b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 22.2%

Time: 34.5s

Alternatives: 6

Speedup: 7.2×

• could not determine a ground truth

  1. a = -2.5460692200975443e-294
  2. b = 3.865433100532432e+226
  3. angle = 5.670658321942746e+44
  4. x-scale = 3.638745891636718e+297
  5. y-scale = 1.2530293938733491e-175

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: 0.1% 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: 22.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|b\right|\right)}^{2}\\ t_1 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;\left|b\right| \leq 2.9 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{-0.25}{\left|b\right|} \cdot \left(\frac{\sqrt{\left(8 \cdot \left(\mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5\right) - \sqrt{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{4}}\right)\right) \cdot {\left(\left|b\right|\right)}^{4}}}{\left|x-scale\right|} \cdot \left(\left(a \cdot x-scale\right) \cdot x-scale\right)\right)}{\left|b\right|}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left(t\_0 \cdot \frac{\sqrt{8 \cdot \left({t\_1}^{2} - \sqrt{{t\_1}^{4}}\right)}}{x-scale}\right)\right)}{t\_0}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (pow (fabs b) 2.0))
        (t_1 (cos (* 0.005555555555555556 (* angle PI)))))
   (if (<= (fabs b) 2.9e-82)
     (/
      (*
       (/ -0.25 (fabs b))
       (*
        (/
         (sqrt
          (*
           (*
            8.0
            (-
             (fma 0.5 (cos (* 0.011111111111111112 (* PI angle))) 0.5)
             (sqrt (pow (cos (* PI (* angle 0.005555555555555556))) 4.0))))
           (pow (fabs b) 4.0)))
         (fabs x-scale))
        (* (* a x-scale) x-scale)))
      (fabs b))
     (*
      -0.25
      (/
       (*
        a
        (*
         (pow x-scale 2.0)
         (*
          t_0
          (/ (sqrt (* 8.0 (- (pow t_1 2.0) (sqrt (pow t_1 4.0))))) x-scale))))
       t_0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = pow(fabs(b), 2.0);
	double t_1 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (fabs(b) <= 2.9e-82) {
		tmp = ((-0.25 / fabs(b)) * ((sqrt(((8.0 * (fma(0.5, cos((0.011111111111111112 * (((double) M_PI) * angle))), 0.5) - sqrt(pow(cos((((double) M_PI) * (angle * 0.005555555555555556))), 4.0)))) * pow(fabs(b), 4.0))) / fabs(x_45_scale)) * ((a * x_45_scale) * x_45_scale))) / fabs(b);
	} else {
		tmp = -0.25 * ((a * (pow(x_45_scale, 2.0) * (t_0 * (sqrt((8.0 * (pow(t_1, 2.0) - sqrt(pow(t_1, 4.0))))) / x_45_scale)))) / t_0);
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(b) ^ 2.0
	t_1 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (abs(b) <= 2.9e-82)
		tmp = Float64(Float64(Float64(-0.25 / abs(b)) * Float64(Float64(sqrt(Float64(Float64(8.0 * Float64(fma(0.5, cos(Float64(0.011111111111111112 * Float64(pi * angle))), 0.5) - sqrt((cos(Float64(pi * Float64(angle * 0.005555555555555556))) ^ 4.0)))) * (abs(b) ^ 4.0))) / abs(x_45_scale)) * Float64(Float64(a * x_45_scale) * x_45_scale))) / abs(b));
	else
		tmp = Float64(-0.25 * Float64(Float64(a * Float64((x_45_scale ^ 2.0) * Float64(t_0 * Float64(sqrt(Float64(8.0 * Float64((t_1 ^ 2.0) - sqrt((t_1 ^ 4.0))))) / x_45_scale)))) / t_0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2.9e-82], N[(N[(N[(-0.25 / N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(N[(8.0 * N[(N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] - N[Sqrt[N[Power[N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[(N[(a * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[b], $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[(a * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[(t$95$0 * 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] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.89999999999999977e-82

   1. Initial program 0.1%

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

   2. Taylor expanded in a around -inf

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 4.4%

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

   9. Applied rewrites 18.7%

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

   11. Applied rewrites 19.4%

       \[\leadsto \frac{\frac{-0.25}{b} \cdot \left(\frac{\sqrt{\left(8 \cdot \left(\mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5\right) - \sqrt{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{4}}\right)\right) \cdot {b}^{4}}}{\left|x-scale\right|} \cdot \left(\left(a \cdot x-scale\right) \cdot x-scale\right)\right)}{\color{blue}{b}} \]

   if 2.89999999999999977e-82 < b

   1. Initial program 0.1%

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

   2. Taylor expanded in a around -inf

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 4.4%

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

   8. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

   10. Applied rewrites 5.2%

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

   11. Taylor expanded in x-scale around 0

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

       1. lower-/.f64 N/A

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

   13. Applied rewrites 11.0%

       \[\leadsto -0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \left({b}^{2} \cdot \frac{\sqrt{8 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}}{x-scale}\right)\right)}{{b}^{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 21.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5\right)\\ t_1 := \left(\left|b\right| \cdot a\right) \cdot \left(\left|b\right| \cdot \left(-a\right)\right)\\ t_2 := \frac{-0.25}{\left|b\right|}\\ t_3 := {\left(\left|b\right|\right)}^{4}\\ t_4 := \frac{4 \cdot t\_1}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \mathbf{if}\;\left|b\right| \leq 2.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{t\_2 \cdot \left(\frac{\sqrt{\left(8 \cdot \left(t\_0 - \sqrt{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{4}}\right)\right) \cdot t\_3}}{\left|x-scale\right|} \cdot \left(\left(a \cdot x-scale\right) \cdot x-scale\right)\right)}{\left|b\right|}\\ \mathbf{elif}\;\left|b\right| \leq 1.55 \cdot 10^{+74}:\\ \;\;\;\;t\_2 \cdot \frac{\left(a \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{\sqrt{8 \cdot \left(\left(\mathsf{fma}\left(\cos \left(\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 2\right), 0.5, 0.5\right) - \sqrt{{t\_0}^{2}}\right) \cdot t\_3\right)}}{\left|x-scale\right|}}{\left|b\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_4\right) \cdot t\_1\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{t\_4}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (fma 0.5 (cos (* 0.011111111111111112 (* PI angle))) 0.5))
        (t_1 (* (* (fabs b) a) (* (fabs b) (- a))))
        (t_2 (/ -0.25 (fabs b)))
        (t_3 (pow (fabs b) 4.0))
        (t_4 (/ (* 4.0 t_1) (pow (* x-scale y-scale) 2.0))))
   (if (<= (fabs b) 2.3e-80)
     (/
      (*
       t_2
       (*
        (/
         (sqrt
          (*
           (*
            8.0
            (-
             t_0
             (sqrt (pow (cos (* PI (* angle 0.005555555555555556))) 4.0))))
           t_3))
         (fabs x-scale))
        (* (* a x-scale) x-scale)))
      (fabs b))
     (if (<= (fabs b) 1.55e+74)
       (*
        t_2
        (/
         (*
          (* a (* x-scale x-scale))
          (/
           (sqrt
            (*
             8.0
             (*
              (-
               (fma
                (cos (* (* (* PI angle) 0.005555555555555556) 2.0))
                0.5
                0.5)
               (sqrt (pow t_0 2.0)))
              t_3)))
           (fabs x-scale)))
         (fabs b)))
       (/
        (-
         (sqrt
          (*
           (* (* 2.0 t_4) t_1)
           (/ (- (pow a 2.0) (sqrt (pow a 4.0))) (pow y-scale 2.0)))))
        t_4)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fma(0.5, cos((0.011111111111111112 * (((double) M_PI) * angle))), 0.5);
	double t_1 = (fabs(b) * a) * (fabs(b) * -a);
	double t_2 = -0.25 / fabs(b);
	double t_3 = pow(fabs(b), 4.0);
	double t_4 = (4.0 * t_1) / pow((x_45_scale * y_45_scale), 2.0);
	double tmp;
	if (fabs(b) <= 2.3e-80) {
		tmp = (t_2 * ((sqrt(((8.0 * (t_0 - sqrt(pow(cos((((double) M_PI) * (angle * 0.005555555555555556))), 4.0)))) * t_3)) / fabs(x_45_scale)) * ((a * x_45_scale) * x_45_scale))) / fabs(b);
	} else if (fabs(b) <= 1.55e+74) {
		tmp = t_2 * (((a * (x_45_scale * x_45_scale)) * (sqrt((8.0 * ((fma(cos((((((double) M_PI) * angle) * 0.005555555555555556) * 2.0)), 0.5, 0.5) - sqrt(pow(t_0, 2.0))) * t_3))) / fabs(x_45_scale))) / fabs(b));
	} else {
		tmp = -sqrt((((2.0 * t_4) * t_1) * ((pow(a, 2.0) - sqrt(pow(a, 4.0))) / pow(y_45_scale, 2.0)))) / t_4;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = fma(0.5, cos(Float64(0.011111111111111112 * Float64(pi * angle))), 0.5)
	t_1 = Float64(Float64(abs(b) * a) * Float64(abs(b) * Float64(-a)))
	t_2 = Float64(-0.25 / abs(b))
	t_3 = abs(b) ^ 4.0
	t_4 = Float64(Float64(4.0 * t_1) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	tmp = 0.0
	if (abs(b) <= 2.3e-80)
		tmp = Float64(Float64(t_2 * Float64(Float64(sqrt(Float64(Float64(8.0 * Float64(t_0 - sqrt((cos(Float64(pi * Float64(angle * 0.005555555555555556))) ^ 4.0)))) * t_3)) / abs(x_45_scale)) * Float64(Float64(a * x_45_scale) * x_45_scale))) / abs(b));
	elseif (abs(b) <= 1.55e+74)
		tmp = Float64(t_2 * Float64(Float64(Float64(a * Float64(x_45_scale * x_45_scale)) * Float64(sqrt(Float64(8.0 * Float64(Float64(fma(cos(Float64(Float64(Float64(pi * angle) * 0.005555555555555556) * 2.0)), 0.5, 0.5) - sqrt((t_0 ^ 2.0))) * t_3))) / abs(x_45_scale))) / abs(b)));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_4) * t_1) * Float64(Float64((a ^ 2.0) - sqrt((a ^ 4.0))) / (y_45_scale ^ 2.0))))) / t_4);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[b], $MachinePrecision] * a), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-0.25 / N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(4.0 * t$95$1), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2.3e-80], N[(N[(t$95$2 * N[(N[(N[Sqrt[N[(N[(8.0 * N[(t$95$0 - N[Sqrt[N[Power[N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[(N[(a * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 1.55e+74], N[(t$95$2 * N[(N[(N[(a * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(8.0 * N[(N[(N[(N[Cos[N[(N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] - N[Sqrt[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$4), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(N[Power[a, 2.0], $MachinePrecision] - N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.2999999999999998e-80

   1. Initial program 0.1%

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

   2. Taylor expanded in a around -inf

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 4.4%

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

   9. Applied rewrites 18.7%

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

   11. Applied rewrites 19.4%

       \[\leadsto \frac{\frac{-0.25}{b} \cdot \left(\frac{\sqrt{\left(8 \cdot \left(\mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5\right) - \sqrt{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{4}}\right)\right) \cdot {b}^{4}}}{\left|x-scale\right|} \cdot \left(\left(a \cdot x-scale\right) \cdot x-scale\right)\right)}{\color{blue}{b}} \]

   if 2.2999999999999998e-80 < b < 1.55000000000000011e74

   1. Initial program 0.1%

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

   2. Taylor expanded in a around -inf

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 4.4%

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

   9. Applied rewrites 18.7%

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

   11. Applied rewrites 19.4%

       \[\leadsto \frac{-0.25}{b} \cdot \frac{\left(a \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{\sqrt{8 \cdot \left(\left(\mathsf{fma}\left(\cos \left(\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot 2\right), 0.5, 0.5\right) - \sqrt{{\left(\mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5\right)\right)}^{2}}\right) \cdot {b}^{4}\right)}}{\left|x-scale\right|}}{b} \]

   if 1.55000000000000011e74 < b

   1. Initial program 0.1%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 2.3

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

   7. Applied rewrites 2.3%

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

Alternative 3: 21.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|b\right| \cdot a\right) \cdot \left(\left|b\right| \cdot \left(-a\right)\right)\\ t_1 := \frac{4 \cdot t\_0}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \mathbf{if}\;\left|b\right| \leq 1.55 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{-0.25}{\left|b\right|} \cdot \left(\frac{\sqrt{\left(8 \cdot \left(\mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5\right) - \sqrt{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{4}}\right)\right) \cdot {\left(\left|b\right|\right)}^{4}}}{\left|x-scale\right|} \cdot \left(\left(a \cdot x-scale\right) \cdot x-scale\right)\right)}{\left|b\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_1\right) \cdot t\_0\right) \cdot \frac{{a}^{2} - \sqrt{{a}^{4}}}{{y-scale}^{2}}}}{t\_1}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* (fabs b) a) (* (fabs b) (- a))))
        (t_1 (/ (* 4.0 t_0) (pow (* x-scale y-scale) 2.0))))
   (if (<= (fabs b) 1.55e+74)
     (/
      (*
       (/ -0.25 (fabs b))
       (*
        (/
         (sqrt
          (*
           (*
            8.0
            (-
             (fma 0.5 (cos (* 0.011111111111111112 (* PI angle))) 0.5)
             (sqrt (pow (cos (* PI (* angle 0.005555555555555556))) 4.0))))
           (pow (fabs b) 4.0)))
         (fabs x-scale))
        (* (* a x-scale) x-scale)))
      (fabs b))
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_1) t_0)
         (/ (- (pow a 2.0) (sqrt (pow a 4.0))) (pow y-scale 2.0)))))
      t_1))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (fabs(b) * a) * (fabs(b) * -a);
	double t_1 = (4.0 * t_0) / pow((x_45_scale * y_45_scale), 2.0);
	double tmp;
	if (fabs(b) <= 1.55e+74) {
		tmp = ((-0.25 / fabs(b)) * ((sqrt(((8.0 * (fma(0.5, cos((0.011111111111111112 * (((double) M_PI) * angle))), 0.5) - sqrt(pow(cos((((double) M_PI) * (angle * 0.005555555555555556))), 4.0)))) * pow(fabs(b), 4.0))) / fabs(x_45_scale)) * ((a * x_45_scale) * x_45_scale))) / fabs(b);
	} else {
		tmp = -sqrt((((2.0 * t_1) * t_0) * ((pow(a, 2.0) - sqrt(pow(a, 4.0))) / pow(y_45_scale, 2.0)))) / t_1;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(abs(b) * a) * Float64(abs(b) * Float64(-a)))
	t_1 = Float64(Float64(4.0 * t_0) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	tmp = 0.0
	if (abs(b) <= 1.55e+74)
		tmp = Float64(Float64(Float64(-0.25 / abs(b)) * Float64(Float64(sqrt(Float64(Float64(8.0 * Float64(fma(0.5, cos(Float64(0.011111111111111112 * Float64(pi * angle))), 0.5) - sqrt((cos(Float64(pi * Float64(angle * 0.005555555555555556))) ^ 4.0)))) * (abs(b) ^ 4.0))) / abs(x_45_scale)) * Float64(Float64(a * x_45_scale) * x_45_scale))) / abs(b));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_1) * t_0) * Float64(Float64((a ^ 2.0) - sqrt((a ^ 4.0))) / (y_45_scale ^ 2.0))))) / t_1);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[Abs[b], $MachinePrecision] * a), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * t$95$0), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 1.55e+74], N[(N[(N[(-0.25 / N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(N[(8.0 * N[(N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] - N[Sqrt[N[Power[N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[(N[(a * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[b], $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[Power[a, 2.0], $MachinePrecision] - N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.55000000000000011e74

   1. Initial program 0.1%

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

   2. Taylor expanded in a around -inf

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 4.4%

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

   9. Applied rewrites 18.7%

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

   11. Applied rewrites 19.4%

       \[\leadsto \frac{\frac{-0.25}{b} \cdot \left(\frac{\sqrt{\left(8 \cdot \left(\mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5\right) - \sqrt{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{4}}\right)\right) \cdot {b}^{4}}}{\left|x-scale\right|} \cdot \left(\left(a \cdot x-scale\right) \cdot x-scale\right)\right)}{\color{blue}{b}} \]

   if 1.55000000000000011e74 < b

   1. Initial program 0.1%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 2.3

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

   7. Applied rewrites 2.3%

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

Alternative 4: 20.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} t_0 := a \cdot \left|b\right|\\ \mathbf{if}\;\left|b\right| \leq 1.55 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{-0.25}{\left|b\right|} \cdot \left(\frac{\sqrt{\left(8 \cdot \left(\mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5\right) - \sqrt{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{4}}\right)\right) \cdot {\left(\left|b\right|\right)}^{4}}}{\left|x-scale\right|} \cdot \left(\left(a \cdot x-scale\right) \cdot x-scale\right)\right)}{\left|b\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sqrt{\left(\left(\left(\left(\left(t\_0 \cdot \left|b\right|\right) \cdot \frac{-a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\right) \cdot 8\right) \cdot \left(\left(-a\right) \cdot \left|b\right|\right)\right) \cdot t\_0\right) \cdot \left(\frac{a}{y-scale \cdot y-scale} \cdot a - \frac{\sqrt{{a}^{4}}}{y-scale \cdot y-scale}\right)}}{t\_0}}{\left(a \cdot 4\right) \cdot \left|b\right|} \cdot \left(\left(x-scale \cdot x-scale\right) \cdot y-scale\right)\right) \cdot y-scale\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* a (fabs b))))
   (if (<= (fabs b) 1.55e+74)
     (/
      (*
       (/ -0.25 (fabs b))
       (*
        (/
         (sqrt
          (*
           (*
            8.0
            (-
             (fma 0.5 (cos (* 0.011111111111111112 (* PI angle))) 0.5)
             (sqrt (pow (cos (* PI (* angle 0.005555555555555556))) 4.0))))
           (pow (fabs b) 4.0)))
         (fabs x-scale))
        (* (* a x-scale) x-scale)))
      (fabs b))
     (*
      (*
       (/
        (/
         (sqrt
          (*
           (*
            (*
             (*
              (*
               (* t_0 (fabs b))
               (/ (- a) (* (* y-scale x-scale) (* y-scale x-scale))))
              8.0)
             (* (- a) (fabs b)))
            t_0)
           (-
            (* (/ a (* y-scale y-scale)) a)
            (/ (sqrt (pow a 4.0)) (* y-scale y-scale)))))
         t_0)
        (* (* a 4.0) (fabs b)))
       (* (* x-scale x-scale) y-scale))
      y-scale))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * fabs(b);
	double tmp;
	if (fabs(b) <= 1.55e+74) {
		tmp = ((-0.25 / fabs(b)) * ((sqrt(((8.0 * (fma(0.5, cos((0.011111111111111112 * (((double) M_PI) * angle))), 0.5) - sqrt(pow(cos((((double) M_PI) * (angle * 0.005555555555555556))), 4.0)))) * pow(fabs(b), 4.0))) / fabs(x_45_scale)) * ((a * x_45_scale) * x_45_scale))) / fabs(b);
	} else {
		tmp = (((sqrt(((((((t_0 * fabs(b)) * (-a / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)))) * 8.0) * (-a * fabs(b))) * t_0) * (((a / (y_45_scale * y_45_scale)) * a) - (sqrt(pow(a, 4.0)) / (y_45_scale * y_45_scale))))) / t_0) / ((a * 4.0) * fabs(b))) * ((x_45_scale * x_45_scale) * y_45_scale)) * y_45_scale;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a * abs(b))
	tmp = 0.0
	if (abs(b) <= 1.55e+74)
		tmp = Float64(Float64(Float64(-0.25 / abs(b)) * Float64(Float64(sqrt(Float64(Float64(8.0 * Float64(fma(0.5, cos(Float64(0.011111111111111112 * Float64(pi * angle))), 0.5) - sqrt((cos(Float64(pi * Float64(angle * 0.005555555555555556))) ^ 4.0)))) * (abs(b) ^ 4.0))) / abs(x_45_scale)) * Float64(Float64(a * x_45_scale) * x_45_scale))) / abs(b));
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(t_0 * abs(b)) * Float64(Float64(-a) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale)))) * 8.0) * Float64(Float64(-a) * abs(b))) * t_0) * Float64(Float64(Float64(a / Float64(y_45_scale * y_45_scale)) * a) - Float64(sqrt((a ^ 4.0)) / Float64(y_45_scale * y_45_scale))))) / t_0) / Float64(Float64(a * 4.0) * abs(b))) * Float64(Float64(x_45_scale * x_45_scale) * y_45_scale)) * y_45_scale);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 1.55e+74], N[(N[(N[(-0.25 / N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(N[(8.0 * N[(N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] - N[Sqrt[N[Power[N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[(N[(a * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[b], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[N[(N[(N[(N[(N[(N[(t$95$0 * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[((-a) / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision] * N[((-a) * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] - N[(N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(a * 4.0), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.55000000000000011e74

   1. Initial program 0.1%

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

   2. Taylor expanded in a around -inf

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 4.4%

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

   9. Applied rewrites 18.7%

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

   11. Applied rewrites 19.4%

       \[\leadsto \frac{\frac{-0.25}{b} \cdot \left(\frac{\sqrt{\left(8 \cdot \left(\mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5\right) - \sqrt{{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{4}}\right)\right) \cdot {b}^{4}}}{\left|x-scale\right|} \cdot \left(\left(a \cdot x-scale\right) \cdot x-scale\right)\right)}{\color{blue}{b}} \]

   if 1.55000000000000011e74 < b

   1. Initial program 0.1%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 2.0

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

   7. Applied rewrites 2.0%

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

   9. Applied rewrites 0.3%

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

   11. Applied rewrites 4.9%

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

Alternative 5: 20.7% accurate, 6.4× speedup

Math

\[\begin{array}{l} t_0 := a \cdot \left|b\right|\\ \mathbf{if}\;\left|b\right| \leq 1.5 \cdot 10^{+74}:\\ \;\;\;\;\frac{-0.25}{\left|b\right|} \cdot \frac{\left(a \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{\sqrt{8 \cdot \left(\left(1 - \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right) \cdot {\left(\left|b\right|\right)}^{4}\right)}}{\left|x-scale\right|}}{\left|b\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sqrt{\left(\left(\left(\left(\left(t\_0 \cdot \left|b\right|\right) \cdot \frac{-a}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}\right) \cdot 8\right) \cdot \left(\left(-a\right) \cdot \left|b\right|\right)\right) \cdot t\_0\right) \cdot \left(\frac{a}{y-scale \cdot y-scale} \cdot a - \frac{\sqrt{{a}^{4}}}{y-scale \cdot y-scale}\right)}}{t\_0}}{\left(a \cdot 4\right) \cdot \left|b\right|} \cdot \left(\left(x-scale \cdot x-scale\right) \cdot y-scale\right)\right) \cdot y-scale\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* a (fabs b))))
   (if (<= (fabs b) 1.5e+74)
     (*
      (/ -0.25 (fabs b))
      (/
       (*
        (* a (* x-scale x-scale))
        (/
         (sqrt
          (*
           8.0
           (*
            (-
             1.0
             (sqrt (pow (cos (* (* PI angle) 0.005555555555555556)) 4.0)))
            (pow (fabs b) 4.0))))
         (fabs x-scale)))
       (fabs b)))
     (*
      (*
       (/
        (/
         (sqrt
          (*
           (*
            (*
             (*
              (*
               (* t_0 (fabs b))
               (/ (- a) (* (* y-scale x-scale) (* y-scale x-scale))))
              8.0)
             (* (- a) (fabs b)))
            t_0)
           (-
            (* (/ a (* y-scale y-scale)) a)
            (/ (sqrt (pow a 4.0)) (* y-scale y-scale)))))
         t_0)
        (* (* a 4.0) (fabs b)))
       (* (* x-scale x-scale) y-scale))
      y-scale))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * fabs(b);
	double tmp;
	if (fabs(b) <= 1.5e+74) {
		tmp = (-0.25 / fabs(b)) * (((a * (x_45_scale * x_45_scale)) * (sqrt((8.0 * ((1.0 - sqrt(pow(cos(((((double) M_PI) * angle) * 0.005555555555555556)), 4.0))) * pow(fabs(b), 4.0)))) / fabs(x_45_scale))) / fabs(b));
	} else {
		tmp = (((sqrt(((((((t_0 * fabs(b)) * (-a / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)))) * 8.0) * (-a * fabs(b))) * t_0) * (((a / (y_45_scale * y_45_scale)) * a) - (sqrt(pow(a, 4.0)) / (y_45_scale * y_45_scale))))) / t_0) / ((a * 4.0) * fabs(b))) * ((x_45_scale * x_45_scale) * y_45_scale)) * y_45_scale;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * Math.abs(b);
	double tmp;
	if (Math.abs(b) <= 1.5e+74) {
		tmp = (-0.25 / Math.abs(b)) * (((a * (x_45_scale * x_45_scale)) * (Math.sqrt((8.0 * ((1.0 - Math.sqrt(Math.pow(Math.cos(((Math.PI * angle) * 0.005555555555555556)), 4.0))) * Math.pow(Math.abs(b), 4.0)))) / Math.abs(x_45_scale))) / Math.abs(b));
	} else {
		tmp = (((Math.sqrt(((((((t_0 * Math.abs(b)) * (-a / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)))) * 8.0) * (-a * Math.abs(b))) * t_0) * (((a / (y_45_scale * y_45_scale)) * a) - (Math.sqrt(Math.pow(a, 4.0)) / (y_45_scale * y_45_scale))))) / t_0) / ((a * 4.0) * Math.abs(b))) * ((x_45_scale * x_45_scale) * y_45_scale)) * y_45_scale;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = a * math.fabs(b)
	tmp = 0
	if math.fabs(b) <= 1.5e+74:
		tmp = (-0.25 / math.fabs(b)) * (((a * (x_45_scale * x_45_scale)) * (math.sqrt((8.0 * ((1.0 - math.sqrt(math.pow(math.cos(((math.pi * angle) * 0.005555555555555556)), 4.0))) * math.pow(math.fabs(b), 4.0)))) / math.fabs(x_45_scale))) / math.fabs(b))
	else:
		tmp = (((math.sqrt(((((((t_0 * math.fabs(b)) * (-a / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)))) * 8.0) * (-a * math.fabs(b))) * t_0) * (((a / (y_45_scale * y_45_scale)) * a) - (math.sqrt(math.pow(a, 4.0)) / (y_45_scale * y_45_scale))))) / t_0) / ((a * 4.0) * math.fabs(b))) * ((x_45_scale * x_45_scale) * y_45_scale)) * y_45_scale
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a * abs(b))
	tmp = 0.0
	if (abs(b) <= 1.5e+74)
		tmp = Float64(Float64(-0.25 / abs(b)) * Float64(Float64(Float64(a * Float64(x_45_scale * x_45_scale)) * Float64(sqrt(Float64(8.0 * Float64(Float64(1.0 - sqrt((cos(Float64(Float64(pi * angle) * 0.005555555555555556)) ^ 4.0))) * (abs(b) ^ 4.0)))) / abs(x_45_scale))) / abs(b)));
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(t_0 * abs(b)) * Float64(Float64(-a) / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale)))) * 8.0) * Float64(Float64(-a) * abs(b))) * t_0) * Float64(Float64(Float64(a / Float64(y_45_scale * y_45_scale)) * a) - Float64(sqrt((a ^ 4.0)) / Float64(y_45_scale * y_45_scale))))) / t_0) / Float64(Float64(a * 4.0) * abs(b))) * Float64(Float64(x_45_scale * x_45_scale) * y_45_scale)) * y_45_scale);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = a * abs(b);
	tmp = 0.0;
	if (abs(b) <= 1.5e+74)
		tmp = (-0.25 / abs(b)) * (((a * (x_45_scale * x_45_scale)) * (sqrt((8.0 * ((1.0 - sqrt((cos(((pi * angle) * 0.005555555555555556)) ^ 4.0))) * (abs(b) ^ 4.0)))) / abs(x_45_scale))) / abs(b));
	else
		tmp = (((sqrt(((((((t_0 * abs(b)) * (-a / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale)))) * 8.0) * (-a * abs(b))) * t_0) * (((a / (y_45_scale * y_45_scale)) * a) - (sqrt((a ^ 4.0)) / (y_45_scale * y_45_scale))))) / t_0) / ((a * 4.0) * abs(b))) * ((x_45_scale * x_45_scale) * y_45_scale)) * y_45_scale;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 1.5e+74], N[(N[(-0.25 / N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(a * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(8.0 * N[(N[(1.0 - N[Sqrt[N[Power[N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[N[(N[(N[(N[(N[(N[(t$95$0 * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[((-a) / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision] * N[((-a) * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] - N[(N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[(a * 4.0), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5e74

   1. Initial program 0.1%

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

   2. Taylor expanded in a around -inf

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 4.4%

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

   9. Applied rewrites 18.7%

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

   10. Taylor expanded in angle around 0

       \[\leadsto \frac{\frac{-1}{4}}{b} \cdot \frac{\left(a \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{\sqrt{8 \cdot \left(\left(1 - \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right) \cdot {b}^{4}\right)}}{\left|x-scale\right|}}{b} \]
   11. Step-by-step derivation

   12. Applied rewrites 19.3%

       \[\leadsto \frac{-0.25}{b} \cdot \frac{\left(a \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{\sqrt{8 \cdot \left(\left(1 - \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right) \cdot {b}^{4}\right)}}{\left|x-scale\right|}}{b} \]

   if 1.5e74 < b

   1. Initial program 0.1%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 2.0

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

   7. Applied rewrites 2.0%

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

   9. Applied rewrites 0.3%

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

   11. Applied rewrites 4.9%

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

Alternative 6: 19.3% accurate, 7.2× speedup

Math

\[\frac{-0.25}{b} \cdot \frac{\left(a \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{\sqrt{8 \cdot \left(\left(1 - \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right) \cdot {b}^{4}\right)}}{\left|x-scale\right|}}{b} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  (/ -0.25 b)
  (/
   (*
    (* a (* x-scale x-scale))
    (/
     (sqrt
      (*
       8.0
       (*
        (- 1.0 (sqrt (pow (cos (* (* PI angle) 0.005555555555555556)) 4.0)))
        (pow b 4.0))))
     (fabs x-scale)))
   b)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (-0.25 / b) * (((a * (x_45_scale * x_45_scale)) * (sqrt((8.0 * ((1.0 - sqrt(pow(cos(((((double) M_PI) * angle) * 0.005555555555555556)), 4.0))) * pow(b, 4.0)))) / fabs(x_45_scale))) / b);
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (-0.25 / b) * (((a * (x_45_scale * x_45_scale)) * (Math.sqrt((8.0 * ((1.0 - Math.sqrt(Math.pow(Math.cos(((Math.PI * angle) * 0.005555555555555556)), 4.0))) * Math.pow(b, 4.0)))) / Math.abs(x_45_scale))) / b);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (-0.25 / b) * (((a * (x_45_scale * x_45_scale)) * (math.sqrt((8.0 * ((1.0 - math.sqrt(math.pow(math.cos(((math.pi * angle) * 0.005555555555555556)), 4.0))) * math.pow(b, 4.0)))) / math.fabs(x_45_scale))) / b)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(-0.25 / b) * Float64(Float64(Float64(a * Float64(x_45_scale * x_45_scale)) * Float64(sqrt(Float64(8.0 * Float64(Float64(1.0 - sqrt((cos(Float64(Float64(pi * angle) * 0.005555555555555556)) ^ 4.0))) * (b ^ 4.0)))) / abs(x_45_scale))) / b))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = (-0.25 / b) * (((a * (x_45_scale * x_45_scale)) * (sqrt((8.0 * ((1.0 - sqrt((cos(((pi * angle) * 0.005555555555555556)) ^ 4.0))) * (b ^ 4.0)))) / abs(x_45_scale))) / b);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(-0.25 / b), $MachinePrecision] * N[(N[(N[(a * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(8.0 * N[(N[(1.0 - N[Sqrt[N[Power[N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1%

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

2. Taylor expanded in a around -inf

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

4. Applied rewrites 0.5%

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

5. Taylor expanded in y-scale around 0

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

   1. lower-sqrt.f64 N/A

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

7. Applied rewrites 4.4%

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

9. Applied rewrites 18.7%

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

10. Taylor expanded in angle around 0

    \[\leadsto \frac{\frac{-1}{4}}{b} \cdot \frac{\left(a \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{\sqrt{8 \cdot \left(\left(1 - \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right) \cdot {b}^{4}\right)}}{\left|x-scale\right|}}{b} \]
11. Step-by-step derivation

12. Applied rewrites 19.3%

    \[\leadsto \frac{-0.25}{b} \cdot \frac{\left(a \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{\sqrt{8 \cdot \left(\left(1 - \sqrt{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}\right) \cdot {b}^{4}\right)}}{\left|x-scale\right|}}{b} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025171 
(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))))