Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.1% → 95.0%

Time: 14.7s

Alternatives: 9

Speedup: 20.4×

• 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: 25.1% 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: 95.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(-a\right) \cdot \frac{b}{x-scale}}{y-scale}\\ t_1 := \frac{angle}{180} \cdot \pi\\ t_2 := \sin t\_1\\ t_3 := \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\\ t_4 := \cos t\_1\\ t_5 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_2\right) \cdot t\_4}{x-scale}}{y-scale}\\ \mathbf{if}\;t\_5 \cdot t\_5 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_4\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_4\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{y-scale}}{y-scale} \leq 10^{+43}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(t\_3 \cdot t\_3\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (* (- a) (/ b x-scale)) y-scale))
        (t_1 (* (/ angle 180.0) PI))
        (t_2 (sin t_1))
        (t_3 (/ (* (- a) b) (* y-scale x-scale)))
        (t_4 (cos t_1))
        (t_5
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_2) t_4) x-scale)
          y-scale)))
   (if (<=
        (-
         (* t_5 t_5)
         (*
          (*
           4.0
           (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_4) 2.0)) x-scale) x-scale))
          (/ (/ (+ (pow (* a t_4) 2.0) (pow (* b t_2) 2.0)) y-scale) y-scale)))
        1e+43)
     (* (* t_0 t_0) -4.0)
     (* (* t_3 t_3) -4.0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (-a * (b / x_45_scale)) / y_45_scale;
	double t_1 = (angle / 180.0) * ((double) M_PI);
	double t_2 = sin(t_1);
	double t_3 = (-a * b) / (y_45_scale * x_45_scale);
	double t_4 = cos(t_1);
	double t_5 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_2) * t_4) / x_45_scale) / y_45_scale;
	double tmp;
	if (((t_5 * t_5) - ((4.0 * (((pow((a * t_2), 2.0) + pow((b * t_4), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_4), 2.0) + pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale))) <= 1e+43) {
		tmp = (t_0 * t_0) * -4.0;
	} else {
		tmp = (t_3 * t_3) * -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 = (-a * (b / x_45_scale)) / y_45_scale;
	double t_1 = (angle / 180.0) * Math.PI;
	double t_2 = Math.sin(t_1);
	double t_3 = (-a * b) / (y_45_scale * x_45_scale);
	double t_4 = Math.cos(t_1);
	double t_5 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_2) * t_4) / x_45_scale) / y_45_scale;
	double tmp;
	if (((t_5 * t_5) - ((4.0 * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_4), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_4), 2.0) + Math.pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale))) <= 1e+43) {
		tmp = (t_0 * t_0) * -4.0;
	} else {
		tmp = (t_3 * t_3) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (-a * (b / x_45_scale)) / y_45_scale
	t_1 = (angle / 180.0) * math.pi
	t_2 = math.sin(t_1)
	t_3 = (-a * b) / (y_45_scale * x_45_scale)
	t_4 = math.cos(t_1)
	t_5 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_2) * t_4) / x_45_scale) / y_45_scale
	tmp = 0
	if ((t_5 * t_5) - ((4.0 * (((math.pow((a * t_2), 2.0) + math.pow((b * t_4), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_4), 2.0) + math.pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale))) <= 1e+43:
		tmp = (t_0 * t_0) * -4.0
	else:
		tmp = (t_3 * t_3) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(-a) * Float64(b / x_45_scale)) / y_45_scale)
	t_1 = Float64(Float64(angle / 180.0) * pi)
	t_2 = sin(t_1)
	t_3 = Float64(Float64(Float64(-a) * b) / Float64(y_45_scale * x_45_scale))
	t_4 = cos(t_1)
	t_5 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_2) * t_4) / x_45_scale) / y_45_scale)
	tmp = 0.0
	if (Float64(Float64(t_5 * t_5) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_4) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_4) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / y_45_scale) / y_45_scale))) <= 1e+43)
		tmp = Float64(Float64(t_0 * t_0) * -4.0);
	else
		tmp = Float64(Float64(t_3 * t_3) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (-a * (b / x_45_scale)) / y_45_scale;
	t_1 = (angle / 180.0) * pi;
	t_2 = sin(t_1);
	t_3 = (-a * b) / (y_45_scale * x_45_scale);
	t_4 = cos(t_1);
	t_5 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_2) * t_4) / x_45_scale) / y_45_scale;
	tmp = 0.0;
	if (((t_5 * t_5) - ((4.0 * (((((a * t_2) ^ 2.0) + ((b * t_4) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_4) ^ 2.0) + ((b * t_2) ^ 2.0)) / y_45_scale) / y_45_scale))) <= 1e+43)
		tmp = (t_0 * t_0) * -4.0;
	else
		tmp = (t_3 * t_3) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[((-a) * N[(b / x$45$scale), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(N[((-a) * b), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$4), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$5 * t$95$5), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$4), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$4), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+43], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(t$95$3 * t$95$3), $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))) < 1.00000000000000001e43

   1. Initial program 25.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 48.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 48.0

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

   6. Applied rewrites 59.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. swap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. sqr-neg-rev N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

   8. Applied rewrites 93.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 88.9

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

   10. Applied rewrites 88.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. times-frac N/A

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

       5. associate-*l/ N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 93.9

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

   12. Applied rewrites 93.9%

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

   if 1.00000000000000001e43 < (-.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 25.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 48.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 48.0

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

   6. Applied rewrites 59.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. swap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. sqr-neg-rev N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

   8. Applied rewrites 93.9%

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

Alternative 2: 93.9% accurate, 18.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.1%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 48.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 48.0

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

6. Applied rewrites 59.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. associate-*r/ N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. swap-sqr N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. sqr-neg-rev N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f64 N/A

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

   16. times-frac N/A

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

   17. lower-*.f64 N/A

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

8. Applied rewrites 93.9%

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

Alternative 3: 93.9% accurate, 20.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.1%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 48.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 48.0

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

6. Applied rewrites 59.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. associate-*r/ N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. swap-sqr N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l* N/A

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

   14. lift-*.f64 N/A

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

   15. times-frac N/A

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

   16. lower-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. associate-/l* N/A

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

   20. lower-*.f64 N/A

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

   21. lower-/.f64 N/A

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

8. Applied rewrites 93.9%

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

Alternative 4: 89.9% accurate, 20.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.1%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 48.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-prod-down N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 64.6

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-pow.f64 N/A

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

   18. pow2 N/A

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

   19. lift-pow.f64 N/A

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

   20. pow2 N/A

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

   21. unswap-sqr N/A

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

   22. lower-*.f64 N/A

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

   23. lower-*.f64 N/A

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

   24. lower-*.f64 83.8

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

6. Applied rewrites 83.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. lift-*.f64 N/A

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

8. Applied rewrites 74.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l/ N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. times-frac N/A

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

   13. remove-double-neg N/A

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

   14. lift-*.f64 N/A

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

   15. distribute-lft-neg-out N/A

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

   16. lift-neg.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. distribute-neg-frac N/A

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

   19. lift-/.f64 N/A

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

10. Applied rewrites 89.9%

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

Alternative 5: 80.8% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.86 \cdot 10^{-174}:\\ \;\;\;\;\left(b \cdot \frac{\left(a \cdot b\right) \cdot a}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{\frac{a}{y-scale \cdot x-scale}}{y-scale \cdot x-scale} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= b 1.86e-174)
   (* (* b (/ (* (* a b) a) (* (* (* y-scale x-scale) x-scale) y-scale))) -4.0)
   (*
    (* (* (* (/ (/ a (* y-scale x-scale)) (* y-scale x-scale)) a) b) b)
    -4.0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 1.86e-174) {
		tmp = (b * (((a * b) * a) / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))) * -4.0;
	} else {
		tmp = (((((a / (y_45_scale * x_45_scale)) / (y_45_scale * x_45_scale)) * a) * b) * b) * -4.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (b <= 1.86d-174) then
        tmp = (b * (((a * b) * a) / (((y_45scale * x_45scale) * x_45scale) * y_45scale))) * (-4.0d0)
    else
        tmp = (((((a / (y_45scale * x_45scale)) / (y_45scale * x_45scale)) * a) * b) * b) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 1.86e-174) {
		tmp = (b * (((a * b) * a) / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))) * -4.0;
	} else {
		tmp = (((((a / (y_45_scale * x_45_scale)) / (y_45_scale * x_45_scale)) * a) * b) * b) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b <= 1.86e-174:
		tmp = (b * (((a * b) * a) / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))) * -4.0
	else:
		tmp = (((((a / (y_45_scale * x_45_scale)) / (y_45_scale * x_45_scale)) * a) * b) * b) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b <= 1.86e-174)
		tmp = Float64(Float64(b * Float64(Float64(Float64(a * b) * a) / Float64(Float64(Float64(y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))) * -4.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(a / Float64(y_45_scale * x_45_scale)) / Float64(y_45_scale * x_45_scale)) * a) * b) * b) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b <= 1.86e-174)
		tmp = (b * (((a * b) * a) / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))) * -4.0;
	else
		tmp = (((((a / (y_45_scale * x_45_scale)) / (y_45_scale * x_45_scale)) * a) * b) * b) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b, 1.86e-174], N[(N[(b * N[(N[(N[(a * b), $MachinePrecision] * a), $MachinePrecision] / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(a / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.86e-174

   1. Initial program 25.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 48.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 48.0

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

   6. Applied rewrites 59.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. swap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.0

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 74.0

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

   8. Applied rewrites 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 74.4

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

   10. Applied rewrites 74.4%

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

   if 1.86e-174 < b

   1. Initial program 25.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 48.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-prod-down N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.6

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-pow.f64 N/A

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

      18. pow2 N/A

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

      19. lift-pow.f64 N/A

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

      20. pow2 N/A

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

      21. unswap-sqr N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 83.8

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

   6. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

   8. Applied rewrites 74.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 83.4

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

   10. Applied rewrites 83.4%

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

Alternative 6: 78.6% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y-scale \leq 5.2 \cdot 10^{-168}:\\ \;\;\;\;\left(b \cdot \frac{\left(a \cdot b\right) \cdot a}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}\right) \cdot -4\\ \mathbf{elif}\;y-scale \leq 1.58 \cdot 10^{+178}:\\ \;\;\;\;\left(\left(a \cdot b\right) \cdot \frac{a \cdot b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= y-scale 5.2e-168)
   (* (* b (/ (* (* a b) a) (* (* (* y-scale x-scale) x-scale) y-scale))) -4.0)
   (if (<= y-scale 1.58e+178)
     (*
      (* (* a b) (/ (* a b) (* (* (* y-scale x-scale) y-scale) x-scale)))
      -4.0)
     (*
      (* (* (* (/ a (* (* y-scale x-scale) (* y-scale x-scale))) a) b) b)
      -4.0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 5.2e-168) {
		tmp = (b * (((a * b) * a) / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))) * -4.0;
	} else if (y_45_scale <= 1.58e+178) {
		tmp = ((a * b) * ((a * b) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))) * -4.0;
	} else {
		tmp = ((((a / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * a) * b) * b) * -4.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (y_45scale <= 5.2d-168) then
        tmp = (b * (((a * b) * a) / (((y_45scale * x_45scale) * x_45scale) * y_45scale))) * (-4.0d0)
    else if (y_45scale <= 1.58d+178) then
        tmp = ((a * b) * ((a * b) / (((y_45scale * x_45scale) * y_45scale) * x_45scale))) * (-4.0d0)
    else
        tmp = ((((a / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * a) * b) * b) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 5.2e-168) {
		tmp = (b * (((a * b) * a) / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))) * -4.0;
	} else if (y_45_scale <= 1.58e+178) {
		tmp = ((a * b) * ((a * b) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))) * -4.0;
	} else {
		tmp = ((((a / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * a) * b) * b) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if y_45_scale <= 5.2e-168:
		tmp = (b * (((a * b) * a) / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))) * -4.0
	elif y_45_scale <= 1.58e+178:
		tmp = ((a * b) * ((a * b) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))) * -4.0
	else:
		tmp = ((((a / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * a) * b) * b) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (y_45_scale <= 5.2e-168)
		tmp = Float64(Float64(b * Float64(Float64(Float64(a * b) * a) / Float64(Float64(Float64(y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))) * -4.0);
	elseif (y_45_scale <= 1.58e+178)
		tmp = Float64(Float64(Float64(a * b) * Float64(Float64(a * b) / Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))) * -4.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * a) * b) * b) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (y_45_scale <= 5.2e-168)
		tmp = (b * (((a * b) * a) / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))) * -4.0;
	elseif (y_45_scale <= 1.58e+178)
		tmp = ((a * b) * ((a * b) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))) * -4.0;
	else
		tmp = ((((a / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * a) * b) * b) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[y$45$scale, 5.2e-168], N[(N[(b * N[(N[(N[(a * b), $MachinePrecision] * a), $MachinePrecision] / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], If[LessEqual[y$45$scale, 1.58e+178], N[(N[(N[(a * b), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(N[(N[(N[(a / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * -4.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 5.2000000000000002e-168

   1. Initial program 25.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 48.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 48.0

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

   6. Applied rewrites 59.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. swap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.0

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 74.0

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

   8. Applied rewrites 74.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 74.4

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

   10. Applied rewrites 74.4%

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

   if 5.2000000000000002e-168 < y-scale < 1.58e178

   1. Initial program 25.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 48.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 48.0

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

   6. Applied rewrites 59.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. swap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 80.4

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 80.4

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

   8. Applied rewrites 80.4%

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

   if 1.58e178 < y-scale

   1. Initial program 25.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 48.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-prod-down N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.6

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-pow.f64 N/A

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

      18. pow2 N/A

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

      19. lift-pow.f64 N/A

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

      20. pow2 N/A

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

      21. unswap-sqr N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 83.8

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

   6. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

   8. Applied rewrites 74.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 77.4

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

   10. Applied rewrites 77.4%

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

Alternative 7: 78.1% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y-scale \leq 1.58 \cdot 10^{+178}:\\ \;\;\;\;\left(\left(a \cdot b\right) \cdot \frac{a \cdot b}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot a\right) \cdot b\right) \cdot b\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= y-scale 1.58e+178)
   (* (* (* a b) (/ (* a b) (* (* (* y-scale x-scale) y-scale) x-scale))) -4.0)
   (*
    (* (* (* (/ a (* (* y-scale x-scale) (* y-scale x-scale))) a) b) b)
    -4.0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 1.58e+178) {
		tmp = ((a * b) * ((a * b) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))) * -4.0;
	} else {
		tmp = ((((a / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * a) * b) * b) * -4.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: tmp
    if (y_45scale <= 1.58d+178) then
        tmp = ((a * b) * ((a * b) / (((y_45scale * x_45scale) * y_45scale) * x_45scale))) * (-4.0d0)
    else
        tmp = ((((a / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * a) * b) * b) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (y_45_scale <= 1.58e+178) {
		tmp = ((a * b) * ((a * b) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))) * -4.0;
	} else {
		tmp = ((((a / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * a) * b) * b) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if y_45_scale <= 1.58e+178:
		tmp = ((a * b) * ((a * b) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))) * -4.0
	else:
		tmp = ((((a / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * a) * b) * b) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (y_45_scale <= 1.58e+178)
		tmp = Float64(Float64(Float64(a * b) * Float64(Float64(a * b) / Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))) * -4.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * a) * b) * b) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (y_45_scale <= 1.58e+178)
		tmp = ((a * b) * ((a * b) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))) * -4.0;
	else
		tmp = ((((a / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * a) * b) * b) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[y$45$scale, 1.58e+178], N[(N[(N[(a * b), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(N[(N[(N[(a / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.58e178

   1. Initial program 25.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 48.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 48.0

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

   6. Applied rewrites 59.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. swap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 80.4

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 80.4

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

   8. Applied rewrites 80.4%

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

   if 1.58e178 < y-scale

   1. Initial program 25.1%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 48.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-prod-down N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.6

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-pow.f64 N/A

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

      18. pow2 N/A

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

      19. lift-pow.f64 N/A

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

      20. pow2 N/A

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

      21. unswap-sqr N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 83.8

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

   6. Applied rewrites 83.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

   8. Applied rewrites 74.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 77.4

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

   10. Applied rewrites 77.4%

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

Alternative 8: 77.4% accurate, 20.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.1%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 48.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-prod-down N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 64.6

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-pow.f64 N/A

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

   18. pow2 N/A

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

   19. lift-pow.f64 N/A

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

   20. pow2 N/A

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

   21. unswap-sqr N/A

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

   22. lower-*.f64 N/A

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

   23. lower-*.f64 N/A

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

   24. lower-*.f64 83.8

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

6. Applied rewrites 83.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. lift-*.f64 N/A

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

8. Applied rewrites 74.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 78.1

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

10. Applied rewrites 78.1%

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

Alternative 9: 77.3% accurate, 20.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.1%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 48.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-prod-down N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 64.6

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-pow.f64 N/A

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

   18. pow2 N/A

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

   19. lift-pow.f64 N/A

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

   20. pow2 N/A

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

   21. unswap-sqr N/A

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

   22. lower-*.f64 N/A

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

   23. lower-*.f64 N/A

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

   24. lower-*.f64 83.8

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

6. Applied rewrites 83.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. lift-*.f64 N/A

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

8. Applied rewrites 74.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lower-*.f64 77.4

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

10. Applied rewrites 77.4%

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

Reproduce

herbie shell --seed 2025156 
(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))))