b from scale-rotated-ellipse

Percentage Accurate: 0.0% → 29.1%

Time: 34.4s

Alternatives: 5

Speedup: 8.0×

• could not determine a ground truth

  1. a = 7.503054192584326e+138
  2. b = 3.804532722141395e-279
  3. angle = 2.124995831850641e-31
  4. x-scale = 8.694998556165309e+302
  5. y-scale = 1.7439670242898022e-192

Specification

Math

\[\begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\\ t_5 := \left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\\ t_6 := \frac{4 \cdot t\_5}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_6\right) \cdot t\_5\right) \cdot \left(\left(t\_4 + t\_3\right) - \sqrt{{\left(t\_4 - t\_3\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\right)}^{2}}\right)}}{t\_6} \end{array} \]

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
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
        (t_5 (* (* b a) (* b (- a))))
        (t_6 (/ (* 4.0 t_5) (pow (* x-scale y-scale) 2.0))))
   (/
    (-
     (sqrt
      (*
       (* (* 2.0 t_6) t_5)
       (-
        (+ t_4 t_3)
        (sqrt
         (+
          (pow (- t_4 t_3) 2.0)
          (pow
           (/
            (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
            y-scale)
           2.0)))))))
    t_6)))

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 = ((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / pow((x_45_scale * y_45_scale), 2.0);
	return -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - sqrt((pow((t_4 - t_3), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

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 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / Math.pow((x_45_scale * y_45_scale), 2.0);
	return -Math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - Math.sqrt((Math.pow((t_4 - t_3), 2.0) + Math.pow((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

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 = ((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale
	t_4 = ((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale
	t_5 = (b * a) * (b * -a)
	t_6 = (4.0 * t_5) / math.pow((x_45_scale * y_45_scale), 2.0)
	return -math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - math.sqrt((math.pow((t_4 - t_3), 2.0) + math.pow((((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6

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(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)
	t_5 = Float64(Float64(b * a) * Float64(b * Float64(-a)))
	t_6 = Float64(Float64(4.0 * t_5) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_6) * t_5) * Float64(Float64(t_4 + t_3) - sqrt(Float64((Float64(t_4 - t_3) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0))))))) / t_6)
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 = ((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_4 = ((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale;
	t_5 = (b * a) * (b * -a);
	t_6 = (4.0 * t_5) / ((x_45_scale * y_45_scale) ^ 2.0);
	tmp = -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - sqrt((((t_4 - t_3) ^ 2.0) + ((((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0)))))) / t_6;
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[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]}, Block[{t$95$4 = 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]}, Block[{t$95$5 = N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(4.0 * t$95$5), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$6), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(t$95$4 + t$95$3), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$4 - t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[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], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]]]]]]]]

Initial Program: 0.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\\ t_5 := \left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\\ t_6 := \frac{4 \cdot t\_5}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_6\right) \cdot t\_5\right) \cdot \left(\left(t\_4 + t\_3\right) - \sqrt{{\left(t\_4 - t\_3\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\right)}^{2}}\right)}}{t\_6} \end{array} \]

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
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale))
        (t_4
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
        (t_5 (* (* b a) (* b (- a))))
        (t_6 (/ (* 4.0 t_5) (pow (* x-scale y-scale) 2.0))))
   (/
    (-
     (sqrt
      (*
       (* (* 2.0 t_6) t_5)
       (-
        (+ t_4 t_3)
        (sqrt
         (+
          (pow (- t_4 t_3) 2.0)
          (pow
           (/
            (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
            y-scale)
           2.0)))))))
    t_6)))

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 = ((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / pow((x_45_scale * y_45_scale), 2.0);
	return -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - sqrt((pow((t_4 - t_3), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

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 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / Math.pow((x_45_scale * y_45_scale), 2.0);
	return -Math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - Math.sqrt((Math.pow((t_4 - t_3), 2.0) + Math.pow((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

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 = ((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale
	t_4 = ((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale
	t_5 = (b * a) * (b * -a)
	t_6 = (4.0 * t_5) / math.pow((x_45_scale * y_45_scale), 2.0)
	return -math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - math.sqrt((math.pow((t_4 - t_3), 2.0) + math.pow((((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6

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(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)
	t_5 = Float64(Float64(b * a) * Float64(b * Float64(-a)))
	t_6 = Float64(Float64(4.0 * t_5) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_6) * t_5) * Float64(Float64(t_4 + t_3) - sqrt(Float64((Float64(t_4 - t_3) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0))))))) / t_6)
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 = ((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_4 = ((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale;
	t_5 = (b * a) * (b * -a);
	t_6 = (4.0 * t_5) / ((x_45_scale * y_45_scale) ^ 2.0);
	tmp = -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - sqrt((((t_4 - t_3) ^ 2.0) + ((((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0)))))) / t_6;
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[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]}, Block[{t$95$4 = 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]}, Block[{t$95$5 = N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(4.0 * t$95$5), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$6), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(t$95$4 + t$95$3), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$4 - t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[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], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]]]]]]]]

Alternative 1: 29.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(y-scale \cdot \left|x-scale\right|\right) \cdot \left|x-scale\right|\right) \cdot y-scale\\ t_1 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_2 := {\left(\left|x-scale\right|\right)}^{2}\\ \mathbf{if}\;\left|x-scale\right| \leq 2.8 \cdot 10^{-162}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(t\_2 \cdot \frac{\sqrt{8 \cdot \left({t\_1}^{2} - \sqrt{{t\_1}^{4}}\right)}}{\left|x-scale\right|}\right)\right)\\ \mathbf{elif}\;\left|x-scale\right| \leq 4 \cdot 10^{+139}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(t\_2 \cdot \left(angle \cdot \sqrt{8 \cdot \frac{3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - \sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}}}{t\_2}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{b} \cdot \frac{t\_0 \cdot \sqrt{\frac{\left(b \cdot \frac{b}{\left|x-scale\right| \cdot \left|x-scale\right|} - \sqrt{{b}^{4} \cdot {\left(\left|x-scale\right|\right)}^{-4}}\right) \cdot {b}^{4}}{t\_0} \cdot 8}}{b}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* (* y-scale (fabs x-scale)) (fabs x-scale)) y-scale))
        (t_1 (sin (* 0.005555555555555556 (* angle PI))))
        (t_2 (pow (fabs x-scale) 2.0)))
   (if (<= (fabs x-scale) 2.8e-162)
     (*
      0.25
      (*
       b
       (*
        t_2
        (/
         (sqrt (* 8.0 (- (pow t_1 2.0) (sqrt (pow t_1 4.0)))))
         (fabs x-scale)))))
     (if (<= (fabs x-scale) 4e+139)
       (*
        0.25
        (*
         b
         (*
          t_2
          (*
           angle
           (sqrt
            (*
             8.0
             (/
              (-
               (* 3.08641975308642e-5 (pow PI 2.0))
               (sqrt (* 9.525986892242036e-10 (pow PI 4.0))))
              t_2)))))))
       (*
        (/ 0.25 b)
        (/
         (*
          t_0
          (sqrt
           (*
            (/
             (*
              (-
               (* b (/ b (* (fabs x-scale) (fabs x-scale))))
               (sqrt (* (pow b 4.0) (pow (fabs x-scale) -4.0))))
              (pow b 4.0))
             t_0)
            8.0)))
         b))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((y_45_scale * fabs(x_45_scale)) * fabs(x_45_scale)) * y_45_scale;
	double t_1 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_2 = pow(fabs(x_45_scale), 2.0);
	double tmp;
	if (fabs(x_45_scale) <= 2.8e-162) {
		tmp = 0.25 * (b * (t_2 * (sqrt((8.0 * (pow(t_1, 2.0) - sqrt(pow(t_1, 4.0))))) / fabs(x_45_scale))));
	} else if (fabs(x_45_scale) <= 4e+139) {
		tmp = 0.25 * (b * (t_2 * (angle * sqrt((8.0 * (((3.08641975308642e-5 * pow(((double) M_PI), 2.0)) - sqrt((9.525986892242036e-10 * pow(((double) M_PI), 4.0)))) / t_2))))));
	} else {
		tmp = (0.25 / b) * ((t_0 * sqrt((((((b * (b / (fabs(x_45_scale) * fabs(x_45_scale)))) - sqrt((pow(b, 4.0) * pow(fabs(x_45_scale), -4.0)))) * pow(b, 4.0)) / t_0) * 8.0))) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((y_45_scale * Math.abs(x_45_scale)) * Math.abs(x_45_scale)) * y_45_scale;
	double t_1 = Math.sin((0.005555555555555556 * (angle * Math.PI)));
	double t_2 = Math.pow(Math.abs(x_45_scale), 2.0);
	double tmp;
	if (Math.abs(x_45_scale) <= 2.8e-162) {
		tmp = 0.25 * (b * (t_2 * (Math.sqrt((8.0 * (Math.pow(t_1, 2.0) - Math.sqrt(Math.pow(t_1, 4.0))))) / Math.abs(x_45_scale))));
	} else if (Math.abs(x_45_scale) <= 4e+139) {
		tmp = 0.25 * (b * (t_2 * (angle * Math.sqrt((8.0 * (((3.08641975308642e-5 * Math.pow(Math.PI, 2.0)) - Math.sqrt((9.525986892242036e-10 * Math.pow(Math.PI, 4.0)))) / t_2))))));
	} else {
		tmp = (0.25 / b) * ((t_0 * Math.sqrt((((((b * (b / (Math.abs(x_45_scale) * Math.abs(x_45_scale)))) - Math.sqrt((Math.pow(b, 4.0) * Math.pow(Math.abs(x_45_scale), -4.0)))) * Math.pow(b, 4.0)) / t_0) * 8.0))) / b);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = ((y_45_scale * math.fabs(x_45_scale)) * math.fabs(x_45_scale)) * y_45_scale
	t_1 = math.sin((0.005555555555555556 * (angle * math.pi)))
	t_2 = math.pow(math.fabs(x_45_scale), 2.0)
	tmp = 0
	if math.fabs(x_45_scale) <= 2.8e-162:
		tmp = 0.25 * (b * (t_2 * (math.sqrt((8.0 * (math.pow(t_1, 2.0) - math.sqrt(math.pow(t_1, 4.0))))) / math.fabs(x_45_scale))))
	elif math.fabs(x_45_scale) <= 4e+139:
		tmp = 0.25 * (b * (t_2 * (angle * math.sqrt((8.0 * (((3.08641975308642e-5 * math.pow(math.pi, 2.0)) - math.sqrt((9.525986892242036e-10 * math.pow(math.pi, 4.0)))) / t_2))))))
	else:
		tmp = (0.25 / b) * ((t_0 * math.sqrt((((((b * (b / (math.fabs(x_45_scale) * math.fabs(x_45_scale)))) - math.sqrt((math.pow(b, 4.0) * math.pow(math.fabs(x_45_scale), -4.0)))) * math.pow(b, 4.0)) / t_0) * 8.0))) / b)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(y_45_scale * abs(x_45_scale)) * abs(x_45_scale)) * y_45_scale)
	t_1 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_2 = abs(x_45_scale) ^ 2.0
	tmp = 0.0
	if (abs(x_45_scale) <= 2.8e-162)
		tmp = Float64(0.25 * Float64(b * Float64(t_2 * Float64(sqrt(Float64(8.0 * Float64((t_1 ^ 2.0) - sqrt((t_1 ^ 4.0))))) / abs(x_45_scale)))));
	elseif (abs(x_45_scale) <= 4e+139)
		tmp = Float64(0.25 * Float64(b * Float64(t_2 * Float64(angle * sqrt(Float64(8.0 * Float64(Float64(Float64(3.08641975308642e-5 * (pi ^ 2.0)) - sqrt(Float64(9.525986892242036e-10 * (pi ^ 4.0)))) / t_2)))))));
	else
		tmp = Float64(Float64(0.25 / b) * Float64(Float64(t_0 * sqrt(Float64(Float64(Float64(Float64(Float64(b * Float64(b / Float64(abs(x_45_scale) * abs(x_45_scale)))) - sqrt(Float64((b ^ 4.0) * (abs(x_45_scale) ^ -4.0)))) * (b ^ 4.0)) / t_0) * 8.0))) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = ((y_45_scale * abs(x_45_scale)) * abs(x_45_scale)) * y_45_scale;
	t_1 = sin((0.005555555555555556 * (angle * pi)));
	t_2 = abs(x_45_scale) ^ 2.0;
	tmp = 0.0;
	if (abs(x_45_scale) <= 2.8e-162)
		tmp = 0.25 * (b * (t_2 * (sqrt((8.0 * ((t_1 ^ 2.0) - sqrt((t_1 ^ 4.0))))) / abs(x_45_scale))));
	elseif (abs(x_45_scale) <= 4e+139)
		tmp = 0.25 * (b * (t_2 * (angle * sqrt((8.0 * (((3.08641975308642e-5 * (pi ^ 2.0)) - sqrt((9.525986892242036e-10 * (pi ^ 4.0)))) / t_2))))));
	else
		tmp = (0.25 / b) * ((t_0 * sqrt((((((b * (b / (abs(x_45_scale) * abs(x_45_scale)))) - sqrt(((b ^ 4.0) * (abs(x_45_scale) ^ -4.0)))) * (b ^ 4.0)) / t_0) * 8.0))) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(y$45$scale * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Abs[x$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 2.8e-162], N[(0.25 * N[(b * N[(t$95$2 * N[(N[Sqrt[N[(8.0 * N[(N[Power[t$95$1, 2.0], $MachinePrecision] - N[Sqrt[N[Power[t$95$1, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 4e+139], N[(0.25 * N[(b * N[(t$95$2 * N[(angle * N[Sqrt[N[(8.0 * N[(N[(N[(3.08641975308642e-5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(9.525986892242036e-10 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / b), $MachinePrecision] * N[(N[(t$95$0 * N[Sqrt[N[(N[(N[(N[(N[(b * N[(b / N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(N[Power[b, 4.0], $MachinePrecision] * N[Power[N[Abs[x$45$scale], $MachinePrecision], -4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < 2.8000000000000002e-162

   1. Initial program 0.0%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 1.7%

      \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in y-scale around 0

      \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]
   9. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

   10. Applied rewrites 12.0%

       \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

   11. Taylor expanded in x-scale around 0

       \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \frac{\sqrt{8 \cdot \left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}}{x-scale}\right)\right) \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \frac{\sqrt{8 \cdot \left({\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right)}}{x-scale}\right)\right) \]

   13. Applied rewrites 24.9%

       \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \frac{\sqrt{8 \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}}{x-scale}\right)\right) \]

   if 2.8000000000000002e-162 < x-scale < 4.0000000000000001e139

   1. Initial program 0.0%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 1.7%

      \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in y-scale around 0

      \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]
   9. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

   10. Applied rewrites 12.0%

       \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

   11. Taylor expanded in angle around 0

       \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\pi}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\pi}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

   13. Applied rewrites 13.5%

       \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - \sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

   if 4.0000000000000001e139 < x-scale

   1. Initial program 0.0%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}}} \]

   5. Taylor expanded in angle around 0

      \[\leadsto 0.25 \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}} \]

   7. Applied rewrites 0.5%

      \[\leadsto 0.25 \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}} \]

   9. Applied rewrites 5.6%

      \[\leadsto \frac{0.25}{b} \cdot \color{blue}{\frac{\left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right) \cdot \sqrt{\frac{\left(b \cdot \frac{b}{x-scale \cdot x-scale} - \sqrt{{b}^{4} \cdot {x-scale}^{-4}}\right) \cdot {b}^{4}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8}}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 26.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} t_0 := \left|x-scale\right| \cdot \left|x-scale\right|\\ t_1 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_2 := {\left(\left|x-scale\right|\right)}^{2}\\ t_3 := \left(\left(y-scale \cdot \left|x-scale\right|\right) \cdot \left|x-scale\right|\right) \cdot y-scale\\ \mathbf{if}\;\left|x-scale\right| \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;0.25 \cdot \left(\left(\frac{\sqrt{8 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot t\_1\right)\right) - \sqrt{{\sin t\_1}^{4}}\right)}}{\left|\left|x-scale\right|\right|} \cdot t\_0\right) \cdot b\right)\\ \mathbf{elif}\;\left|x-scale\right| \leq 4 \cdot 10^{+139}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(t\_2 \cdot \left(angle \cdot \sqrt{8 \cdot \frac{3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - \sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}}}{t\_2}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{b} \cdot \frac{t\_3 \cdot \sqrt{\frac{\left(b \cdot \frac{b}{t\_0} - \sqrt{{b}^{4} \cdot {\left(\left|x-scale\right|\right)}^{-4}}\right) \cdot {b}^{4}}{t\_3} \cdot 8}}{b}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (fabs x-scale) (fabs x-scale)))
        (t_1 (* (* PI angle) 0.005555555555555556))
        (t_2 (pow (fabs x-scale) 2.0))
        (t_3 (* (* (* y-scale (fabs x-scale)) (fabs x-scale)) y-scale)))
   (if (<= (fabs x-scale) 1.6e-162)
     (*
      0.25
      (*
       (*
        (/
         (sqrt
          (*
           8.0
           (- (- 0.5 (* 0.5 (cos (* 2.0 t_1)))) (sqrt (pow (sin t_1) 4.0)))))
         (fabs (fabs x-scale)))
        t_0)
       b))
     (if (<= (fabs x-scale) 4e+139)
       (*
        0.25
        (*
         b
         (*
          t_2
          (*
           angle
           (sqrt
            (*
             8.0
             (/
              (-
               (* 3.08641975308642e-5 (pow PI 2.0))
               (sqrt (* 9.525986892242036e-10 (pow PI 4.0))))
              t_2)))))))
       (*
        (/ 0.25 b)
        (/
         (*
          t_3
          (sqrt
           (*
            (/
             (*
              (-
               (* b (/ b t_0))
               (sqrt (* (pow b 4.0) (pow (fabs x-scale) -4.0))))
              (pow b 4.0))
             t_3)
            8.0)))
         b))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(x_45_scale) * fabs(x_45_scale);
	double t_1 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_2 = pow(fabs(x_45_scale), 2.0);
	double t_3 = ((y_45_scale * fabs(x_45_scale)) * fabs(x_45_scale)) * y_45_scale;
	double tmp;
	if (fabs(x_45_scale) <= 1.6e-162) {
		tmp = 0.25 * (((sqrt((8.0 * ((0.5 - (0.5 * cos((2.0 * t_1)))) - sqrt(pow(sin(t_1), 4.0))))) / fabs(fabs(x_45_scale))) * t_0) * b);
	} else if (fabs(x_45_scale) <= 4e+139) {
		tmp = 0.25 * (b * (t_2 * (angle * sqrt((8.0 * (((3.08641975308642e-5 * pow(((double) M_PI), 2.0)) - sqrt((9.525986892242036e-10 * pow(((double) M_PI), 4.0)))) / t_2))))));
	} else {
		tmp = (0.25 / b) * ((t_3 * sqrt((((((b * (b / t_0)) - sqrt((pow(b, 4.0) * pow(fabs(x_45_scale), -4.0)))) * pow(b, 4.0)) / t_3) * 8.0))) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.abs(x_45_scale) * Math.abs(x_45_scale);
	double t_1 = (Math.PI * angle) * 0.005555555555555556;
	double t_2 = Math.pow(Math.abs(x_45_scale), 2.0);
	double t_3 = ((y_45_scale * Math.abs(x_45_scale)) * Math.abs(x_45_scale)) * y_45_scale;
	double tmp;
	if (Math.abs(x_45_scale) <= 1.6e-162) {
		tmp = 0.25 * (((Math.sqrt((8.0 * ((0.5 - (0.5 * Math.cos((2.0 * t_1)))) - Math.sqrt(Math.pow(Math.sin(t_1), 4.0))))) / Math.abs(Math.abs(x_45_scale))) * t_0) * b);
	} else if (Math.abs(x_45_scale) <= 4e+139) {
		tmp = 0.25 * (b * (t_2 * (angle * Math.sqrt((8.0 * (((3.08641975308642e-5 * Math.pow(Math.PI, 2.0)) - Math.sqrt((9.525986892242036e-10 * Math.pow(Math.PI, 4.0)))) / t_2))))));
	} else {
		tmp = (0.25 / b) * ((t_3 * Math.sqrt((((((b * (b / t_0)) - Math.sqrt((Math.pow(b, 4.0) * Math.pow(Math.abs(x_45_scale), -4.0)))) * Math.pow(b, 4.0)) / t_3) * 8.0))) / b);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.fabs(x_45_scale) * math.fabs(x_45_scale)
	t_1 = (math.pi * angle) * 0.005555555555555556
	t_2 = math.pow(math.fabs(x_45_scale), 2.0)
	t_3 = ((y_45_scale * math.fabs(x_45_scale)) * math.fabs(x_45_scale)) * y_45_scale
	tmp = 0
	if math.fabs(x_45_scale) <= 1.6e-162:
		tmp = 0.25 * (((math.sqrt((8.0 * ((0.5 - (0.5 * math.cos((2.0 * t_1)))) - math.sqrt(math.pow(math.sin(t_1), 4.0))))) / math.fabs(math.fabs(x_45_scale))) * t_0) * b)
	elif math.fabs(x_45_scale) <= 4e+139:
		tmp = 0.25 * (b * (t_2 * (angle * math.sqrt((8.0 * (((3.08641975308642e-5 * math.pow(math.pi, 2.0)) - math.sqrt((9.525986892242036e-10 * math.pow(math.pi, 4.0)))) / t_2))))))
	else:
		tmp = (0.25 / b) * ((t_3 * math.sqrt((((((b * (b / t_0)) - math.sqrt((math.pow(b, 4.0) * math.pow(math.fabs(x_45_scale), -4.0)))) * math.pow(b, 4.0)) / t_3) * 8.0))) / b)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(x_45_scale) * abs(x_45_scale))
	t_1 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_2 = abs(x_45_scale) ^ 2.0
	t_3 = Float64(Float64(Float64(y_45_scale * abs(x_45_scale)) * abs(x_45_scale)) * y_45_scale)
	tmp = 0.0
	if (abs(x_45_scale) <= 1.6e-162)
		tmp = Float64(0.25 * Float64(Float64(Float64(sqrt(Float64(8.0 * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_1)))) - sqrt((sin(t_1) ^ 4.0))))) / abs(abs(x_45_scale))) * t_0) * b));
	elseif (abs(x_45_scale) <= 4e+139)
		tmp = Float64(0.25 * Float64(b * Float64(t_2 * Float64(angle * sqrt(Float64(8.0 * Float64(Float64(Float64(3.08641975308642e-5 * (pi ^ 2.0)) - sqrt(Float64(9.525986892242036e-10 * (pi ^ 4.0)))) / t_2)))))));
	else
		tmp = Float64(Float64(0.25 / b) * Float64(Float64(t_3 * sqrt(Float64(Float64(Float64(Float64(Float64(b * Float64(b / t_0)) - sqrt(Float64((b ^ 4.0) * (abs(x_45_scale) ^ -4.0)))) * (b ^ 4.0)) / t_3) * 8.0))) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(x_45_scale) * abs(x_45_scale);
	t_1 = (pi * angle) * 0.005555555555555556;
	t_2 = abs(x_45_scale) ^ 2.0;
	t_3 = ((y_45_scale * abs(x_45_scale)) * abs(x_45_scale)) * y_45_scale;
	tmp = 0.0;
	if (abs(x_45_scale) <= 1.6e-162)
		tmp = 0.25 * (((sqrt((8.0 * ((0.5 - (0.5 * cos((2.0 * t_1)))) - sqrt((sin(t_1) ^ 4.0))))) / abs(abs(x_45_scale))) * t_0) * b);
	elseif (abs(x_45_scale) <= 4e+139)
		tmp = 0.25 * (b * (t_2 * (angle * sqrt((8.0 * (((3.08641975308642e-5 * (pi ^ 2.0)) - sqrt((9.525986892242036e-10 * (pi ^ 4.0)))) / t_2))))));
	else
		tmp = (0.25 / b) * ((t_3 * sqrt((((((b * (b / t_0)) - sqrt(((b ^ 4.0) * (abs(x_45_scale) ^ -4.0)))) * (b ^ 4.0)) / t_3) * 8.0))) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Abs[x$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(y$45$scale * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision]}, If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 1.6e-162], N[(0.25 * N[(N[(N[(N[Sqrt[N[(8.0 * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[Power[N[Sin[t$95$1], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[Abs[x$45$scale], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 4e+139], N[(0.25 * N[(b * N[(t$95$2 * N[(angle * N[Sqrt[N[(8.0 * N[(N[(N[(3.08641975308642e-5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(9.525986892242036e-10 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / b), $MachinePrecision] * N[(N[(t$95$3 * N[Sqrt[N[(N[(N[(N[(N[(b * N[(b / t$95$0), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(N[Power[b, 4.0], $MachinePrecision] * N[Power[N[Abs[x$45$scale], $MachinePrecision], -4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < 1.5999999999999999e-162

   1. Initial program 0.0%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 1.7%

      \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in y-scale around 0

      \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]
   9. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

   10. Applied rewrites 12.0%

       \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

   12. Applied rewrites 18.5%

       \[\leadsto 0.25 \cdot \left(\left(\frac{\sqrt{8 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) - \sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right)}}{\left|x-scale\right|} \cdot \left(x-scale \cdot x-scale\right)\right) \cdot b\right) \]

   if 1.5999999999999999e-162 < x-scale < 4.0000000000000001e139

   1. Initial program 0.0%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 1.7%

      \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in y-scale around 0

      \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]
   9. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

   10. Applied rewrites 12.0%

       \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

   11. Taylor expanded in angle around 0

       \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\pi}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\pi}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

   13. Applied rewrites 13.5%

       \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - \sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

   if 4.0000000000000001e139 < x-scale

   1. Initial program 0.0%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}}} \]

   5. Taylor expanded in angle around 0

      \[\leadsto 0.25 \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}} \]

   7. Applied rewrites 0.5%

      \[\leadsto 0.25 \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}} \]

   9. Applied rewrites 5.6%

      \[\leadsto \frac{0.25}{b} \cdot \color{blue}{\frac{\left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right) \cdot \sqrt{\frac{\left(b \cdot \frac{b}{x-scale \cdot x-scale} - \sqrt{{b}^{4} \cdot {x-scale}^{-4}}\right) \cdot {b}^{4}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8}}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 16.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(y-scale \cdot \left|x-scale\right|\right) \cdot \left|x-scale\right|\right) \cdot y-scale\\ t_1 := {\left(\left|x-scale\right|\right)}^{2}\\ \mathbf{if}\;\left|x-scale\right| \leq 4 \cdot 10^{+139}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(t\_1 \cdot \left(angle \cdot \sqrt{8 \cdot \frac{3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - \sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}}}{t\_1}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{b} \cdot \frac{t\_0 \cdot \sqrt{\frac{\left(b \cdot \frac{b}{\left|x-scale\right| \cdot \left|x-scale\right|} - \sqrt{{b}^{4} \cdot {\left(\left|x-scale\right|\right)}^{-4}}\right) \cdot {b}^{4}}{t\_0} \cdot 8}}{b}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* (* y-scale (fabs x-scale)) (fabs x-scale)) y-scale))
        (t_1 (pow (fabs x-scale) 2.0)))
   (if (<= (fabs x-scale) 4e+139)
     (*
      0.25
      (*
       b
       (*
        t_1
        (*
         angle
         (sqrt
          (*
           8.0
           (/
            (-
             (* 3.08641975308642e-5 (pow PI 2.0))
             (sqrt (* 9.525986892242036e-10 (pow PI 4.0))))
            t_1)))))))
     (*
      (/ 0.25 b)
      (/
       (*
        t_0
        (sqrt
         (*
          (/
           (*
            (-
             (* b (/ b (* (fabs x-scale) (fabs x-scale))))
             (sqrt (* (pow b 4.0) (pow (fabs x-scale) -4.0))))
            (pow b 4.0))
           t_0)
          8.0)))
       b)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((y_45_scale * fabs(x_45_scale)) * fabs(x_45_scale)) * y_45_scale;
	double t_1 = pow(fabs(x_45_scale), 2.0);
	double tmp;
	if (fabs(x_45_scale) <= 4e+139) {
		tmp = 0.25 * (b * (t_1 * (angle * sqrt((8.0 * (((3.08641975308642e-5 * pow(((double) M_PI), 2.0)) - sqrt((9.525986892242036e-10 * pow(((double) M_PI), 4.0)))) / t_1))))));
	} else {
		tmp = (0.25 / b) * ((t_0 * sqrt((((((b * (b / (fabs(x_45_scale) * fabs(x_45_scale)))) - sqrt((pow(b, 4.0) * pow(fabs(x_45_scale), -4.0)))) * pow(b, 4.0)) / t_0) * 8.0))) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((y_45_scale * Math.abs(x_45_scale)) * Math.abs(x_45_scale)) * y_45_scale;
	double t_1 = Math.pow(Math.abs(x_45_scale), 2.0);
	double tmp;
	if (Math.abs(x_45_scale) <= 4e+139) {
		tmp = 0.25 * (b * (t_1 * (angle * Math.sqrt((8.0 * (((3.08641975308642e-5 * Math.pow(Math.PI, 2.0)) - Math.sqrt((9.525986892242036e-10 * Math.pow(Math.PI, 4.0)))) / t_1))))));
	} else {
		tmp = (0.25 / b) * ((t_0 * Math.sqrt((((((b * (b / (Math.abs(x_45_scale) * Math.abs(x_45_scale)))) - Math.sqrt((Math.pow(b, 4.0) * Math.pow(Math.abs(x_45_scale), -4.0)))) * Math.pow(b, 4.0)) / t_0) * 8.0))) / b);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = ((y_45_scale * math.fabs(x_45_scale)) * math.fabs(x_45_scale)) * y_45_scale
	t_1 = math.pow(math.fabs(x_45_scale), 2.0)
	tmp = 0
	if math.fabs(x_45_scale) <= 4e+139:
		tmp = 0.25 * (b * (t_1 * (angle * math.sqrt((8.0 * (((3.08641975308642e-5 * math.pow(math.pi, 2.0)) - math.sqrt((9.525986892242036e-10 * math.pow(math.pi, 4.0)))) / t_1))))))
	else:
		tmp = (0.25 / b) * ((t_0 * math.sqrt((((((b * (b / (math.fabs(x_45_scale) * math.fabs(x_45_scale)))) - math.sqrt((math.pow(b, 4.0) * math.pow(math.fabs(x_45_scale), -4.0)))) * math.pow(b, 4.0)) / t_0) * 8.0))) / b)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(y_45_scale * abs(x_45_scale)) * abs(x_45_scale)) * y_45_scale)
	t_1 = abs(x_45_scale) ^ 2.0
	tmp = 0.0
	if (abs(x_45_scale) <= 4e+139)
		tmp = Float64(0.25 * Float64(b * Float64(t_1 * Float64(angle * sqrt(Float64(8.0 * Float64(Float64(Float64(3.08641975308642e-5 * (pi ^ 2.0)) - sqrt(Float64(9.525986892242036e-10 * (pi ^ 4.0)))) / t_1)))))));
	else
		tmp = Float64(Float64(0.25 / b) * Float64(Float64(t_0 * sqrt(Float64(Float64(Float64(Float64(Float64(b * Float64(b / Float64(abs(x_45_scale) * abs(x_45_scale)))) - sqrt(Float64((b ^ 4.0) * (abs(x_45_scale) ^ -4.0)))) * (b ^ 4.0)) / t_0) * 8.0))) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = ((y_45_scale * abs(x_45_scale)) * abs(x_45_scale)) * y_45_scale;
	t_1 = abs(x_45_scale) ^ 2.0;
	tmp = 0.0;
	if (abs(x_45_scale) <= 4e+139)
		tmp = 0.25 * (b * (t_1 * (angle * sqrt((8.0 * (((3.08641975308642e-5 * (pi ^ 2.0)) - sqrt((9.525986892242036e-10 * (pi ^ 4.0)))) / t_1))))));
	else
		tmp = (0.25 / b) * ((t_0 * sqrt((((((b * (b / (abs(x_45_scale) * abs(x_45_scale)))) - sqrt(((b ^ 4.0) * (abs(x_45_scale) ^ -4.0)))) * (b ^ 4.0)) / t_0) * 8.0))) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(y$45$scale * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Abs[x$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 4e+139], N[(0.25 * N[(b * N[(t$95$1 * N[(angle * N[Sqrt[N[(8.0 * N[(N[(N[(3.08641975308642e-5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(9.525986892242036e-10 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / b), $MachinePrecision] * N[(N[(t$95$0 * N[Sqrt[N[(N[(N[(N[(N[(b * N[(b / N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(N[Power[b, 4.0], $MachinePrecision] * N[Power[N[Abs[x$45$scale], $MachinePrecision], -4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 4.0000000000000001e139

   1. Initial program 0.0%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 1.7%

      \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in y-scale around 0

      \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]
   9. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

   10. Applied rewrites 12.0%

       \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

   11. Taylor expanded in angle around 0

       \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\pi}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\pi}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

   13. Applied rewrites 13.5%

       \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - \sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

   if 4.0000000000000001e139 < x-scale

   1. Initial program 0.0%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}}} \]

   5. Taylor expanded in angle around 0

      \[\leadsto 0.25 \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}} \]

   7. Applied rewrites 0.5%

      \[\leadsto 0.25 \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}} \]

   9. Applied rewrites 5.6%

      \[\leadsto \frac{0.25}{b} \cdot \color{blue}{\frac{\left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right) \cdot \sqrt{\frac{\left(b \cdot \frac{b}{x-scale \cdot x-scale} - \sqrt{{b}^{4} \cdot {x-scale}^{-4}}\right) \cdot {b}^{4}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8}}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 15.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|x-scale\right|\right)}^{2}\\ \mathbf{if}\;\left|x-scale\right| \leq 2.75 \cdot 10^{+152}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(t\_0 \cdot \left(angle \cdot \sqrt{8 \cdot \frac{3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - \sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}}}{t\_0}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left|x-scale\right| \cdot \left(\left|x-scale\right| \cdot \left(\sqrt{\frac{\left(\frac{b}{\left|x-scale\right| \cdot \left|x-scale\right|} \cdot b - \sqrt{{\left(\left|x-scale\right|\right)}^{-4} \cdot {b}^{4}}\right) \cdot {b}^{4}}{\left(\left(y-scale \cdot \left|x-scale\right|\right) \cdot \left|x-scale\right|\right) \cdot y-scale} \cdot 8} \cdot \frac{y-scale \cdot y-scale}{b \cdot b}\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (pow (fabs x-scale) 2.0)))
   (if (<= (fabs x-scale) 2.75e+152)
     (*
      0.25
      (*
       b
       (*
        t_0
        (*
         angle
         (sqrt
          (*
           8.0
           (/
            (-
             (* 3.08641975308642e-5 (pow PI 2.0))
             (sqrt (* 9.525986892242036e-10 (pow PI 4.0))))
            t_0)))))))
     (*
      0.25
      (*
       (fabs x-scale)
       (*
        (fabs x-scale)
        (*
         (sqrt
          (*
           (/
            (*
             (-
              (* (/ b (* (fabs x-scale) (fabs x-scale))) b)
              (sqrt (* (pow (fabs x-scale) -4.0) (pow b 4.0))))
             (pow b 4.0))
            (* (* (* y-scale (fabs x-scale)) (fabs x-scale)) y-scale))
           8.0))
         (/ (* y-scale y-scale) (* b b)))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = pow(fabs(x_45_scale), 2.0);
	double tmp;
	if (fabs(x_45_scale) <= 2.75e+152) {
		tmp = 0.25 * (b * (t_0 * (angle * sqrt((8.0 * (((3.08641975308642e-5 * pow(((double) M_PI), 2.0)) - sqrt((9.525986892242036e-10 * pow(((double) M_PI), 4.0)))) / t_0))))));
	} else {
		tmp = 0.25 * (fabs(x_45_scale) * (fabs(x_45_scale) * (sqrt(((((((b / (fabs(x_45_scale) * fabs(x_45_scale))) * b) - sqrt((pow(fabs(x_45_scale), -4.0) * pow(b, 4.0)))) * pow(b, 4.0)) / (((y_45_scale * fabs(x_45_scale)) * fabs(x_45_scale)) * y_45_scale)) * 8.0)) * ((y_45_scale * y_45_scale) / (b * b)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.pow(Math.abs(x_45_scale), 2.0);
	double tmp;
	if (Math.abs(x_45_scale) <= 2.75e+152) {
		tmp = 0.25 * (b * (t_0 * (angle * Math.sqrt((8.0 * (((3.08641975308642e-5 * Math.pow(Math.PI, 2.0)) - Math.sqrt((9.525986892242036e-10 * Math.pow(Math.PI, 4.0)))) / t_0))))));
	} else {
		tmp = 0.25 * (Math.abs(x_45_scale) * (Math.abs(x_45_scale) * (Math.sqrt(((((((b / (Math.abs(x_45_scale) * Math.abs(x_45_scale))) * b) - Math.sqrt((Math.pow(Math.abs(x_45_scale), -4.0) * Math.pow(b, 4.0)))) * Math.pow(b, 4.0)) / (((y_45_scale * Math.abs(x_45_scale)) * Math.abs(x_45_scale)) * y_45_scale)) * 8.0)) * ((y_45_scale * y_45_scale) / (b * b)))));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.pow(math.fabs(x_45_scale), 2.0)
	tmp = 0
	if math.fabs(x_45_scale) <= 2.75e+152:
		tmp = 0.25 * (b * (t_0 * (angle * math.sqrt((8.0 * (((3.08641975308642e-5 * math.pow(math.pi, 2.0)) - math.sqrt((9.525986892242036e-10 * math.pow(math.pi, 4.0)))) / t_0))))))
	else:
		tmp = 0.25 * (math.fabs(x_45_scale) * (math.fabs(x_45_scale) * (math.sqrt(((((((b / (math.fabs(x_45_scale) * math.fabs(x_45_scale))) * b) - math.sqrt((math.pow(math.fabs(x_45_scale), -4.0) * math.pow(b, 4.0)))) * math.pow(b, 4.0)) / (((y_45_scale * math.fabs(x_45_scale)) * math.fabs(x_45_scale)) * y_45_scale)) * 8.0)) * ((y_45_scale * y_45_scale) / (b * b)))))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(x_45_scale) ^ 2.0
	tmp = 0.0
	if (abs(x_45_scale) <= 2.75e+152)
		tmp = Float64(0.25 * Float64(b * Float64(t_0 * Float64(angle * sqrt(Float64(8.0 * Float64(Float64(Float64(3.08641975308642e-5 * (pi ^ 2.0)) - sqrt(Float64(9.525986892242036e-10 * (pi ^ 4.0)))) / t_0)))))));
	else
		tmp = Float64(0.25 * Float64(abs(x_45_scale) * Float64(abs(x_45_scale) * Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(b / Float64(abs(x_45_scale) * abs(x_45_scale))) * b) - sqrt(Float64((abs(x_45_scale) ^ -4.0) * (b ^ 4.0)))) * (b ^ 4.0)) / Float64(Float64(Float64(y_45_scale * abs(x_45_scale)) * abs(x_45_scale)) * y_45_scale)) * 8.0)) * Float64(Float64(y_45_scale * y_45_scale) / Float64(b * b))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(x_45_scale) ^ 2.0;
	tmp = 0.0;
	if (abs(x_45_scale) <= 2.75e+152)
		tmp = 0.25 * (b * (t_0 * (angle * sqrt((8.0 * (((3.08641975308642e-5 * (pi ^ 2.0)) - sqrt((9.525986892242036e-10 * (pi ^ 4.0)))) / t_0))))));
	else
		tmp = 0.25 * (abs(x_45_scale) * (abs(x_45_scale) * (sqrt(((((((b / (abs(x_45_scale) * abs(x_45_scale))) * b) - sqrt(((abs(x_45_scale) ^ -4.0) * (b ^ 4.0)))) * (b ^ 4.0)) / (((y_45_scale * abs(x_45_scale)) * abs(x_45_scale)) * y_45_scale)) * 8.0)) * ((y_45_scale * y_45_scale) / (b * b)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Power[N[Abs[x$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 2.75e+152], N[(0.25 * N[(b * N[(t$95$0 * N[(angle * N[Sqrt[N[(8.0 * N[(N[(N[(3.08641975308642e-5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(9.525986892242036e-10 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[Abs[x$45$scale], $MachinePrecision] * N[(N[Abs[x$45$scale], $MachinePrecision] * N[(N[Sqrt[N[(N[(N[(N[(N[(N[(b / N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] - N[Sqrt[N[(N[Power[N[Abs[x$45$scale], $MachinePrecision], -4.0], $MachinePrecision] * N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y$45$scale * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] * N[(N[(y$45$scale * y$45$scale), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2.75e152

   1. Initial program 0.0%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}}} \]

   5. Taylor expanded in b around 0

      \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   7. Applied rewrites 1.7%

      \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

   8. Taylor expanded in y-scale around 0

      \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]
   9. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

   10. Applied rewrites 12.0%

       \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

   11. Taylor expanded in angle around 0

       \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\pi}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\pi}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

       2. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

   13. Applied rewrites 13.5%

       \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - \sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

   if 2.75e152 < x-scale

   1. Initial program 0.0%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 0.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}}} \]

   5. Taylor expanded in angle around 0

      \[\leadsto 0.25 \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}} \]

   7. Applied rewrites 0.5%

      \[\leadsto 0.25 \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\frac{{b}^{2}}{{x-scale}^{2}} - \sqrt{\frac{{b}^{4}}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}} \]

   9. Applied rewrites 1.2%

      \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \color{blue}{\frac{\sqrt{\frac{\left(b \cdot \frac{b}{x-scale \cdot x-scale} - \sqrt{{b}^{4} \cdot {x-scale}^{-4}}\right) \cdot {b}^{4}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8} \cdot \left(y-scale \cdot y-scale\right)}{b \cdot b}}\right) \]

   11. Applied rewrites 2.8%

       \[\leadsto 0.25 \cdot \left(x-scale \cdot \color{blue}{\left(x-scale \cdot \left(\sqrt{\frac{\left(\frac{b}{x-scale \cdot x-scale} \cdot b - \sqrt{{x-scale}^{-4} \cdot {b}^{4}}\right) \cdot {b}^{4}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8} \cdot \frac{y-scale \cdot y-scale}{b \cdot b}\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 13.5% accurate, 8.0× speedup

Math

\[0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - \sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  0.25
  (*
   b
   (*
    (pow x-scale 2.0)
    (*
     angle
     (sqrt
      (*
       8.0
       (/
        (-
         (* 3.08641975308642e-5 (pow PI 2.0))
         (sqrt (* 9.525986892242036e-10 (pow PI 4.0))))
        (pow x-scale 2.0)))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * (b * (pow(x_45_scale, 2.0) * (angle * sqrt((8.0 * (((3.08641975308642e-5 * pow(((double) M_PI), 2.0)) - sqrt((9.525986892242036e-10 * pow(((double) M_PI), 4.0)))) / pow(x_45_scale, 2.0)))))));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * (b * (Math.pow(x_45_scale, 2.0) * (angle * Math.sqrt((8.0 * (((3.08641975308642e-5 * Math.pow(Math.PI, 2.0)) - Math.sqrt((9.525986892242036e-10 * Math.pow(Math.PI, 4.0)))) / Math.pow(x_45_scale, 2.0)))))));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 0.25 * (b * (math.pow(x_45_scale, 2.0) * (angle * math.sqrt((8.0 * (((3.08641975308642e-5 * math.pow(math.pi, 2.0)) - math.sqrt((9.525986892242036e-10 * math.pow(math.pi, 4.0)))) / math.pow(x_45_scale, 2.0)))))))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(0.25 * Float64(b * Float64((x_45_scale ^ 2.0) * Float64(angle * sqrt(Float64(8.0 * Float64(Float64(Float64(3.08641975308642e-5 * (pi ^ 2.0)) - sqrt(Float64(9.525986892242036e-10 * (pi ^ 4.0)))) / (x_45_scale ^ 2.0))))))))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.25 * (b * ((x_45_scale ^ 2.0) * (angle * sqrt((8.0 * (((3.08641975308642e-5 * (pi ^ 2.0)) - sqrt((9.525986892242036e-10 * (pi ^ 4.0)))) / (x_45_scale ^ 2.0)))))));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(0.25 * N[(b * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[(angle * N[Sqrt[N[(8.0 * N[(N[(N[(3.08641975308642e-5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(9.525986892242036e-10 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0%

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

2. Taylor expanded in a around 0

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

4. Applied rewrites 0.7%

   \[\leadsto \color{blue}{0.25 \cdot \frac{{x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)}{{b}^{2}}} \]

5. Taylor expanded in b around 0

   \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{4 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

7. Applied rewrites 1.7%

   \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right) - \sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}\right) \]

8. Taylor expanded in y-scale around 0

   \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]
9. Step-by-step derivation

   1. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

10. Applied rewrites 12.0%

    \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}}{{x-scale}^{2}}}\right)\right) \]

11. Taylor expanded in angle around 0

    \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\pi}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\pi}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

    2. lower-sqrt.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

    3. lower-*.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

    4. lower-/.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} - \sqrt{\frac{1}{1049760000} \cdot {\mathsf{PI}\left(\right)}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]

13. Applied rewrites 13.5%

    \[\leadsto 0.25 \cdot \left(b \cdot \left({x-scale}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - \sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}}}{{x-scale}^{2}}}\right)\right)\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025205 
(FPCore (a b angle x-scale y-scale)
  :name "b from scale-rotated-ellipse"
  :precision binary64
  (/ (- (sqrt (* (* (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))) (* (* b a) (* b (- a)))) (- (+ (/ (/ (+ (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)) (sqrt (+ (pow (- (/ (/ (+ (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)) 2.0) (pow (/ (/ (* (* (* 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))))))) (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))