Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.4% → 94.2%

Time: 16.6s

Alternatives: 11

Speedup: 20.4×

• 66.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 25.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.2% accurate, 20.1× speedup

Math

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

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(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)) * 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[(N[(t$95$0 * t$95$0), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Initial program 25.4%

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

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 59.7%

   \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot \frac{b \cdot b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\right) \cdot -4} \]
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 67.2%

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

   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 74.7%

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

8. Applied rewrites 74.7%

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

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

   4. unpow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. unpow2 N/A

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

   9. associate-*r/ N/A

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

   10. pow-prod-down N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

   17. lift-*.f64 N/A

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

   18. times-frac N/A

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

10. Applied rewrites 94.2%

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

Alternative 2: 88.9% accurate, 15.6× speedup

Math

\[\begin{array}{l} t_0 := a \cdot \left|b\right|\\ \mathbf{if}\;\left|b\right| \leq 4.4 \cdot 10^{-226}:\\ \;\;\;\;\left(t\_0 \cdot \frac{t\_0}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{-4}{y-scale \cdot x-scale} \cdot a\right) \cdot \left|b\right|\right) \cdot \frac{\left|b\right|}{y-scale \cdot x-scale}\right) \cdot a\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* a (fabs b))))
   (if (<= (fabs b) 4.4e-226)
     (* (* t_0 (/ t_0 (* (* (* y-scale x-scale) y-scale) x-scale))) -4.0)
     (*
      (*
       (* (* (/ -4.0 (* y-scale x-scale)) a) (fabs b))
       (/ (fabs b) (* y-scale x-scale)))
      a))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * fabs(b);
	double tmp;
	if (fabs(b) <= 4.4e-226) {
		tmp = (t_0 * (t_0 / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))) * -4.0;
	} else {
		tmp = ((((-4.0 / (y_45_scale * x_45_scale)) * a) * fabs(b)) * (fabs(b) / (y_45_scale * x_45_scale))) * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * abs(b)
    if (abs(b) <= 4.4d-226) then
        tmp = (t_0 * (t_0 / (((y_45scale * x_45scale) * y_45scale) * x_45scale))) * (-4.0d0)
    else
        tmp = (((((-4.0d0) / (y_45scale * x_45scale)) * a) * abs(b)) * (abs(b) / (y_45scale * x_45scale))) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * Math.abs(b);
	double tmp;
	if (Math.abs(b) <= 4.4e-226) {
		tmp = (t_0 * (t_0 / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))) * -4.0;
	} else {
		tmp = ((((-4.0 / (y_45_scale * x_45_scale)) * a) * Math.abs(b)) * (Math.abs(b) / (y_45_scale * x_45_scale))) * a;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = a * math.fabs(b)
	tmp = 0
	if math.fabs(b) <= 4.4e-226:
		tmp = (t_0 * (t_0 / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))) * -4.0
	else:
		tmp = ((((-4.0 / (y_45_scale * x_45_scale)) * a) * math.fabs(b)) * (math.fabs(b) / (y_45_scale * x_45_scale))) * a
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a * abs(b))
	tmp = 0.0
	if (abs(b) <= 4.4e-226)
		tmp = Float64(Float64(t_0 * Float64(t_0 / Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))) * -4.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-4.0 / Float64(y_45_scale * x_45_scale)) * a) * abs(b)) * Float64(abs(b) / Float64(y_45_scale * x_45_scale))) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = a * abs(b);
	tmp = 0.0;
	if (abs(b) <= 4.4e-226)
		tmp = (t_0 * (t_0 / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))) * -4.0;
	else
		tmp = ((((-4.0 / (y_45_scale * x_45_scale)) * a) * abs(b)) * (abs(b) / (y_45_scale * x_45_scale))) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 4.4e-226], N[(N[(t$95$0 * N[(t$95$0 / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(N[(N[(N[(-4.0 / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.4e-226

   1. Initial program 25.4%

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

   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 59.7%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot \frac{b \cdot b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\right) \cdot -4} \]
   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 67.2%

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

      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 74.7%

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

   8. Applied rewrites 74.7%

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

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

      4. unpow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-*r/ N/A

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

      10. pow-prod-down N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 80.7%

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 80.7%

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

   10. Applied rewrites 80.7%

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

   if 4.4e-226 < b

   1. Initial program 25.4%

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

   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 64.8%

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

   6. Applied rewrites 83.8%

      \[\leadsto \frac{-4}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}} \]
   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. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 89.9%

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

   8. Applied rewrites 89.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   10. Applied rewrites 89.6%

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

Alternative 3: 84.2% accurate, 13.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{b}{y-scale \cdot x-scale}\\ t_1 := b \cdot \left|a\right|\\ \mathbf{if}\;\left|a\right| \leq 6 \cdot 10^{-141}:\\ \;\;\;\;\frac{-4 \cdot \left(t\_1 \cdot t\_1\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\\ \mathbf{elif}\;\left|a\right| \leq 5.6 \cdot 10^{+136}:\\ \;\;\;\;\left(\left(\left|a\right| \cdot \left|a\right|\right) \cdot \left(t\_0 \cdot t\_0\right)\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\left|a\right| \cdot b\right) \cdot \left(\left(\frac{\left|a\right|}{y-scale} \cdot b\right) \cdot \frac{-4}{\left(y-scale \cdot x-scale\right) \cdot x-scale}\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* y-scale x-scale))) (t_1 (* b (fabs a))))
   (if (<= (fabs a) 6e-141)
     (/ (* -4.0 (* t_1 t_1)) (* (* y-scale x-scale) (* y-scale x-scale)))
     (if (<= (fabs a) 5.6e+136)
       (* (* (* (fabs a) (fabs a)) (* t_0 t_0)) -4.0)
       (*
        (* (fabs a) b)
        (*
         (* (/ (fabs a) y-scale) b)
         (/ -4.0 (* (* y-scale x-scale) x-scale))))))))

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 t_1 = b * fabs(a);
	double tmp;
	if (fabs(a) <= 6e-141) {
		tmp = (-4.0 * (t_1 * t_1)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale));
	} else if (fabs(a) <= 5.6e+136) {
		tmp = ((fabs(a) * fabs(a)) * (t_0 * t_0)) * -4.0;
	} else {
		tmp = (fabs(a) * b) * (((fabs(a) / y_45_scale) * b) * (-4.0 / ((y_45_scale * x_45_scale) * x_45_scale)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = b / (y_45scale * x_45scale)
    t_1 = b * abs(a)
    if (abs(a) <= 6d-141) then
        tmp = ((-4.0d0) * (t_1 * t_1)) / ((y_45scale * x_45scale) * (y_45scale * x_45scale))
    else if (abs(a) <= 5.6d+136) then
        tmp = ((abs(a) * abs(a)) * (t_0 * t_0)) * (-4.0d0)
    else
        tmp = (abs(a) * b) * (((abs(a) / y_45scale) * b) * ((-4.0d0) / ((y_45scale * x_45scale) * x_45scale)))
    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 t_1 = b * Math.abs(a);
	double tmp;
	if (Math.abs(a) <= 6e-141) {
		tmp = (-4.0 * (t_1 * t_1)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale));
	} else if (Math.abs(a) <= 5.6e+136) {
		tmp = ((Math.abs(a) * Math.abs(a)) * (t_0 * t_0)) * -4.0;
	} else {
		tmp = (Math.abs(a) * b) * (((Math.abs(a) / y_45_scale) * b) * (-4.0 / ((y_45_scale * x_45_scale) * x_45_scale)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = b / (y_45_scale * x_45_scale)
	t_1 = b * math.fabs(a)
	tmp = 0
	if math.fabs(a) <= 6e-141:
		tmp = (-4.0 * (t_1 * t_1)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))
	elif math.fabs(a) <= 5.6e+136:
		tmp = ((math.fabs(a) * math.fabs(a)) * (t_0 * t_0)) * -4.0
	else:
		tmp = (math.fabs(a) * b) * (((math.fabs(a) / y_45_scale) * b) * (-4.0 / ((y_45_scale * x_45_scale) * x_45_scale)))
	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))
	t_1 = Float64(b * abs(a))
	tmp = 0.0
	if (abs(a) <= 6e-141)
		tmp = Float64(Float64(-4.0 * Float64(t_1 * t_1)) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale)));
	elseif (abs(a) <= 5.6e+136)
		tmp = Float64(Float64(Float64(abs(a) * abs(a)) * Float64(t_0 * t_0)) * -4.0);
	else
		tmp = Float64(Float64(abs(a) * b) * Float64(Float64(Float64(abs(a) / y_45_scale) * b) * Float64(-4.0 / Float64(Float64(y_45_scale * x_45_scale) * x_45_scale))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = b / (y_45_scale * x_45_scale);
	t_1 = b * abs(a);
	tmp = 0.0;
	if (abs(a) <= 6e-141)
		tmp = (-4.0 * (t_1 * t_1)) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale));
	elseif (abs(a) <= 5.6e+136)
		tmp = ((abs(a) * abs(a)) * (t_0 * t_0)) * -4.0;
	else
		tmp = (abs(a) * b) * (((abs(a) / y_45_scale) * b) * (-4.0 / ((y_45_scale * x_45_scale) * x_45_scale)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 6e-141], N[(N[(-4.0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[a], $MachinePrecision], 5.6e+136], N[(N[(N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(N[Abs[a], $MachinePrecision] * b), $MachinePrecision] * N[(N[(N[(N[Abs[a], $MachinePrecision] / y$45$scale), $MachinePrecision] * b), $MachinePrecision] * N[(-4.0 / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 5.99999999999999967e-141

   1. Initial program 25.4%

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

   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. lift-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   6. Applied rewrites 75.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 78.1%

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

   8. Applied rewrites 78.1%

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

   if 5.99999999999999967e-141 < a < 5.6000000000000004e136

   1. Initial program 25.4%

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

   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 59.7%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot \frac{b \cdot b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\right) \cdot -4} \]
   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. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 75.0%

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

   8. Applied rewrites 75.0%

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

   if 5.6000000000000004e136 < a

   1. Initial program 25.4%

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

   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 64.8%

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

   6. Applied rewrites 83.8%

      \[\leadsto \frac{-4}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}} \]
   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. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 89.9%

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

   8. Applied rewrites 89.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

   10. Applied rewrites 80.8%

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

Alternative 4: 82.5% accurate, 15.6× speedup

Math

\[\begin{array}{l} t_0 := y-scale \cdot \left|x-scale\right|\\ \mathbf{if}\;\left|x-scale\right| \leq 7.2 \cdot 10^{-209}:\\ \;\;\;\;\left(\left(a \cdot b\right) \cdot \frac{a \cdot b}{\left(t\_0 \cdot y-scale\right) \cdot \left|x-scale\right|}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot b\right) \cdot \left(\left(\frac{a}{y-scale} \cdot b\right) \cdot \frac{-4}{t\_0 \cdot \left|x-scale\right|}\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* y-scale (fabs x-scale))))
   (if (<= (fabs x-scale) 7.2e-209)
     (* (* (* a b) (/ (* a b) (* (* t_0 y-scale) (fabs x-scale)))) -4.0)
     (* (* a b) (* (* (/ a y-scale) b) (/ -4.0 (* t_0 (fabs x-scale))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = y_45_scale * fabs(x_45_scale);
	double tmp;
	if (fabs(x_45_scale) <= 7.2e-209) {
		tmp = ((a * b) * ((a * b) / ((t_0 * y_45_scale) * fabs(x_45_scale)))) * -4.0;
	} else {
		tmp = (a * b) * (((a / y_45_scale) * b) * (-4.0 / (t_0 * fabs(x_45_scale))));
	}
	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 = y_45scale * abs(x_45scale)
    if (abs(x_45scale) <= 7.2d-209) then
        tmp = ((a * b) * ((a * b) / ((t_0 * y_45scale) * abs(x_45scale)))) * (-4.0d0)
    else
        tmp = (a * b) * (((a / y_45scale) * b) * ((-4.0d0) / (t_0 * abs(x_45scale))))
    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 = y_45_scale * Math.abs(x_45_scale);
	double tmp;
	if (Math.abs(x_45_scale) <= 7.2e-209) {
		tmp = ((a * b) * ((a * b) / ((t_0 * y_45_scale) * Math.abs(x_45_scale)))) * -4.0;
	} else {
		tmp = (a * b) * (((a / y_45_scale) * b) * (-4.0 / (t_0 * Math.abs(x_45_scale))));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = y_45_scale * math.fabs(x_45_scale)
	tmp = 0
	if math.fabs(x_45_scale) <= 7.2e-209:
		tmp = ((a * b) * ((a * b) / ((t_0 * y_45_scale) * math.fabs(x_45_scale)))) * -4.0
	else:
		tmp = (a * b) * (((a / y_45_scale) * b) * (-4.0 / (t_0 * math.fabs(x_45_scale))))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(y_45_scale * abs(x_45_scale))
	tmp = 0.0
	if (abs(x_45_scale) <= 7.2e-209)
		tmp = Float64(Float64(Float64(a * b) * Float64(Float64(a * b) / Float64(Float64(t_0 * y_45_scale) * abs(x_45_scale)))) * -4.0);
	else
		tmp = Float64(Float64(a * b) * Float64(Float64(Float64(a / y_45_scale) * b) * Float64(-4.0 / Float64(t_0 * abs(x_45_scale)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = y_45_scale * abs(x_45_scale);
	tmp = 0.0;
	if (abs(x_45_scale) <= 7.2e-209)
		tmp = ((a * b) * ((a * b) / ((t_0 * y_45_scale) * abs(x_45_scale)))) * -4.0;
	else
		tmp = (a * b) * (((a / y_45_scale) * b) * (-4.0 / (t_0 * abs(x_45_scale))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(y$45$scale * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 7.2e-209], N[(N[(N[(a * b), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] / N[(N[(t$95$0 * y$45$scale), $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(a * b), $MachinePrecision] * N[(N[(N[(a / y$45$scale), $MachinePrecision] * b), $MachinePrecision] * N[(-4.0 / N[(t$95$0 * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 7.20000000000000032e-209

   1. Initial program 25.4%

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

   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 59.7%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot \frac{b \cdot b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\right) \cdot -4} \]
   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 67.2%

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

      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 74.7%

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

   8. Applied rewrites 74.7%

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

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

      4. unpow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-*r/ N/A

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

      10. pow-prod-down N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 80.7%

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 80.7%

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

   10. Applied rewrites 80.7%

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

   if 7.20000000000000032e-209 < x-scale

   1. Initial program 25.4%

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

   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 64.8%

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

   6. Applied rewrites 83.8%

      \[\leadsto \frac{-4}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{y-scale \cdot x-scale}} \]
   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. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 89.9%

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

   8. Applied rewrites 89.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

   10. Applied rewrites 80.8%

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

Alternative 5: 81.2% accurate, 15.8× speedup

Math

\[\begin{array}{l} t_0 := \left|y-scale\right| \cdot x-scale\\ \mathbf{if}\;\left|y-scale\right| \leq 5 \cdot 10^{+229}:\\ \;\;\;\;\left(\left(a \cdot b\right) \cdot \frac{a \cdot b}{\left(t\_0 \cdot \left|y-scale\right|\right) \cdot x-scale}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot a\right)\right)}{t\_0 \cdot t\_0}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (fabs y-scale) x-scale)))
   (if (<= (fabs y-scale) 5e+229)
     (* (* (* a b) (/ (* a b) (* (* t_0 (fabs y-scale)) x-scale))) -4.0)
     (/ (* -4.0 (* (* b a) (* b a))) (* t_0 t_0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(y_45_scale) * x_45_scale;
	double tmp;
	if (fabs(y_45_scale) <= 5e+229) {
		tmp = ((a * b) * ((a * b) / ((t_0 * fabs(y_45_scale)) * x_45_scale))) * -4.0;
	} else {
		tmp = (-4.0 * ((b * a) * (b * a))) / (t_0 * t_0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(y_45scale) * x_45scale
    if (abs(y_45scale) <= 5d+229) then
        tmp = ((a * b) * ((a * b) / ((t_0 * abs(y_45scale)) * x_45scale))) * (-4.0d0)
    else
        tmp = ((-4.0d0) * ((b * a) * (b * a))) / (t_0 * t_0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.abs(y_45_scale) * x_45_scale;
	double tmp;
	if (Math.abs(y_45_scale) <= 5e+229) {
		tmp = ((a * b) * ((a * b) / ((t_0 * Math.abs(y_45_scale)) * x_45_scale))) * -4.0;
	} else {
		tmp = (-4.0 * ((b * a) * (b * a))) / (t_0 * t_0);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.fabs(y_45_scale) * x_45_scale
	tmp = 0
	if math.fabs(y_45_scale) <= 5e+229:
		tmp = ((a * b) * ((a * b) / ((t_0 * math.fabs(y_45_scale)) * x_45_scale))) * -4.0
	else:
		tmp = (-4.0 * ((b * a) * (b * a))) / (t_0 * t_0)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(y_45_scale) * x_45_scale)
	tmp = 0.0
	if (abs(y_45_scale) <= 5e+229)
		tmp = Float64(Float64(Float64(a * b) * Float64(Float64(a * b) / Float64(Float64(t_0 * abs(y_45_scale)) * x_45_scale))) * -4.0);
	else
		tmp = Float64(Float64(-4.0 * Float64(Float64(b * a) * Float64(b * a))) / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(y_45_scale) * x_45_scale;
	tmp = 0.0;
	if (abs(y_45_scale) <= 5e+229)
		tmp = ((a * b) * ((a * b) / ((t_0 * abs(y_45_scale)) * x_45_scale))) * -4.0;
	else
		tmp = (-4.0 * ((b * a) * (b * a))) / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[y$45$scale], $MachinePrecision] * x$45$scale), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 5e+229], N[(N[(N[(a * b), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] / N[(N[(t$95$0 * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(-4.0 * N[(N[(b * a), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 5.0000000000000005e229

   1. Initial program 25.4%

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

   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 59.7%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot \frac{b \cdot b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\right) \cdot -4} \]
   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 67.2%

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

      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 74.7%

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

   8. Applied rewrites 74.7%

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

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

      4. unpow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-*r/ N/A

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

      10. pow-prod-down N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 80.7%

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 80.7%

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

   10. Applied rewrites 80.7%

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

   if 5.0000000000000005e229 < y-scale

   1. Initial program 25.4%

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

   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. lift-*.f64 N/A

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

      12. lower-/.f64 N/A

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

   6. Applied rewrites 75.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 78.1%

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

   8. Applied rewrites 78.1%

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

Alternative 6: 80.9% accurate, 15.8× speedup

Math

\[\begin{array}{l} t_0 := \left|y-scale\right| \cdot x-scale\\ \mathbf{if}\;\left|y-scale\right| \leq 2.7 \cdot 10^{+221}:\\ \;\;\;\;\left(\left(a \cdot b\right) \cdot \frac{a \cdot b}{\left(t\_0 \cdot \left|y-scale\right|\right) \cdot x-scale}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\left(\left(a \cdot b\right) \cdot b\right) \cdot a}{\left|y-scale\right| \cdot \left(t\_0 \cdot x-scale\right)}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (fabs y-scale) x-scale)))
   (if (<= (fabs y-scale) 2.7e+221)
     (* (* (* a b) (/ (* a b) (* (* t_0 (fabs y-scale)) x-scale))) -4.0)
     (* -4.0 (/ (* (* (* a b) b) a) (* (fabs y-scale) (* t_0 x-scale)))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(y_45_scale) * x_45_scale;
	double tmp;
	if (fabs(y_45_scale) <= 2.7e+221) {
		tmp = ((a * b) * ((a * b) / ((t_0 * fabs(y_45_scale)) * x_45_scale))) * -4.0;
	} else {
		tmp = -4.0 * ((((a * b) * b) * a) / (fabs(y_45_scale) * (t_0 * x_45_scale)));
	}
	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 = abs(y_45scale) * x_45scale
    if (abs(y_45scale) <= 2.7d+221) then
        tmp = ((a * b) * ((a * b) / ((t_0 * abs(y_45scale)) * x_45scale))) * (-4.0d0)
    else
        tmp = (-4.0d0) * ((((a * b) * b) * a) / (abs(y_45scale) * (t_0 * x_45scale)))
    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 = Math.abs(y_45_scale) * x_45_scale;
	double tmp;
	if (Math.abs(y_45_scale) <= 2.7e+221) {
		tmp = ((a * b) * ((a * b) / ((t_0 * Math.abs(y_45_scale)) * x_45_scale))) * -4.0;
	} else {
		tmp = -4.0 * ((((a * b) * b) * a) / (Math.abs(y_45_scale) * (t_0 * x_45_scale)));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.fabs(y_45_scale) * x_45_scale
	tmp = 0
	if math.fabs(y_45_scale) <= 2.7e+221:
		tmp = ((a * b) * ((a * b) / ((t_0 * math.fabs(y_45_scale)) * x_45_scale))) * -4.0
	else:
		tmp = -4.0 * ((((a * b) * b) * a) / (math.fabs(y_45_scale) * (t_0 * x_45_scale)))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(y_45_scale) * x_45_scale)
	tmp = 0.0
	if (abs(y_45_scale) <= 2.7e+221)
		tmp = Float64(Float64(Float64(a * b) * Float64(Float64(a * b) / Float64(Float64(t_0 * abs(y_45_scale)) * x_45_scale))) * -4.0);
	else
		tmp = Float64(-4.0 * Float64(Float64(Float64(Float64(a * b) * b) * a) / Float64(abs(y_45_scale) * Float64(t_0 * x_45_scale))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(y_45_scale) * x_45_scale;
	tmp = 0.0;
	if (abs(y_45_scale) <= 2.7e+221)
		tmp = ((a * b) * ((a * b) / ((t_0 * abs(y_45_scale)) * x_45_scale))) * -4.0;
	else
		tmp = -4.0 * ((((a * b) * b) * a) / (abs(y_45_scale) * (t_0 * x_45_scale)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[y$45$scale], $MachinePrecision] * x$45$scale), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 2.7e+221], N[(N[(N[(a * b), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] / N[(N[(t$95$0 * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(-4.0 * N[(N[(N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision] / N[(N[Abs[y$45$scale], $MachinePrecision] * N[(t$95$0 * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.7e221

   1. Initial program 25.4%

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

   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 59.7%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot \frac{b \cdot b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\right) \cdot -4} \]
   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 67.2%

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

      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 74.7%

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

   8. Applied rewrites 74.7%

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

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

      4. unpow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-*r/ N/A

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

      10. pow-prod-down N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 80.7%

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 80.7%

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

   10. Applied rewrites 80.7%

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

   if 2.7e221 < y-scale

   1. Initial program 25.4%

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

   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 64.8%

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

   6. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. frac-times N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

   8. Applied rewrites 72.7%

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

Alternative 7: 80.7% accurate, 20.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.4%

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

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 59.7%

   \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot \frac{b \cdot b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\right) \cdot -4} \]
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 67.2%

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

   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 74.7%

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

8. Applied rewrites 74.7%

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

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

   4. unpow2 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. unpow2 N/A

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

   9. associate-*r/ N/A

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

   10. pow-prod-down N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. pow2 N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lower-/.f64 80.7%

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

   20. lift-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lift-*.f64 80.7%

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

10. Applied rewrites 80.7%

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

Alternative 8: 78.7% accurate, 20.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return 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)
end

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := 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]

Derivation

1. Initial program 25.4%

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

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 59.7%

   \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot \frac{b \cdot b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\right) \cdot -4} \]
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 67.2%

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

   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 74.7%

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

8. Applied rewrites 74.7%

   \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\right)\right)\right) \cdot -4 \]
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. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-*.f64 78.7%

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

10. Applied rewrites 78.7%

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

Alternative 9: 78.2% accurate, 20.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.4%

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

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 59.7%

   \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot \frac{b \cdot b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\right) \cdot -4} \]
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 67.2%

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

   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 74.7%

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

8. Applied rewrites 74.7%

   \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\right)\right)\right) \cdot -4 \]
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. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 78.2%

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

10. Applied rewrites 78.2%

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

Alternative 10: 74.7% accurate, 20.4× speedup

Math

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

FPCore

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(a * N[(a * N[(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. Initial program 25.4%

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

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 59.7%

   \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot \frac{b \cdot b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\right) \cdot -4} \]
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 67.2%

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

   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 74.7%

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

8. Applied rewrites 74.7%

   \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\right)\right)\right) \cdot -4 \]
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. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 74.3%

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

10. Applied rewrites 74.3%

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

Alternative 11: 74.3% accurate, 20.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.4%

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

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 59.7%

   \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot \frac{b \cdot b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\right) \cdot -4} \]
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 67.2%

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

   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 74.7%

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

8. Applied rewrites 74.7%

   \[\leadsto \left(a \cdot \left(a \cdot \left(b \cdot \frac{b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\right)\right)\right) \cdot -4 \]
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 74.7%

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

   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. *-commutative N/A

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

   6. lower-*.f64 74.7%

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 74.7%

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

10. Applied rewrites 74.7%

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

Reproduce

herbie shell --seed 2025188 
(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))))