Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.4% → 94.0%

Time: 26.0s

Alternatives: 8

Speedup: 40.5×

• 32.7% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = -1.4411967049100946e-248
  2. b = 1.326555664758651e+306
  3. angle = 6.8274297241312664e+106
  4. x-scale = 2.0155547657819938e-261
  5. y-scale = -6.961447286988036e-142

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.0% accurate, 35.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    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 * ((-4.0d0) * b)) / (x_45scale * y_45scale)) * ((a * b) / (x_45scale * y_45scale))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.6%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. unpow2 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 52.1

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

5. Simplified 52.1%

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. unswap-sqr N/A

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

   4. times-frac N/A

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

   5. *-lowering-*.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. /-lowering-/.f64 N/A

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

   14. *-lowering-*.f64 76.0

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

7. Applied egg-rr 76.0%

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

   1. associate-/l* N/A

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

   2. associate-*l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-lowering-*.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. *-lowering-*.f64 88.6

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

9. Applied egg-rr 88.6%

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

    1. associate-*r/ N/A

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

    2. *-commutative N/A

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

    3. associate-*l* N/A

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

    4. associate-/l* N/A

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

    5. *-lowering-*.f64 N/A

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

    6. associate-*r/ N/A

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

    7. associate-*r/ N/A

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

    8. /-lowering-/.f64 N/A

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

    9. *-lowering-*.f64 N/A

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

    10. *-lowering-*.f64 N/A

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

    11. *-lowering-*.f64 N/A

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

    12. /-lowering-/.f64 N/A

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

    13. *-lowering-*.f64 N/A

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

    14. *-lowering-*.f64 91.6

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

11. Applied egg-rr 91.6%

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

Alternative 2: 76.1% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(-4 \cdot b\right)\\ t_1 := \frac{a \cdot \left(b \cdot t\_0\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \mathbf{if}\;a \leq 8 \cdot 10^{-177}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+103}:\\ \;\;\;\;\left(a \cdot t\_0\right) \cdot \frac{b}{x-scale \cdot \left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* a (* -4.0 b)))
        (t_1 (/ (* a (* b t_0)) (* (* x-scale y-scale) (* x-scale y-scale)))))
   (if (<= a 8e-177)
     t_1
     (if (<= a 1.2e+103)
       (* (* a t_0) (/ b (* x-scale (* y-scale (* x-scale y-scale)))))
       t_1))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * (-4.0 * b);
	double t_1 = (a * (b * t_0)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale));
	double tmp;
	if (a <= 8e-177) {
		tmp = t_1;
	} else if (a <= 1.2e+103) {
		tmp = (a * t_0) * (b / (x_45_scale * (y_45_scale * (x_45_scale * y_45_scale))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    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 = a * ((-4.0d0) * b)
    t_1 = (a * (b * t_0)) / ((x_45scale * y_45scale) * (x_45scale * y_45scale))
    if (a <= 8d-177) then
        tmp = t_1
    else if (a <= 1.2d+103) then
        tmp = (a * t_0) * (b / (x_45scale * (y_45scale * (x_45scale * y_45scale))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * (-4.0 * b);
	double t_1 = (a * (b * t_0)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale));
	double tmp;
	if (a <= 8e-177) {
		tmp = t_1;
	} else if (a <= 1.2e+103) {
		tmp = (a * t_0) * (b / (x_45_scale * (y_45_scale * (x_45_scale * y_45_scale))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = a * (-4.0 * b)
	t_1 = (a * (b * t_0)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))
	tmp = 0
	if a <= 8e-177:
		tmp = t_1
	elif a <= 1.2e+103:
		tmp = (a * t_0) * (b / (x_45_scale * (y_45_scale * (x_45_scale * y_45_scale))))
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a * Float64(-4.0 * b))
	t_1 = Float64(Float64(a * Float64(b * t_0)) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale)))
	tmp = 0.0
	if (a <= 8e-177)
		tmp = t_1;
	elseif (a <= 1.2e+103)
		tmp = Float64(Float64(a * t_0) * Float64(b / Float64(x_45_scale * Float64(y_45_scale * Float64(x_45_scale * y_45_scale)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = a * (-4.0 * b);
	t_1 = (a * (b * t_0)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale));
	tmp = 0.0;
	if (a <= 8e-177)
		tmp = t_1;
	elseif (a <= 1.2e+103)
		tmp = (a * t_0) * (b / (x_45_scale * (y_45_scale * (x_45_scale * y_45_scale))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a * N[(-4.0 * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 8e-177], t$95$1, If[LessEqual[a, 1.2e+103], N[(N[(a * t$95$0), $MachinePrecision] * N[(b / N[(x$45$scale * N[(y$45$scale * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < 7.99999999999999962e-177 or 1.1999999999999999e103 < a

   1. Initial program 26.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 47.5

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

   5. Simplified 47.5%

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

      1. associate-*r* N/A

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

      2. unswap-sqr N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 52.8

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

   7. Applied egg-rr 52.8%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 72.6

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

   9. Applied egg-rr 72.6%

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

   if 7.99999999999999962e-177 < a < 1.1999999999999999e103

   1. Initial program 38.2%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 63.1

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

   5. Simplified 63.1%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. +-lft-identity N/A

          \[\leadsto \left(\left(a \cdot a\right) \cdot \left(-4 \cdot b\right)\right) \cdot \frac{b}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot \color{blue}{\left(0 + y-scale\right)}\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \left(\left(a \cdot a\right) \cdot \left(-4 \cdot b\right)\right) \cdot \frac{b}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot \color{blue}{\left(y-scale + 0\right)}\right)\right)} \]

      14. distribute-rgt-out N/A

          \[\leadsto \left(\left(a \cdot a\right) \cdot \left(-4 \cdot b\right)\right) \cdot \frac{b}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale + 0 \cdot y-scale\right)}\right)} \]

      15. mul0-lft N/A

          \[\leadsto \left(\left(a \cdot a\right) \cdot \left(-4 \cdot b\right)\right) \cdot \frac{b}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale + \color{blue}{0}\right)\right)} \]

      16. accelerator-lowering-fma.f64 70.3

          \[\leadsto \left(\left(a \cdot a\right) \cdot \left(-4 \cdot b\right)\right) \cdot \frac{b}{x-scale \cdot \left(x-scale \cdot \color{blue}{\mathsf{fma}\left(y-scale, y-scale, 0\right)}\right)} \]

   7. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot \left(-4 \cdot b\right)\right) \cdot \frac{b}{x-scale \cdot \left(x-scale \cdot \mathsf{fma}\left(y-scale, y-scale, 0\right)\right)}} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 80.8

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

   9. Applied egg-rr 80.8%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 82.1

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

   11. Applied egg-rr 82.1%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8 \cdot 10^{-177}:\\ \;\;\;\;\frac{a \cdot \left(b \cdot \left(a \cdot \left(-4 \cdot b\right)\right)\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+103}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(-4 \cdot b\right)\right)\right) \cdot \frac{b}{x-scale \cdot \left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(b \cdot \left(a \cdot \left(-4 \cdot b\right)\right)\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.7% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{x-scale \cdot y-scale}\\ \mathbf{if}\;a \leq 1.5 \cdot 10^{+240}:\\ \;\;\;\;t\_0 \cdot \left(a \cdot \left(a \cdot \left(-4 \cdot t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{x-scale \cdot y-scale} \cdot \frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{x-scale \cdot y-scale}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* x-scale y-scale))))
   (if (<= a 1.5e+240)
     (* t_0 (* a (* a (* -4.0 t_0))))
     (*
      (/ -4.0 (* x-scale y-scale))
      (/ (* (* a b) (* a b)) (* x-scale y-scale))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (x_45_scale * y_45_scale);
	double tmp;
	if (a <= 1.5e+240) {
		tmp = t_0 * (a * (a * (-4.0 * t_0)));
	} else {
		tmp = (-4.0 / (x_45_scale * y_45_scale)) * (((a * b) * (a * b)) / (x_45_scale * y_45_scale));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b / (x_45scale * y_45scale)
    if (a <= 1.5d+240) then
        tmp = t_0 * (a * (a * ((-4.0d0) * t_0)))
    else
        tmp = ((-4.0d0) / (x_45scale * y_45scale)) * (((a * b) * (a * b)) / (x_45scale * y_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 / (x_45_scale * y_45_scale);
	double tmp;
	if (a <= 1.5e+240) {
		tmp = t_0 * (a * (a * (-4.0 * t_0)));
	} else {
		tmp = (-4.0 / (x_45_scale * y_45_scale)) * (((a * b) * (a * b)) / (x_45_scale * y_45_scale));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = b / (x_45_scale * y_45_scale)
	tmp = 0
	if a <= 1.5e+240:
		tmp = t_0 * (a * (a * (-4.0 * t_0)))
	else:
		tmp = (-4.0 / (x_45_scale * y_45_scale)) * (((a * b) * (a * b)) / (x_45_scale * y_45_scale))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(x_45_scale * y_45_scale))
	tmp = 0.0
	if (a <= 1.5e+240)
		tmp = Float64(t_0 * Float64(a * Float64(a * Float64(-4.0 * t_0))));
	else
		tmp = Float64(Float64(-4.0 / Float64(x_45_scale * y_45_scale)) * Float64(Float64(Float64(a * b) * Float64(a * b)) / Float64(x_45_scale * y_45_scale)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = b / (x_45_scale * y_45_scale);
	tmp = 0.0;
	if (a <= 1.5e+240)
		tmp = t_0 * (a * (a * (-4.0 * t_0)));
	else
		tmp = (-4.0 / (x_45_scale * y_45_scale)) * (((a * b) * (a * b)) / (x_45_scale * y_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[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 1.5e+240], N[(t$95$0 * N[(a * N[(a * N[(-4.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(a * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.4999999999999999e240

   1. Initial program 30.6%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 53.1

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

   5. Simplified 53.1%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. unswap-sqr N/A

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

      4. times-frac N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 77.5

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

   7. Applied egg-rr 77.5%

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

      1. associate-/l* N/A

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

      2. associate-*l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 89.7

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

   9. Applied egg-rr 89.7%

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

   if 1.4999999999999999e240 < a

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 22.2

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

   5. Simplified 22.2%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. unswap-sqr N/A

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

      4. times-frac N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 45.3

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

   7. Applied egg-rr 45.3%

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

      1. associate-*r* N/A

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

      2. unswap-sqr N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 88.4

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

   9. Applied egg-rr 88.4%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{+240}:\\ \;\;\;\;\frac{b}{x-scale \cdot y-scale} \cdot \left(a \cdot \left(a \cdot \left(-4 \cdot \frac{b}{x-scale \cdot y-scale}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{x-scale \cdot y-scale} \cdot \frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{x-scale \cdot y-scale}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 35.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(-4 \cdot b\right)\\ \mathbf{if}\;a \leq 2.2 \cdot 10^{+161}:\\ \;\;\;\;t\_0 \cdot \left(a \cdot \frac{b}{\mathsf{fma}\left(x-scale \cdot y-scale, x-scale \cdot y-scale, 0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(b \cdot t\_0\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* a (* -4.0 b))))
   (if (<= a 2.2e+161)
     (* t_0 (* a (/ b (fma (* x-scale y-scale) (* x-scale y-scale) 0.0))))
     (/ (* a (* b t_0)) (* (* x-scale y-scale) (* x-scale y-scale))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * (-4.0 * b);
	double tmp;
	if (a <= 2.2e+161) {
		tmp = t_0 * (a * (b / fma((x_45_scale * y_45_scale), (x_45_scale * y_45_scale), 0.0)));
	} else {
		tmp = (a * (b * t_0)) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a * Float64(-4.0 * b))
	tmp = 0.0
	if (a <= 2.2e+161)
		tmp = Float64(t_0 * Float64(a * Float64(b / fma(Float64(x_45_scale * y_45_scale), Float64(x_45_scale * y_45_scale), 0.0))));
	else
		tmp = Float64(Float64(a * Float64(b * t_0)) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a * N[(-4.0 * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 2.2e+161], N[(t$95$0 * N[(a * N[(b / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.2e161

   1. Initial program 32.3%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 53.5

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

   5. Simplified 53.5%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. unswap-sqr N/A

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

      4. times-frac N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 77.9

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

   7. Applied egg-rr 77.9%

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

      1. frac-times N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. swap-sqr N/A

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

      5. associate-*r* N/A

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

      6. +-rgt-identity N/A

         \[\leadsto \frac{b}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale + 0\right)}\right)} \cdot \left(\left(a \cdot a\right) \cdot \left(-4 \cdot b\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \frac{b}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale + 0\right)\right)} \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(-4 \cdot b\right)\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{b}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale + 0\right)\right)} \cdot a\right) \cdot \left(a \cdot \left(-4 \cdot b\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{b}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale + 0\right)\right)} \cdot a\right) \cdot \left(a \cdot \left(-4 \cdot b\right)\right)} \]

   9. Applied egg-rr 77.5%

      \[\leadsto \color{blue}{\left(\frac{b}{\mathsf{fma}\left(x-scale \cdot y-scale, x-scale \cdot y-scale, 0\right)} \cdot a\right) \cdot \left(a \cdot \left(-4 \cdot b\right)\right)} \]

   if 2.2e161 < a

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 36.4

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

   5. Simplified 36.4%

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

      1. associate-*r* N/A

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

      2. unswap-sqr N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 36.4

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

   7. Applied egg-rr 36.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 76.0

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

   9. Applied egg-rr 76.0%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.2 \cdot 10^{+161}:\\ \;\;\;\;\left(a \cdot \left(-4 \cdot b\right)\right) \cdot \left(a \cdot \frac{b}{\mathsf{fma}\left(x-scale \cdot y-scale, x-scale \cdot y-scale, 0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(b \cdot \left(a \cdot \left(-4 \cdot b\right)\right)\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.1% accurate, 35.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.8 \cdot 10^{+87}:\\ \;\;\;\;\left(a \cdot -4\right) \cdot \left(a \cdot \left(b \cdot \frac{b}{\mathsf{fma}\left(x-scale \cdot y-scale, x-scale \cdot y-scale, 0\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(b \cdot \left(a \cdot \left(-4 \cdot b\right)\right)\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= a 5.8e+87)
   (*
    (* a -4.0)
    (* a (* b (/ b (fma (* x-scale y-scale) (* x-scale y-scale) 0.0)))))
   (/
    (* a (* b (* a (* -4.0 b))))
    (* (* x-scale y-scale) (* x-scale y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 5.8e+87) {
		tmp = (a * -4.0) * (a * (b * (b / fma((x_45_scale * y_45_scale), (x_45_scale * y_45_scale), 0.0))));
	} else {
		tmp = (a * (b * (a * (-4.0 * b)))) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a <= 5.8e+87)
		tmp = Float64(Float64(a * -4.0) * Float64(a * Float64(b * Float64(b / fma(Float64(x_45_scale * y_45_scale), Float64(x_45_scale * y_45_scale), 0.0)))));
	else
		tmp = Float64(Float64(a * Float64(b * Float64(a * Float64(-4.0 * b)))) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a, 5.8e+87], N[(N[(a * -4.0), $MachinePrecision] * N[(a * N[(b * N[(b / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(b * N[(a * N[(-4.0 * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.7999999999999996e87

   1. Initial program 33.1%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 53.7

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

   5. Simplified 53.7%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. unswap-sqr N/A

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

      4. times-frac N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-lowering-*.f64 77.9

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

   7. Applied egg-rr 77.9%

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

      1. frac-times N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. pow2 N/A

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

      14. associate-/l* N/A

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

      15. swap-sqr N/A

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

      16. associate-*r* N/A

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

      17. +-rgt-identity N/A

          \[\leadsto \left(a \cdot -4\right) \cdot \left(a \cdot \left(b \cdot \frac{b}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale + 0\right)}\right)}\right)\right) \]

   9. Applied egg-rr 73.1%

      \[\leadsto \color{blue}{\left(a \cdot -4\right) \cdot \left(a \cdot \left(b \cdot \frac{b}{\mathsf{fma}\left(x-scale \cdot y-scale, x-scale \cdot y-scale, 0\right)}\right)\right)} \]

   if 5.7999999999999996e87 < a

   1. Initial program 8.2%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 42.1

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

   5. Simplified 42.1%

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

      1. associate-*r* N/A

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

      2. unswap-sqr N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 44.4

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

   7. Applied egg-rr 44.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 71.5

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

   9. Applied egg-rr 71.5%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.8 \cdot 10^{+87}:\\ \;\;\;\;\left(a \cdot -4\right) \cdot \left(a \cdot \left(b \cdot \frac{b}{\mathsf{fma}\left(x-scale \cdot y-scale, x-scale \cdot y-scale, 0\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(b \cdot \left(a \cdot \left(-4 \cdot b\right)\right)\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.5% accurate, 35.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.6%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. unpow2 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 52.1

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

5. Simplified 52.1%

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. unswap-sqr N/A

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

   4. times-frac N/A

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

   5. *-lowering-*.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. /-lowering-/.f64 N/A

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

   14. *-lowering-*.f64 76.0

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

7. Applied egg-rr 76.0%

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

   1. associate-/l* N/A

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

   2. associate-*l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-lowering-*.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. *-lowering-*.f64 88.6

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

9. Applied egg-rr 88.6%

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

10. Final simplification 88.6%

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

Alternative 7: 73.8% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    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 * ((-4.0d0) * b))) * (b / (x_45scale * (y_45scale * (x_45scale * y_45scale))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.6%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. unpow2 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 52.1

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

5. Simplified 52.1%

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

   1. associate-*r* N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-lowering-*.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. +-lft-identity N/A

       \[\leadsto \left(\left(a \cdot a\right) \cdot \left(-4 \cdot b\right)\right) \cdot \frac{b}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot \color{blue}{\left(0 + y-scale\right)}\right)\right)} \]

   13. +-commutative N/A

       \[\leadsto \left(\left(a \cdot a\right) \cdot \left(-4 \cdot b\right)\right) \cdot \frac{b}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot \color{blue}{\left(y-scale + 0\right)}\right)\right)} \]

   14. distribute-rgt-out N/A

       \[\leadsto \left(\left(a \cdot a\right) \cdot \left(-4 \cdot b\right)\right) \cdot \frac{b}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale + 0 \cdot y-scale\right)}\right)} \]

   15. mul0-lft N/A

       \[\leadsto \left(\left(a \cdot a\right) \cdot \left(-4 \cdot b\right)\right) \cdot \frac{b}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale + \color{blue}{0}\right)\right)} \]

   16. accelerator-lowering-fma.f64 60.9

       \[\leadsto \left(\left(a \cdot a\right) \cdot \left(-4 \cdot b\right)\right) \cdot \frac{b}{x-scale \cdot \left(x-scale \cdot \color{blue}{\mathsf{fma}\left(y-scale, y-scale, 0\right)}\right)} \]

7. Applied egg-rr 60.9%

   \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot \left(-4 \cdot b\right)\right) \cdot \frac{b}{x-scale \cdot \left(x-scale \cdot \mathsf{fma}\left(y-scale, y-scale, 0\right)\right)}} \]
8. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. *-lowering-*.f64 67.8

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

9. Applied egg-rr 67.8%

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

    1. associate-*l* N/A

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

    2. *-commutative N/A

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

    3. *-lowering-*.f64 N/A

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

    4. *-lowering-*.f64 N/A

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

    5. *-lowering-*.f64 71.1

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

11. Applied egg-rr 71.1%

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

12. Final simplification 71.1%

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

Alternative 8: 68.3% accurate, 40.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.6%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. unpow2 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 52.1

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

5. Simplified 52.1%

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. unswap-sqr N/A

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

   4. times-frac N/A

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

   5. *-lowering-*.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. /-lowering-/.f64 N/A

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

   14. *-lowering-*.f64 76.0

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

7. Applied egg-rr 76.0%

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

   1. frac-times N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. pow2 N/A

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

   6. associate-/l* N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. pow2 N/A

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

   12. associate-/l* N/A

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

   13. swap-sqr N/A

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

   14. associate-*r* N/A

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

   15. +-rgt-identity N/A

       \[\leadsto \left(\left(a \cdot a\right) \cdot -4\right) \cdot \left(b \cdot \frac{b}{x-scale \cdot \left(x-scale \cdot \color{blue}{\left(y-scale \cdot y-scale + 0\right)}\right)}\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \left(\left(a \cdot a\right) \cdot -4\right) \cdot \color{blue}{\left(b \cdot \frac{b}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale + 0\right)\right)}\right)} \]

   17. +-rgt-identity N/A

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

9. Applied egg-rr 68.0%

   \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot -4\right) \cdot \left(b \cdot \frac{b}{\mathsf{fma}\left(x-scale \cdot y-scale, x-scale \cdot y-scale, 0\right)}\right)} \]
10. Step-by-step derivation

    1. +-rgt-identity N/A

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

    2. *-lowering-*.f64 N/A

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

    3. *-lowering-*.f64 N/A

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

    4. *-lowering-*.f64 68.0

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

11. Applied egg-rr 68.0%

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

12. Final simplification 68.0%

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

Reproduce

herbie shell --seed 2024197 
(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))))