Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.8% → 89.2%

Time: 30.8s

Alternatives: 5

Speedup: 1905.0×

• 32.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = 4.849961991997741e+305
  2. b = -2.399933791491091e+101
  3. angle = -9.32838301844921e+219
  4. x-scale = 3.7044223789502366e-298
  5. y-scale = 1.1036683768816112e-206

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: 24.8% 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: 89.2% accurate, 35.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((b * -4.0) * ((b * a) / (x_45_scale * y_45_scale))) * (a / (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 = ((b * (-4.0d0)) * ((b * a) / (x_45scale * y_45scale))) * (a / (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 ((b * -4.0) * ((b * a) / (x_45_scale * y_45_scale))) * (a / (x_45_scale * y_45_scale));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 26.1%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
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. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 55.4

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

5. Applied rewrites 55.4%

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

7. Applied rewrites 65.8%

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

9. Applied rewrites 83.0%

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

11. Applied rewrites 90.5%

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

12. Final simplification 90.5%

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

Alternative 2: 48.6% accurate, 35.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.1 \cdot 10^{-195}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot -4\right) \cdot \left(b \cdot \frac{a \cdot a}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= b 6.1e-195)
   0.0
   (*
    (* b -4.0)
    (* b (/ (* a a) (* x-scale (* x-scale (* y-scale y-scale))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 6.1e-195) {
		tmp = 0.0;
	} else {
		tmp = (b * -4.0) * (b * ((a * a) / (x_45_scale * (x_45_scale * (y_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) :: tmp
    if (b <= 6.1d-195) then
        tmp = 0.0d0
    else
        tmp = (b * (-4.0d0)) * (b * ((a * a) / (x_45scale * (x_45scale * (y_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 tmp;
	if (b <= 6.1e-195) {
		tmp = 0.0;
	} else {
		tmp = (b * -4.0) * (b * ((a * a) / (x_45_scale * (x_45_scale * (y_45_scale * y_45_scale)))));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b <= 6.1e-195:
		tmp = 0.0
	else:
		tmp = (b * -4.0) * (b * ((a * a) / (x_45_scale * (x_45_scale * (y_45_scale * y_45_scale)))))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b <= 6.1e-195)
		tmp = 0.0;
	else
		tmp = Float64(Float64(b * -4.0) * Float64(b * Float64(Float64(a * a) / Float64(x_45_scale * Float64(x_45_scale * Float64(y_45_scale * y_45_scale))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b <= 6.1e-195)
		tmp = 0.0;
	else
		tmp = (b * -4.0) * (b * ((a * a) / (x_45_scale * (x_45_scale * (y_45_scale * y_45_scale)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b, 6.1e-195], 0.0, N[(N[(b * -4.0), $MachinePrecision] * N[(b * N[(N[(a * a), $MachinePrecision] / N[(x$45$scale * N[(x$45$scale * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.1000000000000003e-195

   1. Initial program 27.5%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{1}{8100} \cdot \frac{{\mathsf{PI}\left(\right)}^{2} \cdot {\left({b}^{2} - {a}^{2}\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} - 4 \cdot \left(\frac{{a}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{b}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right) - 4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   5. Applied rewrites 16.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \mathsf{fma}\left(0.0001234567901234568, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \pi\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}, -4 \cdot \mathsf{fma}\left(b, b \cdot \frac{\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, a \cdot a, 3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}, \frac{\left(a \cdot a\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, b \cdot b, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}\right)\right), \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto {angle}^{2} \cdot \color{blue}{\left(\frac{-1}{8100} \cdot \frac{{a}^{4} \cdot {\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{1}{8100} \cdot \frac{{a}^{4} \cdot {\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 27.4%

      \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{0} \]

   9. Taylor expanded in angle around 0

      \[\leadsto 0 \]
   10. Step-by-step derivation

   11. Applied rewrites 35.0%

       \[\leadsto 0 \]

   if 6.1000000000000003e-195 < b

   1. Initial program 23.8%

      \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   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. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 58.5

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

   5. Applied rewrites 58.5%

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

   7. Applied rewrites 70.4%

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

   9. Applied rewrites 65.8%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.1 \cdot 10^{-195}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot -4\right) \cdot \left(b \cdot \frac{a \cdot a}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.0% accurate, 40.5× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (* b -4.0) (* (* b a) (/ a (* 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 (b * -4.0) * ((b * a) * (a / (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 = (b * (-4.0d0)) * ((b * a) * (a / (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 (b * -4.0) * ((b * a) * (a / (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 (b * -4.0) * ((b * a) * (a / (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(b * -4.0) * Float64(Float64(b * a) * Float64(a / 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 = (b * -4.0) * ((b * a) * (a / (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[(b * -4.0), $MachinePrecision] * N[(N[(b * a), $MachinePrecision] * N[(a / N[(x$45$scale * N[(y$45$scale * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.1%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
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. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 55.4

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

5. Applied rewrites 55.4%

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

7. Applied rewrites 65.8%

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

9. Applied rewrites 83.0%

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

11. Applied rewrites 79.0%

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

12. Final simplification 79.0%

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

Alternative 4: 66.1% accurate, 40.5× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (* b -4.0) (* b (/ (* a a) (* 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 (b * -4.0) * (b * ((a * a) / (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 = (b * (-4.0d0)) * (b * ((a * a) / (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 (b * -4.0) * (b * ((a * a) / (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 (b * -4.0) * (b * ((a * a) / (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(b * -4.0) * Float64(b * Float64(Float64(a * a) / 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 = (b * -4.0) * (b * ((a * a) / (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[(b * -4.0), $MachinePrecision] * N[(b * N[(N[(a * a), $MachinePrecision] / N[(x$45$scale * N[(y$45$scale * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.1%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
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. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 55.4

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

5. Applied rewrites 55.4%

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

7. Applied rewrites 65.8%

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

9. Applied rewrites 83.0%

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

11. Applied rewrites 69.4%

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

12. Final simplification 69.4%

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

Alternative 5: 35.7% accurate, 1905.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale) :precision binary64 0.0)

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.0;
}

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
    code = 0.0d0
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.0;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 0.0

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return 0.0
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := 0.0

Derivation

1. Initial program 26.1%

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

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{1}{8100} \cdot \frac{{\mathsf{PI}\left(\right)}^{2} \cdot {\left({b}^{2} - {a}^{2}\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} - 4 \cdot \left(\frac{{a}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{{b}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)\right) - 4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

5. Applied rewrites 15.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \mathsf{fma}\left(0.0001234567901234568, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot \pi\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}, -4 \cdot \mathsf{fma}\left(b, b \cdot \frac{\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, a \cdot a, 3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}, \frac{\left(a \cdot a\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, b \cdot b, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}\right)\right), \frac{\left(-4 \cdot \left(b \cdot b\right)\right) \cdot \left(a \cdot a\right)}{x-scale \cdot \left(x-scale \cdot \left(y-scale \cdot y-scale\right)\right)}\right)} \]

6. Taylor expanded in b around 0

   \[\leadsto {angle}^{2} \cdot \color{blue}{\left(\frac{-1}{8100} \cdot \frac{{a}^{4} \cdot {\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + \frac{1}{8100} \cdot \frac{{a}^{4} \cdot {\mathsf{PI}\left(\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 25.2%

   \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{0} \]

9. Taylor expanded in angle around 0

   \[\leadsto 0 \]
10. Step-by-step derivation

11. Applied rewrites 35.2%

    \[\leadsto 0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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))))