Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.8% → 78.2%

Time: 32.3s

Alternatives: 5

Speedup: 13.8×

• 33.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 78.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\\ \mathbf{if}\;t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \leq 0:\\ \;\;\;\;{\left(\frac{\frac{\left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\right)}^{2} - \left(4 \cdot \left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)\right) \cdot \frac{\frac{{a}^{2}}{y-scale}}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(a \cdot b\right)}^{2}}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot -4\\ \end{array} \end{array} \]

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)))
   (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) y-scale)))
        0.0)
     (-
      (pow (/ (/ (* (* (* 2.0 (* b b)) t_1) t_2) x-scale) y-scale) 2.0)
      (*
       (* 4.0 (* (/ b x-scale) (/ b x-scale)))
       (/ (/ (pow a 2.0) y-scale) y-scale)))
     (*
      (/ (pow (* a b) 2.0) (* (* y-scale x-scale) (* y-scale x-scale)))
      -4.0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	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;
	double tmp;
	if (((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))) <= 0.0) {
		tmp = pow((((((2.0 * (b * b)) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0) - ((4.0 * ((b / x_45_scale) * (b / x_45_scale))) * ((pow(a, 2.0) / y_45_scale) / y_45_scale));
	} else {
		tmp = (pow((a * b), 2.0) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0;
	}
	return tmp;
}

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;
	double tmp;
	if (((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))) <= 0.0) {
		tmp = Math.pow((((((2.0 * (b * b)) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0) - ((4.0 * ((b / x_45_scale) * (b / x_45_scale))) * ((Math.pow(a, 2.0) / y_45_scale) / y_45_scale));
	} else {
		tmp = (Math.pow((a * b), 2.0) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0;
	}
	return tmp;
}

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
	tmp = 0
	if ((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))) <= 0.0:
		tmp = math.pow((((((2.0 * (b * b)) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0) - ((4.0 * ((b / x_45_scale) * (b / x_45_scale))) * ((math.pow(a, 2.0) / y_45_scale) / y_45_scale))
	else:
		tmp = (math.pow((a * b), 2.0) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0
	return tmp

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)
	tmp = 0.0
	if (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))) <= 0.0)
		tmp = Float64((Float64(Float64(Float64(Float64(Float64(2.0 * Float64(b * b)) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0) - Float64(Float64(4.0 * Float64(Float64(b / x_45_scale) * Float64(b / x_45_scale))) * Float64(Float64((a ^ 2.0) / y_45_scale) / y_45_scale)));
	else
		tmp = Float64(Float64((Float64(a * b) ^ 2.0) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = 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 = 0.0;
	if (((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))) <= 0.0)
		tmp = ((((((2.0 * (b * b)) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0) - ((4.0 * ((b / x_45_scale) * (b / x_45_scale))) * (((a ^ 2.0) / y_45_scale) / y_45_scale));
	else
		tmp = (((a * b) ^ 2.0) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0;
	end
	tmp_2 = tmp;
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]}, If[LessEqual[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], 0.0], N[(N[Power[N[(N[(N[(N[(N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision] - N[(N[(4.0 * N[(N[(b / x$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[a, 2.0], $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]]]]

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))) < 0.0

   1. Initial program 76.9%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \color{blue}{\frac{{b}^{2}}{{x-scale}^{2}}}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 68.7

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

   4. Applied rewrites 68.7%

      \[\leadsto \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \color{blue}{\frac{b \cdot b}{x-scale \cdot x-scale}}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]
   5. Step-by-step derivation

   6. Applied rewrites 75.2%

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

   7. Taylor expanded in angle around 0

      \[\leadsto {\left(\frac{\frac{\left(\left(2 \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2} - \left(4 \cdot \left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)\right) \cdot \frac{\frac{\color{blue}{{a}^{2}}}{y-scale}}{y-scale} \]
   8. Step-by-step derivation

      1. lift-pow.f64 74.7

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

   9. Applied rewrites 74.7%

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

   10. Taylor expanded in a around 0

       \[\leadsto {\left(\frac{\frac{\left(\left(2 \cdot \color{blue}{{b}^{2}}\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2} - \left(4 \cdot \left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right)\right) \cdot \frac{\frac{{a}^{2}}{y-scale}}{y-scale} \]
   11. Step-by-step derivation

       1. pow2 N/A

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

       2. pow2 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 82.3

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

   12. Applied rewrites 82.3%

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

   if 0.0 < (-.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 0.6%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 76.3

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

   6. Applied rewrites 76.3%

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

Alternative 2: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_3 := \cos t\_0\\ t_4 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_3}{x-scale}}{y-scale}\\ \mathbf{if}\;t\_4 \cdot t\_4 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_3\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_3\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \leq -1 \cdot 10^{-259}:\\ \;\;\;\;{\left(\frac{2 \cdot \left({b}^{2} \cdot \left(\cos t\_2 \cdot \sin t\_2\right)\right)}{y-scale \cdot x-scale}\right)}^{2} - \frac{a \cdot a}{y-scale \cdot y-scale} \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(a \cdot b\right)}^{2}}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot -4\\ \end{array} \end{array} \]

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 (* 0.005555555555555556 (* angle PI)))
        (t_3 (cos t_0))
        (t_4
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_3) x-scale)
          y-scale)))
   (if (<=
        (-
         (* t_4 t_4)
         (*
          (*
           4.0
           (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_3) 2.0)) x-scale) x-scale))
          (/ (/ (+ (pow (* a t_3) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))
        -1e-259)
     (-
      (pow
       (/ (* 2.0 (* (pow b 2.0) (* (cos t_2) (sin t_2)))) (* y-scale x-scale))
       2.0)
      (*
       (/ (* a a) (* y-scale y-scale))
       (* (* (/ b x-scale) (/ b x-scale)) 4.0)))
     (*
      (/ (pow (* a b) 2.0) (* (* y-scale x-scale) (* y-scale x-scale)))
      -4.0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_3 = cos(t_0);
	double t_4 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_3) / x_45_scale) / y_45_scale;
	double tmp;
	if (((t_4 * t_4) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_3), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_3), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))) <= -1e-259) {
		tmp = pow(((2.0 * (pow(b, 2.0) * (cos(t_2) * sin(t_2)))) / (y_45_scale * x_45_scale)), 2.0) - (((a * a) / (y_45_scale * y_45_scale)) * (((b / x_45_scale) * (b / x_45_scale)) * 4.0));
	} else {
		tmp = (pow((a * b), 2.0) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0;
	}
	return tmp;
}

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 = 0.005555555555555556 * (angle * Math.PI);
	double t_3 = Math.cos(t_0);
	double t_4 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_3) / x_45_scale) / y_45_scale;
	double tmp;
	if (((t_4 * t_4) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_3), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_3), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))) <= -1e-259) {
		tmp = Math.pow(((2.0 * (Math.pow(b, 2.0) * (Math.cos(t_2) * Math.sin(t_2)))) / (y_45_scale * x_45_scale)), 2.0) - (((a * a) / (y_45_scale * y_45_scale)) * (((b / x_45_scale) * (b / x_45_scale)) * 4.0));
	} else {
		tmp = (Math.pow((a * b), 2.0) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0;
	}
	return tmp;
}

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 = 0.005555555555555556 * (angle * math.pi)
	t_3 = math.cos(t_0)
	t_4 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_3) / x_45_scale) / y_45_scale
	tmp = 0
	if ((t_4 * t_4) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_3), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_3), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))) <= -1e-259:
		tmp = math.pow(((2.0 * (math.pow(b, 2.0) * (math.cos(t_2) * math.sin(t_2)))) / (y_45_scale * x_45_scale)), 2.0) - (((a * a) / (y_45_scale * y_45_scale)) * (((b / x_45_scale) * (b / x_45_scale)) * 4.0))
	else:
		tmp = (math.pow((a * b), 2.0) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0
	return tmp

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 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_3 = cos(t_0)
	t_4 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_3) / x_45_scale) / y_45_scale)
	tmp = 0.0
	if (Float64(Float64(t_4 * t_4) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_3) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_3) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale))) <= -1e-259)
		tmp = Float64((Float64(Float64(2.0 * Float64((b ^ 2.0) * Float64(cos(t_2) * sin(t_2)))) / Float64(y_45_scale * x_45_scale)) ^ 2.0) - Float64(Float64(Float64(a * a) / Float64(y_45_scale * y_45_scale)) * Float64(Float64(Float64(b / x_45_scale) * Float64(b / x_45_scale)) * 4.0)));
	else
		tmp = Float64(Float64((Float64(a * b) ^ 2.0) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = 0.005555555555555556 * (angle * pi);
	t_3 = cos(t_0);
	t_4 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_3) / x_45_scale) / y_45_scale;
	tmp = 0.0;
	if (((t_4 * t_4) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_3) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_3) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale))) <= -1e-259)
		tmp = (((2.0 * ((b ^ 2.0) * (cos(t_2) * sin(t_2)))) / (y_45_scale * x_45_scale)) ^ 2.0) - (((a * a) / (y_45_scale * y_45_scale)) * (((b / x_45_scale) * (b / x_45_scale)) * 4.0));
	else
		tmp = (((a * b) ^ 2.0) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0;
	end
	tmp_2 = tmp;
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[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$4 = 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$3), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$4 * t$95$4), $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$3), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$3), $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], -1e-259], N[(N[Power[N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] * N[(N[Cos[t$95$2], $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - N[(N[(N[(a * a), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(b / x$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]]]]]

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))) < -1.0000000000000001e-259

   1. Initial program 39.9%

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

   3. Applied rewrites 29.1%

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

   4. Taylor expanded in angle around 0

      \[\leadsto {\left(\frac{\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot 2\right)\right)}{y-scale \cdot x-scale}\right)}^{2} - \frac{{\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot b\right)}^{2} + {\left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot a\right)}^{2}}{y-scale \cdot y-scale} \cdot \left(\color{blue}{\frac{{b}^{2}}{{x-scale}^{2}}} \cdot 4\right) \]
   5. Step-by-step derivation

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 31.2

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

   6. Applied rewrites 31.2%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sin.f64 N/A

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

   9. Applied rewrites 42.9%

      \[\leadsto {\left(\frac{\color{blue}{2 \cdot \left({b}^{2} \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{y-scale \cdot x-scale}\right)}^{2} - \frac{{\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot b\right)}^{2} + {\left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot a\right)}^{2}}{y-scale \cdot y-scale} \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot 4\right) \]

   10. Taylor expanded in angle around 0

       \[\leadsto {\left(\frac{2 \cdot \left({b}^{2} \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{y-scale \cdot x-scale}\right)}^{2} - \frac{\color{blue}{{a}^{2}}}{y-scale \cdot y-scale} \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot 4\right) \]
   11. Step-by-step derivation

       1. pow2 N/A

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

       2. lift-*.f64 46.9

          \[\leadsto {\left(\frac{2 \cdot \left({b}^{2} \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{y-scale \cdot x-scale}\right)}^{2} - \frac{a \cdot \color{blue}{a}}{y-scale \cdot y-scale} \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot 4\right) \]

   12. Applied rewrites 46.9%

       \[\leadsto {\left(\frac{2 \cdot \left({b}^{2} \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{y-scale \cdot x-scale}\right)}^{2} - \frac{\color{blue}{a \cdot a}}{y-scale \cdot y-scale} \cdot \left(\left(\frac{b}{x-scale} \cdot \frac{b}{x-scale}\right) \cdot 4\right) \]

   if -1.0000000000000001e-259 < (-.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 23.6%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 79.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 79.0

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

   6. Applied rewrites 79.0%

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

Alternative 3: 77.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\\ \mathbf{if}\;t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \leq -1 \cdot 10^{-259}:\\ \;\;\;\;\frac{4}{x-scale} \cdot \frac{{a}^{2} \cdot \left(-1 \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right)}{x-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(a \cdot b\right)}^{2}}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot -4\\ \end{array} \end{array} \]

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)))
   (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) y-scale)))
        -1e-259)
     (*
      (/ 4.0 x-scale)
      (/ (* (pow a 2.0) (* -1.0 (/ (* b b) (* y-scale y-scale)))) x-scale))
     (*
      (/ (pow (* a b) 2.0) (* (* y-scale x-scale) (* y-scale x-scale)))
      -4.0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	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;
	double tmp;
	if (((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))) <= -1e-259) {
		tmp = (4.0 / x_45_scale) * ((pow(a, 2.0) * (-1.0 * ((b * b) / (y_45_scale * y_45_scale)))) / x_45_scale);
	} else {
		tmp = (pow((a * b), 2.0) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0;
	}
	return tmp;
}

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;
	double tmp;
	if (((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))) <= -1e-259) {
		tmp = (4.0 / x_45_scale) * ((Math.pow(a, 2.0) * (-1.0 * ((b * b) / (y_45_scale * y_45_scale)))) / x_45_scale);
	} else {
		tmp = (Math.pow((a * b), 2.0) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0;
	}
	return tmp;
}

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
	tmp = 0
	if ((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))) <= -1e-259:
		tmp = (4.0 / x_45_scale) * ((math.pow(a, 2.0) * (-1.0 * ((b * b) / (y_45_scale * y_45_scale)))) / x_45_scale)
	else:
		tmp = (math.pow((a * b), 2.0) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0
	return tmp

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)
	tmp = 0.0
	if (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))) <= -1e-259)
		tmp = Float64(Float64(4.0 / x_45_scale) * Float64(Float64((a ^ 2.0) * Float64(-1.0 * Float64(Float64(b * b) / Float64(y_45_scale * y_45_scale)))) / x_45_scale));
	else
		tmp = Float64(Float64((Float64(a * b) ^ 2.0) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = 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 = 0.0;
	if (((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))) <= -1e-259)
		tmp = (4.0 / x_45_scale) * (((a ^ 2.0) * (-1.0 * ((b * b) / (y_45_scale * y_45_scale)))) / x_45_scale);
	else
		tmp = (((a * b) ^ 2.0) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0;
	end
	tmp_2 = tmp;
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]}, If[LessEqual[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], -1e-259], N[(N[(4.0 / x$45$scale), $MachinePrecision] * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[(-1.0 * N[(N[(b * b), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]]]]]

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))) < -1.0000000000000001e-259

   1. Initial program 39.9%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 25.9%

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

   6. Applied rewrites 28.6%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 51.6%

      \[\leadsto \frac{4}{x-scale} \cdot \frac{{a}^{2} \cdot \left(-2 \cdot \frac{{b}^{2} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{y-scale}^{2}} - \left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{2}}\right)\right)}{x-scale} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \frac{4}{x-scale} \cdot \frac{{a}^{2} \cdot \left(-1 \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)}{x-scale} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{4}{x-scale} \cdot \frac{{a}^{2} \cdot \left(-1 \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)}{x-scale} \]

       2. lower-/.f64 N/A

          \[\leadsto \frac{4}{x-scale} \cdot \frac{{a}^{2} \cdot \left(-1 \cdot \frac{{b}^{2}}{{y-scale}^{2}}\right)}{x-scale} \]

       3. pow2 N/A

          \[\leadsto \frac{4}{x-scale} \cdot \frac{{a}^{2} \cdot \left(-1 \cdot \frac{b \cdot b}{{y-scale}^{2}}\right)}{x-scale} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{4}{x-scale} \cdot \frac{{a}^{2} \cdot \left(-1 \cdot \frac{b \cdot b}{{y-scale}^{2}}\right)}{x-scale} \]

       5. pow2 N/A

          \[\leadsto \frac{4}{x-scale} \cdot \frac{{a}^{2} \cdot \left(-1 \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right)}{x-scale} \]

       6. lift-*.f64 57.8

          \[\leadsto \frac{4}{x-scale} \cdot \frac{{a}^{2} \cdot \left(-1 \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right)}{x-scale} \]

   12. Applied rewrites 57.8%

       \[\leadsto \frac{4}{x-scale} \cdot \frac{{a}^{2} \cdot \left(-1 \cdot \frac{b \cdot b}{y-scale \cdot y-scale}\right)}{x-scale} \]

   if -1.0000000000000001e-259 < (-.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 23.6%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 79.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 79.0

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

   6. Applied rewrites 79.0%

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

Alternative 4: 78.1% accurate, 13.8× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (/ (pow (* a b) 2.0) (* (* y-scale x-scale) (* y-scale x-scale))) -4.0))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (pow((a * b), 2.0) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (((a * b) ** 2.0d0) / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * (-4.0d0)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (Math.pow((a * b), 2.0) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (math.pow((a * b), 2.0) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64((Float64(a * b) ^ 2.0) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * -4.0)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = (((a * b) ^ 2.0) / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * -4.0;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]

Derivation

1. Initial program 24.8%

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

2. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 78.1%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 78.1

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

6. Applied rewrites 78.1%

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

Alternative 5: 61.6% accurate, 13.8× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (* -4.0 (/ (* a a) (pow (* x-scale y-scale) 2.0))) (* b b)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (-4.0 * ((a * a) / pow((x_45_scale * y_45_scale), 2.0))) * (b * b);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = ((-4.0d0) * ((a * a) / ((x_45scale * y_45scale) ** 2.0d0))) * (b * b)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (-4.0 * ((a * a) / Math.pow((x_45_scale * y_45_scale), 2.0))) * (b * b);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (-4.0 * ((a * a) / math.pow((x_45_scale * y_45_scale), 2.0))) * (b * b)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(-4.0 * Float64(Float64(a * a) / (Float64(x_45_scale * y_45_scale) ^ 2.0))) * Float64(b * b))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = (-4.0 * ((a * a) / ((x_45_scale * y_45_scale) ^ 2.0))) * (b * b);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(-4.0 * N[(N[(a * a), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.8%

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

2. Taylor expanded in b around 0

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

4. Applied rewrites 54.5%

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

5. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. pow-prod-down N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-*.f64 61.6

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

7. Applied rewrites 61.6%

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

Reproduce

herbie shell --seed 2025092 
(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))))