Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 23.6% → 92.6%

Time: 20.0s

Alternatives: 9

Speedup: 40.5×

• 34.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

Initial Program: 23.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 92.6% accurate, 26.8× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \frac{a \cdot b}{y-scale\_m \cdot x-scale}\\ \mathbf{if}\;y-scale\_m \leq 2.7 \cdot 10^{+48}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b \cdot \frac{a}{y-scale\_m}\right) \cdot \left(\frac{a}{y-scale\_m} \cdot \left(\frac{b}{x-scale} \cdot -4\right)\right)}{x-scale}\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (let* ((t_0 (/ (* a b) (* y-scale_m x-scale))))
   (if (<= y-scale_m 2.7e+48)
     (* (* t_0 t_0) -4.0)
     (/
      (* (* b (/ a y-scale_m)) (* (/ a y-scale_m) (* (/ b x-scale) -4.0)))
      x-scale))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = (a * b) / (y_45_scale_m * x_45_scale);
	double tmp;
	if (y_45_scale_m <= 2.7e+48) {
		tmp = (t_0 * t_0) * -4.0;
	} else {
		tmp = ((b * (a / y_45_scale_m)) * ((a / y_45_scale_m) * ((b / x_45_scale) * -4.0))) / x_45_scale;
	}
	return tmp;
}

Fortran

y-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a * b) / (y_45scale_m * x_45scale)
    if (y_45scale_m <= 2.7d+48) then
        tmp = (t_0 * t_0) * (-4.0d0)
    else
        tmp = ((b * (a / y_45scale_m)) * ((a / y_45scale_m) * ((b / x_45scale) * (-4.0d0)))) / x_45scale
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = (a * b) / (y_45_scale_m * x_45_scale);
	double tmp;
	if (y_45_scale_m <= 2.7e+48) {
		tmp = (t_0 * t_0) * -4.0;
	} else {
		tmp = ((b * (a / y_45_scale_m)) * ((a / y_45_scale_m) * ((b / x_45_scale) * -4.0))) / x_45_scale;
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	t_0 = (a * b) / (y_45_scale_m * x_45_scale)
	tmp = 0
	if y_45_scale_m <= 2.7e+48:
		tmp = (t_0 * t_0) * -4.0
	else:
		tmp = ((b * (a / y_45_scale_m)) * ((a / y_45_scale_m) * ((b / x_45_scale) * -4.0))) / x_45_scale
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = Float64(Float64(a * b) / Float64(y_45_scale_m * x_45_scale))
	tmp = 0.0
	if (y_45_scale_m <= 2.7e+48)
		tmp = Float64(Float64(t_0 * t_0) * -4.0);
	else
		tmp = Float64(Float64(Float64(b * Float64(a / y_45_scale_m)) * Float64(Float64(a / y_45_scale_m) * Float64(Float64(b / x_45_scale) * -4.0))) / x_45_scale);
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = (a * b) / (y_45_scale_m * x_45_scale);
	tmp = 0.0;
	if (y_45_scale_m <= 2.7e+48)
		tmp = (t_0 * t_0) * -4.0;
	else
		tmp = ((b * (a / y_45_scale_m)) * ((a / y_45_scale_m) * ((b / x_45_scale) * -4.0))) / x_45_scale;
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := Block[{t$95$0 = N[(N[(a * b), $MachinePrecision] / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 2.7e+48], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(N[(b * N[(a / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(a / y$45$scale$95$m), $MachinePrecision] * N[(N[(b / x$45$scale), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.70000000000000004e48

   1. Initial program 21.9%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 55.8%

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

   7. Applied rewrites 79.3%

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

   9. Applied rewrites 93.5%

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

   if 2.70000000000000004e48 < y-scale

   1. Initial program 42.0%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 66.1%

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

   7. Applied rewrites 66.3%

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

   9. Applied rewrites 91.0%

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

   11. Applied rewrites 94.0%

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

Alternative 2: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale\_m}\\ \mathbf{if}\;t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale\_m}}{y-scale\_m} \leq 10^{+116}:\\ \;\;\;\;\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{\frac{b \cdot b}{x-scale \cdot y-scale\_m}}{y-scale\_m \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{\left(x-scale \cdot y-scale\_m\right) \cdot \left(x-scale \cdot y-scale\_m\right)} \cdot -4\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :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_m)))
   (if (<=
        (-
         (* 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_m)
           y-scale_m)))
        1e+116)
     (*
      (* -4.0 (* a a))
      (/ (/ (* b b) (* x-scale y-scale_m)) (* y-scale_m x-scale)))
     (*
      (/ (* (* b a) (* b a)) (* (* x-scale y-scale_m) (* x-scale y-scale_m)))
      -4.0))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale)) (*.f64 (*.f64 #s(literal 4 binary64) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale))) < 1.00000000000000002e116

   1. Initial program 71.9%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 71.3%

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

   7. Applied rewrites 87.9%

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

   9. Applied rewrites 88.0%

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

   if 1.00000000000000002e116 < (-.f64 (*.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale)) (*.f64 (*.f64 #s(literal 4 binary64) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale)) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)))

   1. Initial program 0.0%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 50.2%

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

   7. Applied rewrites 75.4%

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

   9. Applied rewrites 92.4%

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

   11. Applied rewrites 75.4%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{y-scale}}{y-scale} \leq 10^{+116}:\\ \;\;\;\;\left(-4 \cdot \left(a \cdot a\right)\right) \cdot \frac{\frac{b \cdot b}{x-scale \cdot y-scale}}{y-scale \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)} \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.7% accurate, 26.8× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \frac{\frac{b}{x-scale}}{y-scale\_m}\\ t_1 := \frac{a \cdot b}{y-scale\_m \cdot x-scale}\\ \mathbf{if}\;x-scale \leq 3.6 \cdot 10^{+137}:\\ \;\;\;\;\left(t\_1 \cdot t\_1\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(t\_0 \cdot \left(t\_0 \cdot a\right)\right)\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (let* ((t_0 (/ (/ b x-scale) y-scale_m))
        (t_1 (/ (* a b) (* y-scale_m x-scale))))
   (if (<= x-scale 3.6e+137)
     (* (* t_1 t_1) -4.0)
     (* (* a (* t_0 (* t_0 a))) -4.0))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = (b / x_45_scale) / y_45_scale_m;
	double t_1 = (a * b) / (y_45_scale_m * x_45_scale);
	double tmp;
	if (x_45_scale <= 3.6e+137) {
		tmp = (t_1 * t_1) * -4.0;
	} else {
		tmp = (a * (t_0 * (t_0 * a))) * -4.0;
	}
	return tmp;
}

Fortran

y-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b / x_45scale) / y_45scale_m
    t_1 = (a * b) / (y_45scale_m * x_45scale)
    if (x_45scale <= 3.6d+137) then
        tmp = (t_1 * t_1) * (-4.0d0)
    else
        tmp = (a * (t_0 * (t_0 * a))) * (-4.0d0)
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = (b / x_45_scale) / y_45_scale_m;
	double t_1 = (a * b) / (y_45_scale_m * x_45_scale);
	double tmp;
	if (x_45_scale <= 3.6e+137) {
		tmp = (t_1 * t_1) * -4.0;
	} else {
		tmp = (a * (t_0 * (t_0 * a))) * -4.0;
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	t_0 = (b / x_45_scale) / y_45_scale_m
	t_1 = (a * b) / (y_45_scale_m * x_45_scale)
	tmp = 0
	if x_45_scale <= 3.6e+137:
		tmp = (t_1 * t_1) * -4.0
	else:
		tmp = (a * (t_0 * (t_0 * a))) * -4.0
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = Float64(Float64(b / x_45_scale) / y_45_scale_m)
	t_1 = Float64(Float64(a * b) / Float64(y_45_scale_m * x_45_scale))
	tmp = 0.0
	if (x_45_scale <= 3.6e+137)
		tmp = Float64(Float64(t_1 * t_1) * -4.0);
	else
		tmp = Float64(Float64(a * Float64(t_0 * Float64(t_0 * a))) * -4.0);
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = (b / x_45_scale) / y_45_scale_m;
	t_1 = (a * b) / (y_45_scale_m * x_45_scale);
	tmp = 0.0;
	if (x_45_scale <= 3.6e+137)
		tmp = (t_1 * t_1) * -4.0;
	else
		tmp = (a * (t_0 * (t_0 * a))) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := Block[{t$95$0 = N[(N[(b / x$45$scale), $MachinePrecision] / y$45$scale$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, 3.6e+137], N[(N[(t$95$1 * t$95$1), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(a * N[(t$95$0 * N[(t$95$0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 3.6e137

   1. Initial program 22.5%

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

   3. Taylor expanded in 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

   5. Applied rewrites 58.7%

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

   7. Applied rewrites 76.8%

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

   9. Applied rewrites 94.2%

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

   if 3.6e137 < x-scale

   1. Initial program 44.9%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 52.4%

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

   7. Applied rewrites 76.4%

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

   9. Applied rewrites 86.2%

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

   11. Applied rewrites 93.8%

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

Alternative 4: 62.7% accurate, 32.3× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := -4 \cdot \left(a \cdot a\right)\\ \mathbf{if}\;x-scale \leq 5.5 \cdot 10^{-169} \lor \neg \left(x-scale \leq 2.05 \cdot 10^{+167}\right):\\ \;\;\;\;\frac{t\_0}{\left(y-scale\_m \cdot x-scale\right) \cdot \left(y-scale\_m \cdot x-scale\right)} \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot b}{\left(\left(x-scale \cdot x-scale\right) \cdot y-scale\_m\right) \cdot y-scale\_m} \cdot b\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (let* ((t_0 (* -4.0 (* a a))))
   (if (or (<= x-scale 5.5e-169) (not (<= x-scale 2.05e+167)))
     (* (/ t_0 (* (* y-scale_m x-scale) (* y-scale_m x-scale))) (* b b))
     (* (/ (* t_0 b) (* (* (* x-scale x-scale) y-scale_m) y-scale_m)) b))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = -4.0 * (a * a);
	double tmp;
	if ((x_45_scale <= 5.5e-169) || !(x_45_scale <= 2.05e+167)) {
		tmp = (t_0 / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b);
	} else {
		tmp = ((t_0 * b) / (((x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)) * b;
	}
	return tmp;
}

Fortran

y-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-4.0d0) * (a * a)
    if ((x_45scale <= 5.5d-169) .or. (.not. (x_45scale <= 2.05d+167))) then
        tmp = (t_0 / ((y_45scale_m * x_45scale) * (y_45scale_m * x_45scale))) * (b * b)
    else
        tmp = ((t_0 * b) / (((x_45scale * x_45scale) * y_45scale_m) * y_45scale_m)) * b
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = -4.0 * (a * a);
	double tmp;
	if ((x_45_scale <= 5.5e-169) || !(x_45_scale <= 2.05e+167)) {
		tmp = (t_0 / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b);
	} else {
		tmp = ((t_0 * b) / (((x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)) * b;
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	t_0 = -4.0 * (a * a)
	tmp = 0
	if (x_45_scale <= 5.5e-169) or not (x_45_scale <= 2.05e+167):
		tmp = (t_0 / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b)
	else:
		tmp = ((t_0 * b) / (((x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)) * b
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = Float64(-4.0 * Float64(a * a))
	tmp = 0.0
	if ((x_45_scale <= 5.5e-169) || !(x_45_scale <= 2.05e+167))
		tmp = Float64(Float64(t_0 / Float64(Float64(y_45_scale_m * x_45_scale) * Float64(y_45_scale_m * x_45_scale))) * Float64(b * b));
	else
		tmp = Float64(Float64(Float64(t_0 * b) / Float64(Float64(Float64(x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)) * b);
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = -4.0 * (a * a);
	tmp = 0.0;
	if ((x_45_scale <= 5.5e-169) || ~((x_45_scale <= 2.05e+167)))
		tmp = (t_0 / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b);
	else
		tmp = ((t_0 * b) / (((x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)) * b;
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := Block[{t$95$0 = N[(-4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x$45$scale, 5.5e-169], N[Not[LessEqual[x$45$scale, 2.05e+167]], $MachinePrecision]], N[(N[(t$95$0 / N[(N[(y$45$scale$95$m * x$45$scale), $MachinePrecision] * N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * b), $MachinePrecision] / N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale$95$m), $MachinePrecision] * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 5.4999999999999994e-169 or 2.05e167 < x-scale

   1. Initial program 27.6%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 54.4%

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

   10. Applied rewrites 59.6%

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

   if 5.4999999999999994e-169 < x-scale < 2.05e167

   1. Initial program 20.7%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 61.9%

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

   10. Applied rewrites 86.9%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 75.3%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 5.5 \cdot 10^{-169} \lor \neg \left(x-scale \leq 2.05 \cdot 10^{+167}\right):\\ \;\;\;\;\frac{-4 \cdot \left(a \cdot a\right)}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-4 \cdot \left(a \cdot a\right)\right) \cdot b}{\left(\left(x-scale \cdot x-scale\right) \cdot y-scale\right) \cdot y-scale} \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.6% accurate, 32.3× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := -4 \cdot \left(a \cdot a\right)\\ \mathbf{if}\;x-scale \leq 5.5 \cdot 10^{-169}:\\ \;\;\;\;\frac{t\_0}{\left(y-scale\_m \cdot x-scale\right) \cdot \left(y-scale\_m \cdot x-scale\right)} \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;x-scale \leq 2.1 \cdot 10^{+167}:\\ \;\;\;\;\frac{t\_0 \cdot b}{\left(\left(x-scale \cdot x-scale\right) \cdot y-scale\_m\right) \cdot y-scale\_m} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(\left(y-scale\_m \cdot x-scale\right) \cdot y-scale\_m\right) \cdot x-scale} \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (let* ((t_0 (* -4.0 (* a a))))
   (if (<= x-scale 5.5e-169)
     (* (/ t_0 (* (* y-scale_m x-scale) (* y-scale_m x-scale))) (* b b))
     (if (<= x-scale 2.1e+167)
       (* (/ (* t_0 b) (* (* (* x-scale x-scale) y-scale_m) y-scale_m)) b)
       (* (/ t_0 (* (* (* y-scale_m x-scale) y-scale_m) x-scale)) (* b b))))))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = -4.0 * (a * a);
	double tmp;
	if (x_45_scale <= 5.5e-169) {
		tmp = (t_0 / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b);
	} else if (x_45_scale <= 2.1e+167) {
		tmp = ((t_0 * b) / (((x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)) * b;
	} else {
		tmp = (t_0 / (((y_45_scale_m * x_45_scale) * y_45_scale_m) * x_45_scale)) * (b * b);
	}
	return tmp;
}

Fortran

y-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-4.0d0) * (a * a)
    if (x_45scale <= 5.5d-169) then
        tmp = (t_0 / ((y_45scale_m * x_45scale) * (y_45scale_m * x_45scale))) * (b * b)
    else if (x_45scale <= 2.1d+167) then
        tmp = ((t_0 * b) / (((x_45scale * x_45scale) * y_45scale_m) * y_45scale_m)) * b
    else
        tmp = (t_0 / (((y_45scale_m * x_45scale) * y_45scale_m) * x_45scale)) * (b * b)
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = -4.0 * (a * a);
	double tmp;
	if (x_45_scale <= 5.5e-169) {
		tmp = (t_0 / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b);
	} else if (x_45_scale <= 2.1e+167) {
		tmp = ((t_0 * b) / (((x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)) * b;
	} else {
		tmp = (t_0 / (((y_45_scale_m * x_45_scale) * y_45_scale_m) * x_45_scale)) * (b * b);
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	t_0 = -4.0 * (a * a)
	tmp = 0
	if x_45_scale <= 5.5e-169:
		tmp = (t_0 / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b)
	elif x_45_scale <= 2.1e+167:
		tmp = ((t_0 * b) / (((x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)) * b
	else:
		tmp = (t_0 / (((y_45_scale_m * x_45_scale) * y_45_scale_m) * x_45_scale)) * (b * b)
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = Float64(-4.0 * Float64(a * a))
	tmp = 0.0
	if (x_45_scale <= 5.5e-169)
		tmp = Float64(Float64(t_0 / Float64(Float64(y_45_scale_m * x_45_scale) * Float64(y_45_scale_m * x_45_scale))) * Float64(b * b));
	elseif (x_45_scale <= 2.1e+167)
		tmp = Float64(Float64(Float64(t_0 * b) / Float64(Float64(Float64(x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)) * b);
	else
		tmp = Float64(Float64(t_0 / Float64(Float64(Float64(y_45_scale_m * x_45_scale) * y_45_scale_m) * x_45_scale)) * Float64(b * b));
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = -4.0 * (a * a);
	tmp = 0.0;
	if (x_45_scale <= 5.5e-169)
		tmp = (t_0 / ((y_45_scale_m * x_45_scale) * (y_45_scale_m * x_45_scale))) * (b * b);
	elseif (x_45_scale <= 2.1e+167)
		tmp = ((t_0 * b) / (((x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)) * b;
	else
		tmp = (t_0 / (((y_45_scale_m * x_45_scale) * y_45_scale_m) * x_45_scale)) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := Block[{t$95$0 = N[(-4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, 5.5e-169], N[(N[(t$95$0 / N[(N[(y$45$scale$95$m * x$45$scale), $MachinePrecision] * N[(y$45$scale$95$m * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 2.1e+167], N[(N[(N[(t$95$0 * b), $MachinePrecision] / N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale$95$m), $MachinePrecision] * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[(t$95$0 / N[(N[(N[(y$45$scale$95$m * x$45$scale), $MachinePrecision] * y$45$scale$95$m), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < 5.4999999999999994e-169

   1. Initial program 25.6%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 46.6%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 55.3%

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

   10. Applied rewrites 58.5%

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

   if 5.4999999999999994e-169 < x-scale < 2.0999999999999999e167

   1. Initial program 20.7%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 60.8%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 61.9%

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

   10. Applied rewrites 86.9%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 75.3%

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

   if 2.0999999999999999e167 < x-scale

   1. Initial program 37.7%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 49.8%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 49.9%

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

   10. Applied rewrites 65.5%

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

Alternative 6: 94.0% accurate, 35.9× speedup

Math

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

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (let* ((t_0 (/ (* a b) (* y-scale_m x-scale)))) (* (* t_0 t_0) -4.0)))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = (a * b) / (y_45_scale_m * x_45_scale);
	return (t_0 * t_0) * -4.0;
}

Fortran

y-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    t_0 = (a * b) / (y_45scale_m * x_45scale)
    code = (t_0 * t_0) * (-4.0d0)
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = (a * b) / (y_45_scale_m * x_45_scale);
	return (t_0 * t_0) * -4.0;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	t_0 = (a * b) / (y_45_scale_m * x_45_scale)
	return (t_0 * t_0) * -4.0

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = Float64(Float64(a * b) / Float64(y_45_scale_m * x_45_scale))
	return Float64(Float64(t_0 * t_0) * -4.0)
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale, y_45_scale_m)
	t_0 = (a * b) / (y_45_scale_m * x_45_scale);
	tmp = (t_0 * t_0) * -4.0;
end

Wolfram

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

Derivation

1. Initial program 25.8%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 57.8%

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

7. Applied rewrites 76.8%

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

9. Applied rewrites 93.1%

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

Alternative 7: 77.8% accurate, 40.5× speedup

Math

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

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (*
  (/ (* (* b a) (* b a)) (* (* x-scale y-scale_m) (* x-scale y-scale_m)))
  -4.0))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return (((b * a) * (b * a)) / ((x_45_scale * y_45_scale_m) * (x_45_scale * y_45_scale_m))) * -4.0;
}

Fortran

y-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    code = (((b * a) * (b * a)) / ((x_45scale * y_45scale_m) * (x_45scale * y_45scale_m))) * (-4.0d0)
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return (((b * a) * (b * a)) / ((x_45_scale * y_45_scale_m) * (x_45_scale * y_45_scale_m))) * -4.0;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	return (((b * a) * (b * a)) / ((x_45_scale * y_45_scale_m) * (x_45_scale * y_45_scale_m))) * -4.0

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	return Float64(Float64(Float64(Float64(b * a) * Float64(b * a)) / Float64(Float64(x_45_scale * y_45_scale_m) * Float64(x_45_scale * y_45_scale_m))) * -4.0)
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = (((b * a) * (b * a)) / ((x_45_scale * y_45_scale_m) * (x_45_scale * y_45_scale_m))) * -4.0;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := N[(N[(N[(N[(b * a), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale$95$m), $MachinePrecision] * N[(x$45$scale * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]

Derivation

1. Initial program 25.8%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 57.8%

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

7. Applied rewrites 76.8%

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

9. Applied rewrites 93.1%

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

11. Applied rewrites 76.8%

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

12. Final simplification 76.8%

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

Alternative 8: 68.6% accurate, 40.5× speedup

Math

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

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (*
  (* (* (/ (* a a) (* (* x-scale y-scale_m) (* x-scale y-scale_m))) -4.0) b)
  b))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return ((((a * a) / ((x_45_scale * y_45_scale_m) * (x_45_scale * y_45_scale_m))) * -4.0) * b) * b;
}

Fortran

y-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    code = ((((a * a) / ((x_45scale * y_45scale_m) * (x_45scale * y_45scale_m))) * (-4.0d0)) * b) * b
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return ((((a * a) / ((x_45_scale * y_45_scale_m) * (x_45_scale * y_45_scale_m))) * -4.0) * b) * b;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	return ((((a * a) / ((x_45_scale * y_45_scale_m) * (x_45_scale * y_45_scale_m))) * -4.0) * b) * b

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	return Float64(Float64(Float64(Float64(Float64(a * a) / Float64(Float64(x_45_scale * y_45_scale_m) * Float64(x_45_scale * y_45_scale_m))) * -4.0) * b) * b)
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = ((((a * a) / ((x_45_scale * y_45_scale_m) * (x_45_scale * y_45_scale_m))) * -4.0) * b) * b;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := N[(N[(N[(N[(N[(a * a), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale$95$m), $MachinePrecision] * N[(x$45$scale * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]

Derivation

1. Initial program 25.8%

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

3. Taylor expanded in b around 0

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

5. Applied rewrites 50.7%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 56.3%

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

10. Applied rewrites 85.1%

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

12. Applied rewrites 68.3%

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

13. Final simplification 68.3%

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

Alternative 9: 60.8% accurate, 40.5× speedup

Math

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

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (*
  (/ (* (* -4.0 (* a a)) b) (* (* (* x-scale x-scale) y-scale_m) y-scale_m))
  b))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return (((-4.0 * (a * a)) * b) / (((x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)) * b;
}

Fortran

y-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    code = ((((-4.0d0) * (a * a)) * b) / (((x_45scale * x_45scale) * y_45scale_m) * y_45scale_m)) * b
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return (((-4.0 * (a * a)) * b) / (((x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)) * b;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	return (((-4.0 * (a * a)) * b) / (((x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)) * b

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	return Float64(Float64(Float64(Float64(-4.0 * Float64(a * a)) * b) / Float64(Float64(Float64(x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)) * b)
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = (((-4.0 * (a * a)) * b) / (((x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)) * b;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := N[(N[(N[(N[(-4.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale$95$m), $MachinePrecision] * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]

Derivation

1. Initial program 25.8%

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

3. Taylor expanded in b around 0

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

5. Applied rewrites 50.7%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 56.3%

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

10. Applied rewrites 85.1%

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

11. Taylor expanded in angle around 0

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

13. Applied rewrites 62.5%

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

Reproduce

herbie shell --seed 2025021 
(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))))