) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[y$45$scale, 1.72e+189], N[(N[(N[(a * b), $MachinePrecision] * N[(N[(b / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(a * N[(a * N[(b * N[(b / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.71999999999999994e189

   1. Initial program 25.0%

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

   2. Taylor expanded in angle around 0

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

      1. 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 48.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 48.3

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

   6. Applied rewrites 60.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 66.9

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 74.3

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 78.0

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

   10. Applied rewrites 78.0%

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

   if 1.71999999999999994e189 < y-scale

   1. Initial program 25.0%

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

   2. Taylor expanded in angle around 0

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

      1. 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 48.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 48.3

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

   6. Applied rewrites 60.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 66.9

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 74.3

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 74.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.2

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

   10. Applied rewrites 74.2%

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

Alternative 5: 78.0% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y-scale \leq 1.82 \cdot 10^{+189}:\\ \;\;\;\;\left(\left(\left(a \cdot \frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\right) \cdot b\right) \cdot a\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}\right)\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= y-scale 1.82e+189)
   (* (* (* (* a (/ b (* (* (* y-scale x-scale) y-scale) x-scale))) b) a) -4.0)
   (*
    (* a (* a (* b (/ b (* (* (* y-scale x-scale) x-scale) y-scale)))))
    -4.0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 1.82e+189) {
		tmp = (((a * (b / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))) * b) * a) * -4.0;
	} else {
		tmp = (a * (a * (b * (b / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))))) * -4.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (y_45scale <= 1.82d+189) then
        tmp = (((a * (b / (((y_45scale * x_45scale) * y_45scale) * x_45scale))) * b) * a) * (-4.0d0)
    else
        tmp = (a * (a * (b * (b / (((y_45scale * x_45scale) * x_45scale) * y_45scale))))) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 1.82e+189) {
		tmp = (((a * (b / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))) * b) * a) * -4.0;
	} else {
		tmp = (a * (a * (b * (b / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))))) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if y_45_scale <= 1.82e+189:
		tmp = (((a * (b / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))) * b) * a) * -4.0
	else:
		tmp = (a * (a * (b * (b / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))))) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (y_45_scale <= 1.82e+189)
		tmp = Float64(Float64(Float64(Float64(a * Float64(b / Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))) * b) * a) * -4.0);
	else
		tmp = Float64(Float64(a * Float64(a * Float64(b * Float64(b / Float64(Float64(Float64(y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))))) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (y_45_scale <= 1.82e+189)
		tmp = (((a * (b / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))) * b) * a) * -4.0;
	else
		tmp = (a * (a * (b * (b / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))))) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[y$45$scale, 1.82e+189], N[(N[(N[(N[(a * N[(b / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(a * N[(a * N[(b * N[(b / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.82000000000000007e189

   1. Initial program 25.0%

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

   2. Taylor expanded in angle around 0

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

      1. 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 48.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 48.3

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

   6. Applied rewrites 60.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 66.9

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 74.3

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 74.3

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 77.5

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

   10. Applied rewrites 77.5%

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

   if 1.82000000000000007e189 < y-scale

   1. Initial program 25.0%

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

   2. Taylor expanded in angle around 0

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

      1. 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 48.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 48.3

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

   6. Applied rewrites 60.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 66.9

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 74.3

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 74.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.2

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

   10. Applied rewrites 74.2%

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

Alternative 6: 77.5% accurate, 20.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.0%

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

2. Taylor expanded in angle around 0

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

   1. 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 48.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 48.3

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

6. Applied rewrites 60.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 66.9

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 74.3

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

8. Applied rewrites 74.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 74.3

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 77.5

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

10. Applied rewrites 77.5%

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

Reproduce

herbie shell --seed 2025159 
(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))))