a from scale-rotated-ellipse

Percentage Accurate: 2.7% → 24.7%

Time: 26.3s

Alternatives: 13

Speedup: 8.4×

• 70.1% of points produce a very large (infinite) output. You may want to add a precondition.

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: 2.7% 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: 24.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := b \cdot \left|a\right|\\ t_2 := x-scale \cdot \left|y-scale\right|\\ t_3 := \left|t\_2\right|\\ t_4 := \left|y-scale\right| \cdot x-scale\\ t_5 := \frac{\frac{{\left(\left|a\right|\right)}^{2}}{\left|y-scale\right|}}{\left|y-scale\right|}\\ t_6 := \left(t\_1 \cdot b\right) \cdot \left(-\left|a\right|\right)\\ t_7 := \cos t\_0\\ t_8 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ t_9 := \frac{\mathsf{fma}\left(\left(0.5 - t\_7 \cdot 0.5\right) \cdot \left|a\right|, \left|a\right|, \mathsf{fma}\left(t\_7, 0.5, 0.5\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\\ \mathbf{if}\;\left|y-scale\right| \leq 3.9 \cdot 10^{+89}:\\ \;\;\;\;\left(\left(0.25 \cdot \frac{\frac{\left|a\right| \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + t\_8\right)}^{2}} + t\_8\right)\right)}}{\left|y-scale\right|}}{t\_3}\right) \cdot t\_4\right) \cdot t\_4\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\left(\left(t\_6 \cdot 8\right) \cdot \left(\mathsf{hypot}\left(t\_5 - t\_9, \frac{\sin t\_0 \cdot \left(\left(b - \left|a\right|\right) \cdot \left(b + \left|a\right|\right)\right)}{t\_2}\right) + \left(t\_5 + t\_9\right)\right)\right) \cdot t\_6}}{t\_3}}{4 \cdot t\_1}}{t\_1} \cdot t\_4\right) \cdot t\_4\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (* 2.0 PI) (* angle 0.005555555555555556)))
       (t_1 (* b (fabs a)))
       (t_2 (* x-scale (fabs y-scale)))
       (t_3 (fabs t_2))
       (t_4 (* (fabs y-scale) x-scale))
       (t_5 (/ (/ (pow (fabs a) 2.0) (fabs y-scale)) (fabs y-scale)))
       (t_6 (* (* t_1 b) (- (fabs a))))
       (t_7 (cos t_0))
       (t_8 (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
       (t_9
        (/
         (fma
          (* (- 0.5 (* t_7 0.5)) (fabs a))
          (fabs a)
          (* (fma t_7 0.5 0.5) (* b b)))
         (* x-scale x-scale))))
  (if (<= (fabs y-scale) 3.9e+89)
    (*
     (*
      (*
       0.25
       (/
        (/
         (*
          (fabs a)
          (sqrt (* 8.0 (+ 0.5 (+ (sqrt (pow (+ 0.5 t_8) 2.0)) t_8)))))
         (fabs y-scale))
        t_3))
      t_4)
     t_4)
    (*
     (*
      (/
       (/
        (/
         (sqrt
          (*
           (*
            (* t_6 8.0)
            (+
             (hypot
              (- t_5 t_9)
              (/ (* (sin t_0) (* (- b (fabs a)) (+ b (fabs a)))) t_2))
             (+ t_5 t_9)))
           t_6))
         t_3)
        (* 4.0 t_1))
       t_1)
      t_4)
     t_4))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (2.0 * ((double) M_PI)) * (angle * 0.005555555555555556);
	double t_1 = b * fabs(a);
	double t_2 = x_45_scale * fabs(y_45_scale);
	double t_3 = fabs(t_2);
	double t_4 = fabs(y_45_scale) * x_45_scale;
	double t_5 = (pow(fabs(a), 2.0) / fabs(y_45_scale)) / fabs(y_45_scale);
	double t_6 = (t_1 * b) * -fabs(a);
	double t_7 = cos(t_0);
	double t_8 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double t_9 = fma(((0.5 - (t_7 * 0.5)) * fabs(a)), fabs(a), (fma(t_7, 0.5, 0.5) * (b * b))) / (x_45_scale * x_45_scale);
	double tmp;
	if (fabs(y_45_scale) <= 3.9e+89) {
		tmp = ((0.25 * (((fabs(a) * sqrt((8.0 * (0.5 + (sqrt(pow((0.5 + t_8), 2.0)) + t_8))))) / fabs(y_45_scale)) / t_3)) * t_4) * t_4;
	} else {
		tmp = ((((sqrt((((t_6 * 8.0) * (hypot((t_5 - t_9), ((sin(t_0) * ((b - fabs(a)) * (b + fabs(a)))) / t_2)) + (t_5 + t_9))) * t_6)) / t_3) / (4.0 * t_1)) / t_1) * t_4) * t_4;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(2.0 * pi) * Float64(angle * 0.005555555555555556))
	t_1 = Float64(b * abs(a))
	t_2 = Float64(x_45_scale * abs(y_45_scale))
	t_3 = abs(t_2)
	t_4 = Float64(abs(y_45_scale) * x_45_scale)
	t_5 = Float64(Float64((abs(a) ^ 2.0) / abs(y_45_scale)) / abs(y_45_scale))
	t_6 = Float64(Float64(t_1 * b) * Float64(-abs(a)))
	t_7 = cos(t_0)
	t_8 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	t_9 = Float64(fma(Float64(Float64(0.5 - Float64(t_7 * 0.5)) * abs(a)), abs(a), Float64(fma(t_7, 0.5, 0.5) * Float64(b * b))) / Float64(x_45_scale * x_45_scale))
	tmp = 0.0
	if (abs(y_45_scale) <= 3.9e+89)
		tmp = Float64(Float64(Float64(0.25 * Float64(Float64(Float64(abs(a) * sqrt(Float64(8.0 * Float64(0.5 + Float64(sqrt((Float64(0.5 + t_8) ^ 2.0)) + t_8))))) / abs(y_45_scale)) / t_3)) * t_4) * t_4);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64(t_6 * 8.0) * Float64(hypot(Float64(t_5 - t_9), Float64(Float64(sin(t_0) * Float64(Float64(b - abs(a)) * Float64(b + abs(a)))) / t_2)) + Float64(t_5 + t_9))) * t_6)) / t_3) / Float64(4.0 * t_1)) / t_1) * t_4) * t_4);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(2.0 * Pi), $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x$45$scale * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Abs[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[y$45$scale], $MachinePrecision] * x$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$1 * b), $MachinePrecision] * (-N[Abs[a], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$7 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$8 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(N[(N[(N[(0.5 - N[(t$95$7 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision] + N[(N[(t$95$7 * 0.5 + 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 3.9e+89], N[(N[(N[(0.25 * N[(N[(N[(N[Abs[a], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(0.5 + N[(N[Sqrt[N[Power[N[(0.5 + t$95$8), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] * t$95$4), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[(t$95$6 * 8.0), $MachinePrecision] * N[(N[Sqrt[N[(t$95$5 - t$95$9), $MachinePrecision] ^ 2 + N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[(N[(b - N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(b + N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] ^ 2], $MachinePrecision] + N[(t$95$5 + t$95$9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$6), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision] / N[(4.0 * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * t$95$4), $MachinePrecision] * t$95$4), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 3.9000000000000001e89

   1. Initial program 2.7%

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

   3. Applied rewrites 6.7%

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

   4. Taylor expanded in b around 0

      \[\leadsto \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   6. Applied rewrites 2.5%

      \[\leadsto \left(\color{blue}{\left(0.25 \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   7. Taylor expanded in a around 0

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} - \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Applied rewrites 7.2%

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(0.5 \cdot \frac{1}{{x-scale}^{2}} - \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   10. Taylor expanded in y-scale around 0

       \[\leadsto \left(\left(0.25 \cdot \frac{\frac{a \cdot \sqrt{8 \cdot \left(\frac{1}{2} + \left(\sqrt{{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{y-scale}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{1}{4} \cdot \frac{\frac{a \cdot \sqrt{8 \cdot \left(\frac{1}{2} + \left(\sqrt{{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}}{y-scale}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   12. Applied rewrites 9.9%

       \[\leadsto \left(\left(0.25 \cdot \frac{\frac{a \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{y-scale}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   if 3.9000000000000001e89 < y-scale

   1. Initial program 2.7%

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

   3. Applied rewrites 6.7%

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

   5. Applied rewrites 10.5%

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

   7. Applied rewrites 10.5%

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

   9. Applied rewrites 13.6%

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

   10. Taylor expanded in angle around 0

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\color{blue}{\frac{{a}^{2}}{y-scale}}}{y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(0.5, \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right), 0.5\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)}{y-scale}}{y-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. lower-pow.f64 13.6%

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

   12. Applied rewrites 13.6%

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

   13. Taylor expanded in angle around 0

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\frac{{a}^{2}}{y-scale}}{y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\color{blue}{\frac{{a}^{2}}{y-scale}}}{y-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   14. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. lower-pow.f64 15.5%

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

   15. Applied rewrites 15.5%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\frac{{a}^{2}}{y-scale}}{y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\color{blue}{\frac{{a}^{2}}{y-scale}}}{y-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 22.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} t_0 := b \cdot \left|a\right|\\ t_1 := \left(t\_0 \cdot b\right) \cdot \left(-\left|a\right|\right)\\ t_2 := \left|x-scale \cdot \left|y-scale\right|\right|\\ t_3 := \left|y-scale\right| \cdot x-scale\\ t_4 := \frac{{\left(\left|a\right|\right)}^{2}}{{\left(\left|y-scale\right|\right)}^{2}}\\ t_5 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ t_6 := \frac{{b}^{2}}{{x-scale}^{2}}\\ \mathbf{if}\;\left|y-scale\right| \leq 1.26 \cdot 10^{+94}:\\ \;\;\;\;\left(\left(0.25 \cdot \frac{\frac{\left|a\right| \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + t\_5\right)}^{2}} + t\_5\right)\right)}}{\left|y-scale\right|}}{t\_2}\right) \cdot t\_3\right) \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\left(\left(t\_1 \cdot 8\right) \cdot \left(\sqrt{{\left(t\_4 - t\_6\right)}^{2}} + \left(t\_4 + t\_6\right)\right)\right) \cdot t\_1}}{t\_2}}{4 \cdot t\_0}}{t\_0} \cdot t\_3\right) \cdot t\_3\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* b (fabs a)))
       (t_1 (* (* t_0 b) (- (fabs a))))
       (t_2 (fabs (* x-scale (fabs y-scale))))
       (t_3 (* (fabs y-scale) x-scale))
       (t_4 (/ (pow (fabs a) 2.0) (pow (fabs y-scale) 2.0)))
       (t_5 (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
       (t_6 (/ (pow b 2.0) (pow x-scale 2.0))))
  (if (<= (fabs y-scale) 1.26e+94)
    (*
     (*
      (*
       0.25
       (/
        (/
         (*
          (fabs a)
          (sqrt (* 8.0 (+ 0.5 (+ (sqrt (pow (+ 0.5 t_5) 2.0)) t_5)))))
         (fabs y-scale))
        t_2))
      t_3)
     t_3)
    (*
     (*
      (/
       (/
        (/
         (sqrt
          (*
           (*
            (* t_1 8.0)
            (+ (sqrt (pow (- t_4 t_6) 2.0)) (+ t_4 t_6)))
           t_1))
         t_2)
        (* 4.0 t_0))
       t_0)
      t_3)
     t_3))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b * fabs(a);
	double t_1 = (t_0 * b) * -fabs(a);
	double t_2 = fabs((x_45_scale * fabs(y_45_scale)));
	double t_3 = fabs(y_45_scale) * x_45_scale;
	double t_4 = pow(fabs(a), 2.0) / pow(fabs(y_45_scale), 2.0);
	double t_5 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double t_6 = pow(b, 2.0) / pow(x_45_scale, 2.0);
	double tmp;
	if (fabs(y_45_scale) <= 1.26e+94) {
		tmp = ((0.25 * (((fabs(a) * sqrt((8.0 * (0.5 + (sqrt(pow((0.5 + t_5), 2.0)) + t_5))))) / fabs(y_45_scale)) / t_2)) * t_3) * t_3;
	} else {
		tmp = ((((sqrt((((t_1 * 8.0) * (sqrt(pow((t_4 - t_6), 2.0)) + (t_4 + t_6))) * t_1)) / t_2) / (4.0 * t_0)) / t_0) * t_3) * t_3;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b * Math.abs(a);
	double t_1 = (t_0 * b) * -Math.abs(a);
	double t_2 = Math.abs((x_45_scale * Math.abs(y_45_scale)));
	double t_3 = Math.abs(y_45_scale) * x_45_scale;
	double t_4 = Math.pow(Math.abs(a), 2.0) / Math.pow(Math.abs(y_45_scale), 2.0);
	double t_5 = 0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI)));
	double t_6 = Math.pow(b, 2.0) / Math.pow(x_45_scale, 2.0);
	double tmp;
	if (Math.abs(y_45_scale) <= 1.26e+94) {
		tmp = ((0.25 * (((Math.abs(a) * Math.sqrt((8.0 * (0.5 + (Math.sqrt(Math.pow((0.5 + t_5), 2.0)) + t_5))))) / Math.abs(y_45_scale)) / t_2)) * t_3) * t_3;
	} else {
		tmp = ((((Math.sqrt((((t_1 * 8.0) * (Math.sqrt(Math.pow((t_4 - t_6), 2.0)) + (t_4 + t_6))) * t_1)) / t_2) / (4.0 * t_0)) / t_0) * t_3) * t_3;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = b * math.fabs(a)
	t_1 = (t_0 * b) * -math.fabs(a)
	t_2 = math.fabs((x_45_scale * math.fabs(y_45_scale)))
	t_3 = math.fabs(y_45_scale) * x_45_scale
	t_4 = math.pow(math.fabs(a), 2.0) / math.pow(math.fabs(y_45_scale), 2.0)
	t_5 = 0.5 * math.cos((0.011111111111111112 * (angle * math.pi)))
	t_6 = math.pow(b, 2.0) / math.pow(x_45_scale, 2.0)
	tmp = 0
	if math.fabs(y_45_scale) <= 1.26e+94:
		tmp = ((0.25 * (((math.fabs(a) * math.sqrt((8.0 * (0.5 + (math.sqrt(math.pow((0.5 + t_5), 2.0)) + t_5))))) / math.fabs(y_45_scale)) / t_2)) * t_3) * t_3
	else:
		tmp = ((((math.sqrt((((t_1 * 8.0) * (math.sqrt(math.pow((t_4 - t_6), 2.0)) + (t_4 + t_6))) * t_1)) / t_2) / (4.0 * t_0)) / t_0) * t_3) * t_3
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b * abs(a))
	t_1 = Float64(Float64(t_0 * b) * Float64(-abs(a)))
	t_2 = abs(Float64(x_45_scale * abs(y_45_scale)))
	t_3 = Float64(abs(y_45_scale) * x_45_scale)
	t_4 = Float64((abs(a) ^ 2.0) / (abs(y_45_scale) ^ 2.0))
	t_5 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	t_6 = Float64((b ^ 2.0) / (x_45_scale ^ 2.0))
	tmp = 0.0
	if (abs(y_45_scale) <= 1.26e+94)
		tmp = Float64(Float64(Float64(0.25 * Float64(Float64(Float64(abs(a) * sqrt(Float64(8.0 * Float64(0.5 + Float64(sqrt((Float64(0.5 + t_5) ^ 2.0)) + t_5))))) / abs(y_45_scale)) / t_2)) * t_3) * t_3);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64(t_1 * 8.0) * Float64(sqrt((Float64(t_4 - t_6) ^ 2.0)) + Float64(t_4 + t_6))) * t_1)) / t_2) / Float64(4.0 * t_0)) / t_0) * t_3) * t_3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = b * abs(a);
	t_1 = (t_0 * b) * -abs(a);
	t_2 = abs((x_45_scale * abs(y_45_scale)));
	t_3 = abs(y_45_scale) * x_45_scale;
	t_4 = (abs(a) ^ 2.0) / (abs(y_45_scale) ^ 2.0);
	t_5 = 0.5 * cos((0.011111111111111112 * (angle * pi)));
	t_6 = (b ^ 2.0) / (x_45_scale ^ 2.0);
	tmp = 0.0;
	if (abs(y_45_scale) <= 1.26e+94)
		tmp = ((0.25 * (((abs(a) * sqrt((8.0 * (0.5 + (sqrt(((0.5 + t_5) ^ 2.0)) + t_5))))) / abs(y_45_scale)) / t_2)) * t_3) * t_3;
	else
		tmp = ((((sqrt((((t_1 * 8.0) * (sqrt(((t_4 - t_6) ^ 2.0)) + (t_4 + t_6))) * t_1)) / t_2) / (4.0 * t_0)) / t_0) * t_3) * t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * b), $MachinePrecision] * (-N[Abs[a], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(x$45$scale * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[y$45$scale], $MachinePrecision] * x$45$scale), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Power[b, 2.0], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 1.26e+94], N[(N[(N[(0.25 * N[(N[(N[(N[Abs[a], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(0.5 + N[(N[Sqrt[N[Power[N[(0.5 + t$95$5), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[(t$95$1 * 8.0), $MachinePrecision] * N[(N[Sqrt[N[Power[N[(t$95$4 - t$95$6), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + N[(t$95$4 + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision] / N[(4.0 * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.26e94

   1. Initial program 2.7%

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

   3. Applied rewrites 6.7%

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

   4. Taylor expanded in b around 0

      \[\leadsto \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   6. Applied rewrites 2.5%

      \[\leadsto \left(\color{blue}{\left(0.25 \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   7. Taylor expanded in a around 0

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} - \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Applied rewrites 7.2%

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(0.5 \cdot \frac{1}{{x-scale}^{2}} - \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   10. Taylor expanded in y-scale around 0

       \[\leadsto \left(\left(0.25 \cdot \frac{\frac{a \cdot \sqrt{8 \cdot \left(\frac{1}{2} + \left(\sqrt{{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{y-scale}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{1}{4} \cdot \frac{\frac{a \cdot \sqrt{8 \cdot \left(\frac{1}{2} + \left(\sqrt{{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}}{y-scale}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   12. Applied rewrites 9.9%

       \[\leadsto \left(\left(0.25 \cdot \frac{\frac{a \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{y-scale}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   if 1.26e94 < y-scale

   1. Initial program 2.7%

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

   3. Applied rewrites 6.7%

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

   5. Applied rewrites 10.5%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{a}^{2}}{{y-scale}^{2}} - \frac{{b}^{2}}{{x-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 13.3%

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

Alternative 3: 21.4% accurate, 4.6× speedup

Math

\[\begin{array}{l} t_0 := \left|x-scale \cdot \left|y-scale\right|\right|\\ t_1 := \left|y-scale\right| \cdot x-scale\\ t_2 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;\left|y-scale\right| \leq 3.2 \cdot 10^{+70}:\\ \;\;\;\;\left(\left(0.25 \cdot \frac{\frac{\left|a\right| \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + t\_2\right)}^{2}} + t\_2\right)\right)}}{\left|y-scale\right|}}{t\_0}\right) \cdot t\_1\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 \cdot \frac{\left|a\right| \cdot \sqrt{8 \cdot \frac{\left(0.5 + \sqrt{{\left(0.5 - t\_2\right)}^{2}}\right) - t\_2}{{x-scale}^{2}}}}{t\_0}\right) \cdot t\_1\right) \cdot t\_1\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (fabs (* x-scale (fabs y-scale))))
       (t_1 (* (fabs y-scale) x-scale))
       (t_2 (* 0.5 (cos (* 0.011111111111111112 (* angle PI))))))
  (if (<= (fabs y-scale) 3.2e+70)
    (*
     (*
      (*
       0.25
       (/
        (/
         (*
          (fabs a)
          (sqrt (* 8.0 (+ 0.5 (+ (sqrt (pow (+ 0.5 t_2) 2.0)) t_2)))))
         (fabs y-scale))
        t_0))
      t_1)
     t_1)
    (*
     (*
      (*
       0.25
       (/
        (*
         (fabs a)
         (sqrt
          (*
           8.0
           (/
            (- (+ 0.5 (sqrt (pow (- 0.5 t_2) 2.0))) t_2)
            (pow x-scale 2.0)))))
        t_0))
      t_1)
     t_1))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs((x_45_scale * fabs(y_45_scale)));
	double t_1 = fabs(y_45_scale) * x_45_scale;
	double t_2 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double tmp;
	if (fabs(y_45_scale) <= 3.2e+70) {
		tmp = ((0.25 * (((fabs(a) * sqrt((8.0 * (0.5 + (sqrt(pow((0.5 + t_2), 2.0)) + t_2))))) / fabs(y_45_scale)) / t_0)) * t_1) * t_1;
	} else {
		tmp = ((0.25 * ((fabs(a) * sqrt((8.0 * (((0.5 + sqrt(pow((0.5 - t_2), 2.0))) - t_2) / pow(x_45_scale, 2.0))))) / t_0)) * t_1) * t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.abs((x_45_scale * Math.abs(y_45_scale)));
	double t_1 = Math.abs(y_45_scale) * x_45_scale;
	double t_2 = 0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI)));
	double tmp;
	if (Math.abs(y_45_scale) <= 3.2e+70) {
		tmp = ((0.25 * (((Math.abs(a) * Math.sqrt((8.0 * (0.5 + (Math.sqrt(Math.pow((0.5 + t_2), 2.0)) + t_2))))) / Math.abs(y_45_scale)) / t_0)) * t_1) * t_1;
	} else {
		tmp = ((0.25 * ((Math.abs(a) * Math.sqrt((8.0 * (((0.5 + Math.sqrt(Math.pow((0.5 - t_2), 2.0))) - t_2) / Math.pow(x_45_scale, 2.0))))) / t_0)) * t_1) * t_1;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.fabs((x_45_scale * math.fabs(y_45_scale)))
	t_1 = math.fabs(y_45_scale) * x_45_scale
	t_2 = 0.5 * math.cos((0.011111111111111112 * (angle * math.pi)))
	tmp = 0
	if math.fabs(y_45_scale) <= 3.2e+70:
		tmp = ((0.25 * (((math.fabs(a) * math.sqrt((8.0 * (0.5 + (math.sqrt(math.pow((0.5 + t_2), 2.0)) + t_2))))) / math.fabs(y_45_scale)) / t_0)) * t_1) * t_1
	else:
		tmp = ((0.25 * ((math.fabs(a) * math.sqrt((8.0 * (((0.5 + math.sqrt(math.pow((0.5 - t_2), 2.0))) - t_2) / math.pow(x_45_scale, 2.0))))) / t_0)) * t_1) * t_1
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(Float64(x_45_scale * abs(y_45_scale)))
	t_1 = Float64(abs(y_45_scale) * x_45_scale)
	t_2 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	tmp = 0.0
	if (abs(y_45_scale) <= 3.2e+70)
		tmp = Float64(Float64(Float64(0.25 * Float64(Float64(Float64(abs(a) * sqrt(Float64(8.0 * Float64(0.5 + Float64(sqrt((Float64(0.5 + t_2) ^ 2.0)) + t_2))))) / abs(y_45_scale)) / t_0)) * t_1) * t_1);
	else
		tmp = Float64(Float64(Float64(0.25 * Float64(Float64(abs(a) * sqrt(Float64(8.0 * Float64(Float64(Float64(0.5 + sqrt((Float64(0.5 - t_2) ^ 2.0))) - t_2) / (x_45_scale ^ 2.0))))) / t_0)) * t_1) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs((x_45_scale * abs(y_45_scale)));
	t_1 = abs(y_45_scale) * x_45_scale;
	t_2 = 0.5 * cos((0.011111111111111112 * (angle * pi)));
	tmp = 0.0;
	if (abs(y_45_scale) <= 3.2e+70)
		tmp = ((0.25 * (((abs(a) * sqrt((8.0 * (0.5 + (sqrt(((0.5 + t_2) ^ 2.0)) + t_2))))) / abs(y_45_scale)) / t_0)) * t_1) * t_1;
	else
		tmp = ((0.25 * ((abs(a) * sqrt((8.0 * (((0.5 + sqrt(((0.5 - t_2) ^ 2.0))) - t_2) / (x_45_scale ^ 2.0))))) / t_0)) * t_1) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Abs[N[(x$45$scale * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[y$45$scale], $MachinePrecision] * x$45$scale), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 3.2e+70], N[(N[(N[(0.25 * N[(N[(N[(N[Abs[a], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(0.5 + N[(N[Sqrt[N[Power[N[(0.5 + t$95$2), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(0.25 * N[(N[(N[Abs[a], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[(0.5 + N[Sqrt[N[Power[N[(0.5 - t$95$2), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 3.2000000000000002e70

   1. Initial program 2.7%

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

   3. Applied rewrites 6.7%

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

   4. Taylor expanded in b around 0

      \[\leadsto \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   6. Applied rewrites 2.5%

      \[\leadsto \left(\color{blue}{\left(0.25 \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   7. Taylor expanded in a around 0

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} - \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Applied rewrites 7.2%

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(0.5 \cdot \frac{1}{{x-scale}^{2}} - \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   10. Taylor expanded in y-scale around 0

       \[\leadsto \left(\left(0.25 \cdot \frac{\frac{a \cdot \sqrt{8 \cdot \left(\frac{1}{2} + \left(\sqrt{{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{y-scale}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{1}{4} \cdot \frac{\frac{a \cdot \sqrt{8 \cdot \left(\frac{1}{2} + \left(\sqrt{{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}}{y-scale}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   12. Applied rewrites 9.9%

       \[\leadsto \left(\left(0.25 \cdot \frac{\frac{a \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{y-scale}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   if 3.2000000000000002e70 < y-scale

   1. Initial program 2.7%

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

   3. Applied rewrites 6.7%

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

   4. Taylor expanded in b around 0

      \[\leadsto \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   6. Applied rewrites 2.5%

      \[\leadsto \left(\color{blue}{\left(0.25 \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   7. Taylor expanded in a around 0

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} - \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Applied rewrites 7.2%

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(0.5 \cdot \frac{1}{{x-scale}^{2}} - \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   10. Taylor expanded in x-scale around 0

       \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \frac{\left(\frac{1}{2} + \sqrt{{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}\right) - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{1}{4} \cdot \frac{a \cdot \sqrt{8 \cdot \frac{\left(\frac{1}{2} + \sqrt{{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}}\right) - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{{x-scale}^{2}}}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   12. Applied rewrites 4.8%

       \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \frac{\left(0.5 + \sqrt{{\left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}\right) - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 20.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} t_0 := \left|\left|x-scale\right| \cdot \left|y-scale\right|\right|\\ t_1 := \left|y-scale\right| \cdot \left|x-scale\right|\\ t_2 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;\left|y-scale\right| \leq 2.2 \cdot 10^{+69}:\\ \;\;\;\;\left(\left(0.25 \cdot \frac{\frac{\left|a\right| \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + t\_2\right)}^{2}} + t\_2\right)\right)}}{\left|y-scale\right|}}{t\_0}\right) \cdot t\_1\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 \cdot \frac{\left|a\right| \cdot \frac{\sqrt{8 \cdot \left(\left(0.5 + \sqrt{{\left(0.5 - t\_2\right)}^{2}}\right) - t\_2\right)}}{\left|x-scale\right|}}{t\_0}\right) \cdot t\_1\right) \cdot t\_1\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (fabs (* (fabs x-scale) (fabs y-scale))))
       (t_1 (* (fabs y-scale) (fabs x-scale)))
       (t_2 (* 0.5 (cos (* 0.011111111111111112 (* angle PI))))))
  (if (<= (fabs y-scale) 2.2e+69)
    (*
     (*
      (*
       0.25
       (/
        (/
         (*
          (fabs a)
          (sqrt (* 8.0 (+ 0.5 (+ (sqrt (pow (+ 0.5 t_2) 2.0)) t_2)))))
         (fabs y-scale))
        t_0))
      t_1)
     t_1)
    (*
     (*
      (*
       0.25
       (/
        (*
         (fabs a)
         (/
          (sqrt (* 8.0 (- (+ 0.5 (sqrt (pow (- 0.5 t_2) 2.0))) t_2)))
          (fabs x-scale)))
        t_0))
      t_1)
     t_1))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs((fabs(x_45_scale) * fabs(y_45_scale)));
	double t_1 = fabs(y_45_scale) * fabs(x_45_scale);
	double t_2 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double tmp;
	if (fabs(y_45_scale) <= 2.2e+69) {
		tmp = ((0.25 * (((fabs(a) * sqrt((8.0 * (0.5 + (sqrt(pow((0.5 + t_2), 2.0)) + t_2))))) / fabs(y_45_scale)) / t_0)) * t_1) * t_1;
	} else {
		tmp = ((0.25 * ((fabs(a) * (sqrt((8.0 * ((0.5 + sqrt(pow((0.5 - t_2), 2.0))) - t_2))) / fabs(x_45_scale))) / t_0)) * t_1) * t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.abs((Math.abs(x_45_scale) * Math.abs(y_45_scale)));
	double t_1 = Math.abs(y_45_scale) * Math.abs(x_45_scale);
	double t_2 = 0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI)));
	double tmp;
	if (Math.abs(y_45_scale) <= 2.2e+69) {
		tmp = ((0.25 * (((Math.abs(a) * Math.sqrt((8.0 * (0.5 + (Math.sqrt(Math.pow((0.5 + t_2), 2.0)) + t_2))))) / Math.abs(y_45_scale)) / t_0)) * t_1) * t_1;
	} else {
		tmp = ((0.25 * ((Math.abs(a) * (Math.sqrt((8.0 * ((0.5 + Math.sqrt(Math.pow((0.5 - t_2), 2.0))) - t_2))) / Math.abs(x_45_scale))) / t_0)) * t_1) * t_1;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.fabs((math.fabs(x_45_scale) * math.fabs(y_45_scale)))
	t_1 = math.fabs(y_45_scale) * math.fabs(x_45_scale)
	t_2 = 0.5 * math.cos((0.011111111111111112 * (angle * math.pi)))
	tmp = 0
	if math.fabs(y_45_scale) <= 2.2e+69:
		tmp = ((0.25 * (((math.fabs(a) * math.sqrt((8.0 * (0.5 + (math.sqrt(math.pow((0.5 + t_2), 2.0)) + t_2))))) / math.fabs(y_45_scale)) / t_0)) * t_1) * t_1
	else:
		tmp = ((0.25 * ((math.fabs(a) * (math.sqrt((8.0 * ((0.5 + math.sqrt(math.pow((0.5 - t_2), 2.0))) - t_2))) / math.fabs(x_45_scale))) / t_0)) * t_1) * t_1
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(Float64(abs(x_45_scale) * abs(y_45_scale)))
	t_1 = Float64(abs(y_45_scale) * abs(x_45_scale))
	t_2 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	tmp = 0.0
	if (abs(y_45_scale) <= 2.2e+69)
		tmp = Float64(Float64(Float64(0.25 * Float64(Float64(Float64(abs(a) * sqrt(Float64(8.0 * Float64(0.5 + Float64(sqrt((Float64(0.5 + t_2) ^ 2.0)) + t_2))))) / abs(y_45_scale)) / t_0)) * t_1) * t_1);
	else
		tmp = Float64(Float64(Float64(0.25 * Float64(Float64(abs(a) * Float64(sqrt(Float64(8.0 * Float64(Float64(0.5 + sqrt((Float64(0.5 - t_2) ^ 2.0))) - t_2))) / abs(x_45_scale))) / t_0)) * t_1) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs((abs(x_45_scale) * abs(y_45_scale)));
	t_1 = abs(y_45_scale) * abs(x_45_scale);
	t_2 = 0.5 * cos((0.011111111111111112 * (angle * pi)));
	tmp = 0.0;
	if (abs(y_45_scale) <= 2.2e+69)
		tmp = ((0.25 * (((abs(a) * sqrt((8.0 * (0.5 + (sqrt(((0.5 + t_2) ^ 2.0)) + t_2))))) / abs(y_45_scale)) / t_0)) * t_1) * t_1;
	else
		tmp = ((0.25 * ((abs(a) * (sqrt((8.0 * ((0.5 + sqrt(((0.5 - t_2) ^ 2.0))) - t_2))) / abs(x_45_scale))) / t_0)) * t_1) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Abs[N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 2.2e+69], N[(N[(N[(0.25 * N[(N[(N[(N[Abs[a], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(0.5 + N[(N[Sqrt[N[Power[N[(0.5 + t$95$2), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(0.25 * N[(N[(N[Abs[a], $MachinePrecision] * N[(N[Sqrt[N[(8.0 * N[(N[(0.5 + N[Sqrt[N[Power[N[(0.5 - t$95$2), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.2000000000000002e69

   1. Initial program 2.7%

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

   3. Applied rewrites 6.7%

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

   4. Taylor expanded in b around 0

      \[\leadsto \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   6. Applied rewrites 2.5%

      \[\leadsto \left(\color{blue}{\left(0.25 \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   7. Taylor expanded in a around 0

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} - \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Applied rewrites 7.2%

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(0.5 \cdot \frac{1}{{x-scale}^{2}} - \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   10. Taylor expanded in y-scale around 0

       \[\leadsto \left(\left(0.25 \cdot \frac{\frac{a \cdot \sqrt{8 \cdot \left(\frac{1}{2} + \left(\sqrt{{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{y-scale}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{1}{4} \cdot \frac{\frac{a \cdot \sqrt{8 \cdot \left(\frac{1}{2} + \left(\sqrt{{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}}{y-scale}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   12. Applied rewrites 9.9%

       \[\leadsto \left(\left(0.25 \cdot \frac{\frac{a \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{y-scale}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   if 2.2000000000000002e69 < y-scale

   1. Initial program 2.7%

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

   3. Applied rewrites 6.7%

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

   4. Taylor expanded in b around 0

      \[\leadsto \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   6. Applied rewrites 2.5%

      \[\leadsto \left(\color{blue}{\left(0.25 \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   7. Taylor expanded in a around 0

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} - \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Applied rewrites 7.2%

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(0.5 \cdot \frac{1}{{x-scale}^{2}} - \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   10. Taylor expanded in x-scale around 0

       \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \frac{\sqrt{8 \cdot \left(\left(\frac{1}{2} + \sqrt{{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}\right) - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}}{x-scale}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{1}{4} \cdot \frac{a \cdot \frac{\sqrt{8 \cdot \left(\left(\frac{1}{2} + \sqrt{{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}}\right) - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}{x-scale}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   12. Applied rewrites 4.8%

       \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \frac{\sqrt{8 \cdot \left(\left(0.5 + \sqrt{{\left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}\right) - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}}{x-scale}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 20.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} t_0 := \left|\left|x-scale\right| \cdot \left|y-scale\right|\right|\\ t_1 := \left|y-scale\right| \cdot \left|x-scale\right|\\ t_2 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;\left|y-scale\right| \leq 8.5 \cdot 10^{+73}:\\ \;\;\;\;\left(\left(0.25 \cdot \frac{\frac{\left|a\right| \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + t\_2\right)}^{2}} + t\_2\right)\right)}}{\left|y-scale\right|}}{t\_0}\right) \cdot t\_1\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 \cdot \frac{\left|a\right| \cdot \sqrt{8 \cdot \left(\left(0.5 + \sqrt{{\left(0.5 - t\_2\right)}^{2}}\right) - t\_2\right)}}{\left|x-scale\right| \cdot t\_0}\right) \cdot t\_1\right) \cdot t\_1\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (fabs (* (fabs x-scale) (fabs y-scale))))
       (t_1 (* (fabs y-scale) (fabs x-scale)))
       (t_2 (* 0.5 (cos (* 0.011111111111111112 (* angle PI))))))
  (if (<= (fabs y-scale) 8.5e+73)
    (*
     (*
      (*
       0.25
       (/
        (/
         (*
          (fabs a)
          (sqrt (* 8.0 (+ 0.5 (+ (sqrt (pow (+ 0.5 t_2) 2.0)) t_2)))))
         (fabs y-scale))
        t_0))
      t_1)
     t_1)
    (*
     (*
      (*
       0.25
       (/
        (*
         (fabs a)
         (sqrt (* 8.0 (- (+ 0.5 (sqrt (pow (- 0.5 t_2) 2.0))) t_2))))
        (* (fabs x-scale) t_0)))
      t_1)
     t_1))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs((fabs(x_45_scale) * fabs(y_45_scale)));
	double t_1 = fabs(y_45_scale) * fabs(x_45_scale);
	double t_2 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double tmp;
	if (fabs(y_45_scale) <= 8.5e+73) {
		tmp = ((0.25 * (((fabs(a) * sqrt((8.0 * (0.5 + (sqrt(pow((0.5 + t_2), 2.0)) + t_2))))) / fabs(y_45_scale)) / t_0)) * t_1) * t_1;
	} else {
		tmp = ((0.25 * ((fabs(a) * sqrt((8.0 * ((0.5 + sqrt(pow((0.5 - t_2), 2.0))) - t_2)))) / (fabs(x_45_scale) * t_0))) * t_1) * t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.abs((Math.abs(x_45_scale) * Math.abs(y_45_scale)));
	double t_1 = Math.abs(y_45_scale) * Math.abs(x_45_scale);
	double t_2 = 0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI)));
	double tmp;
	if (Math.abs(y_45_scale) <= 8.5e+73) {
		tmp = ((0.25 * (((Math.abs(a) * Math.sqrt((8.0 * (0.5 + (Math.sqrt(Math.pow((0.5 + t_2), 2.0)) + t_2))))) / Math.abs(y_45_scale)) / t_0)) * t_1) * t_1;
	} else {
		tmp = ((0.25 * ((Math.abs(a) * Math.sqrt((8.0 * ((0.5 + Math.sqrt(Math.pow((0.5 - t_2), 2.0))) - t_2)))) / (Math.abs(x_45_scale) * t_0))) * t_1) * t_1;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.fabs((math.fabs(x_45_scale) * math.fabs(y_45_scale)))
	t_1 = math.fabs(y_45_scale) * math.fabs(x_45_scale)
	t_2 = 0.5 * math.cos((0.011111111111111112 * (angle * math.pi)))
	tmp = 0
	if math.fabs(y_45_scale) <= 8.5e+73:
		tmp = ((0.25 * (((math.fabs(a) * math.sqrt((8.0 * (0.5 + (math.sqrt(math.pow((0.5 + t_2), 2.0)) + t_2))))) / math.fabs(y_45_scale)) / t_0)) * t_1) * t_1
	else:
		tmp = ((0.25 * ((math.fabs(a) * math.sqrt((8.0 * ((0.5 + math.sqrt(math.pow((0.5 - t_2), 2.0))) - t_2)))) / (math.fabs(x_45_scale) * t_0))) * t_1) * t_1
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(Float64(abs(x_45_scale) * abs(y_45_scale)))
	t_1 = Float64(abs(y_45_scale) * abs(x_45_scale))
	t_2 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	tmp = 0.0
	if (abs(y_45_scale) <= 8.5e+73)
		tmp = Float64(Float64(Float64(0.25 * Float64(Float64(Float64(abs(a) * sqrt(Float64(8.0 * Float64(0.5 + Float64(sqrt((Float64(0.5 + t_2) ^ 2.0)) + t_2))))) / abs(y_45_scale)) / t_0)) * t_1) * t_1);
	else
		tmp = Float64(Float64(Float64(0.25 * Float64(Float64(abs(a) * sqrt(Float64(8.0 * Float64(Float64(0.5 + sqrt((Float64(0.5 - t_2) ^ 2.0))) - t_2)))) / Float64(abs(x_45_scale) * t_0))) * t_1) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs((abs(x_45_scale) * abs(y_45_scale)));
	t_1 = abs(y_45_scale) * abs(x_45_scale);
	t_2 = 0.5 * cos((0.011111111111111112 * (angle * pi)));
	tmp = 0.0;
	if (abs(y_45_scale) <= 8.5e+73)
		tmp = ((0.25 * (((abs(a) * sqrt((8.0 * (0.5 + (sqrt(((0.5 + t_2) ^ 2.0)) + t_2))))) / abs(y_45_scale)) / t_0)) * t_1) * t_1;
	else
		tmp = ((0.25 * ((abs(a) * sqrt((8.0 * ((0.5 + sqrt(((0.5 - t_2) ^ 2.0))) - t_2)))) / (abs(x_45_scale) * t_0))) * t_1) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Abs[N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 8.5e+73], N[(N[(N[(0.25 * N[(N[(N[(N[Abs[a], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(0.5 + N[(N[Sqrt[N[Power[N[(0.5 + t$95$2), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(0.25 * N[(N[(N[Abs[a], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(0.5 + N[Sqrt[N[Power[N[(0.5 - t$95$2), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[x$45$scale], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 8.4999999999999998e73

   1. Initial program 2.7%

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

   3. Applied rewrites 6.7%

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

   4. Taylor expanded in b around 0

      \[\leadsto \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   6. Applied rewrites 2.5%

      \[\leadsto \left(\color{blue}{\left(0.25 \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   7. Taylor expanded in a around 0

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} - \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Applied rewrites 7.2%

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(0.5 \cdot \frac{1}{{x-scale}^{2}} - \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   10. Taylor expanded in y-scale around 0

       \[\leadsto \left(\left(0.25 \cdot \frac{\frac{a \cdot \sqrt{8 \cdot \left(\frac{1}{2} + \left(\sqrt{{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{y-scale}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{1}{4} \cdot \frac{\frac{a \cdot \sqrt{8 \cdot \left(\frac{1}{2} + \left(\sqrt{{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}}{y-scale}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   12. Applied rewrites 9.9%

       \[\leadsto \left(\left(0.25 \cdot \frac{\frac{a \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{y-scale}}{\left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   if 8.4999999999999998e73 < y-scale

   1. Initial program 2.7%

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

   3. Applied rewrites 6.7%

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

   4. Taylor expanded in b around 0

      \[\leadsto \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   6. Applied rewrites 2.5%

      \[\leadsto \left(\color{blue}{\left(0.25 \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   7. Taylor expanded in a around 0

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} - \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Applied rewrites 7.2%

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(0.5 \cdot \frac{1}{{x-scale}^{2}} - \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   10. Taylor expanded in x-scale around 0

       \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\frac{1}{2} + \sqrt{{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}\right) - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}}{x-scale \cdot \color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{1}{4} \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\frac{1}{2} + \sqrt{{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}}\right) - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}{x-scale \cdot \left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   12. Applied rewrites 4.2%

       \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(0.5 + \sqrt{{\left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}\right) - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}}{x-scale \cdot \color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 19.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} t_0 := \left|\left|x-scale\right| \cdot \left|y-scale\right|\right|\\ t_1 := \left|y-scale\right| \cdot \left|x-scale\right|\\ t_2 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;\left|y-scale\right| \leq 8.5 \cdot 10^{+73}:\\ \;\;\;\;\left(\left(0.25 \cdot \frac{\left|a\right| \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + t\_2\right)}^{2}} + t\_2\right)\right)}}{\left|y-scale\right| \cdot t\_0}\right) \cdot t\_1\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(0.25 \cdot \frac{\left|a\right| \cdot \sqrt{8 \cdot \left(\left(0.5 + \sqrt{{\left(0.5 - t\_2\right)}^{2}}\right) - t\_2\right)}}{\left|x-scale\right| \cdot t\_0}\right) \cdot t\_1\right) \cdot t\_1\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (fabs (* (fabs x-scale) (fabs y-scale))))
       (t_1 (* (fabs y-scale) (fabs x-scale)))
       (t_2 (* 0.5 (cos (* 0.011111111111111112 (* angle PI))))))
  (if (<= (fabs y-scale) 8.5e+73)
    (*
     (*
      (*
       0.25
       (/
        (*
         (fabs a)
         (sqrt (* 8.0 (+ 0.5 (+ (sqrt (pow (+ 0.5 t_2) 2.0)) t_2)))))
        (* (fabs y-scale) t_0)))
      t_1)
     t_1)
    (*
     (*
      (*
       0.25
       (/
        (*
         (fabs a)
         (sqrt (* 8.0 (- (+ 0.5 (sqrt (pow (- 0.5 t_2) 2.0))) t_2))))
        (* (fabs x-scale) t_0)))
      t_1)
     t_1))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs((fabs(x_45_scale) * fabs(y_45_scale)));
	double t_1 = fabs(y_45_scale) * fabs(x_45_scale);
	double t_2 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double tmp;
	if (fabs(y_45_scale) <= 8.5e+73) {
		tmp = ((0.25 * ((fabs(a) * sqrt((8.0 * (0.5 + (sqrt(pow((0.5 + t_2), 2.0)) + t_2))))) / (fabs(y_45_scale) * t_0))) * t_1) * t_1;
	} else {
		tmp = ((0.25 * ((fabs(a) * sqrt((8.0 * ((0.5 + sqrt(pow((0.5 - t_2), 2.0))) - t_2)))) / (fabs(x_45_scale) * t_0))) * t_1) * t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.abs((Math.abs(x_45_scale) * Math.abs(y_45_scale)));
	double t_1 = Math.abs(y_45_scale) * Math.abs(x_45_scale);
	double t_2 = 0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI)));
	double tmp;
	if (Math.abs(y_45_scale) <= 8.5e+73) {
		tmp = ((0.25 * ((Math.abs(a) * Math.sqrt((8.0 * (0.5 + (Math.sqrt(Math.pow((0.5 + t_2), 2.0)) + t_2))))) / (Math.abs(y_45_scale) * t_0))) * t_1) * t_1;
	} else {
		tmp = ((0.25 * ((Math.abs(a) * Math.sqrt((8.0 * ((0.5 + Math.sqrt(Math.pow((0.5 - t_2), 2.0))) - t_2)))) / (Math.abs(x_45_scale) * t_0))) * t_1) * t_1;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.fabs((math.fabs(x_45_scale) * math.fabs(y_45_scale)))
	t_1 = math.fabs(y_45_scale) * math.fabs(x_45_scale)
	t_2 = 0.5 * math.cos((0.011111111111111112 * (angle * math.pi)))
	tmp = 0
	if math.fabs(y_45_scale) <= 8.5e+73:
		tmp = ((0.25 * ((math.fabs(a) * math.sqrt((8.0 * (0.5 + (math.sqrt(math.pow((0.5 + t_2), 2.0)) + t_2))))) / (math.fabs(y_45_scale) * t_0))) * t_1) * t_1
	else:
		tmp = ((0.25 * ((math.fabs(a) * math.sqrt((8.0 * ((0.5 + math.sqrt(math.pow((0.5 - t_2), 2.0))) - t_2)))) / (math.fabs(x_45_scale) * t_0))) * t_1) * t_1
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(Float64(abs(x_45_scale) * abs(y_45_scale)))
	t_1 = Float64(abs(y_45_scale) * abs(x_45_scale))
	t_2 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	tmp = 0.0
	if (abs(y_45_scale) <= 8.5e+73)
		tmp = Float64(Float64(Float64(0.25 * Float64(Float64(abs(a) * sqrt(Float64(8.0 * Float64(0.5 + Float64(sqrt((Float64(0.5 + t_2) ^ 2.0)) + t_2))))) / Float64(abs(y_45_scale) * t_0))) * t_1) * t_1);
	else
		tmp = Float64(Float64(Float64(0.25 * Float64(Float64(abs(a) * sqrt(Float64(8.0 * Float64(Float64(0.5 + sqrt((Float64(0.5 - t_2) ^ 2.0))) - t_2)))) / Float64(abs(x_45_scale) * t_0))) * t_1) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs((abs(x_45_scale) * abs(y_45_scale)));
	t_1 = abs(y_45_scale) * abs(x_45_scale);
	t_2 = 0.5 * cos((0.011111111111111112 * (angle * pi)));
	tmp = 0.0;
	if (abs(y_45_scale) <= 8.5e+73)
		tmp = ((0.25 * ((abs(a) * sqrt((8.0 * (0.5 + (sqrt(((0.5 + t_2) ^ 2.0)) + t_2))))) / (abs(y_45_scale) * t_0))) * t_1) * t_1;
	else
		tmp = ((0.25 * ((abs(a) * sqrt((8.0 * ((0.5 + sqrt(((0.5 - t_2) ^ 2.0))) - t_2)))) / (abs(x_45_scale) * t_0))) * t_1) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Abs[N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 8.5e+73], N[(N[(N[(0.25 * N[(N[(N[Abs[a], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(0.5 + N[(N[Sqrt[N[Power[N[(0.5 + t$95$2), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[y$45$scale], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(0.25 * N[(N[(N[Abs[a], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(0.5 + N[Sqrt[N[Power[N[(0.5 - t$95$2), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[x$45$scale], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 8.4999999999999998e73

   1. Initial program 2.7%

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

   3. Applied rewrites 6.7%

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

   4. Taylor expanded in b around 0

      \[\leadsto \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   6. Applied rewrites 2.5%

      \[\leadsto \left(\color{blue}{\left(0.25 \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   7. Taylor expanded in a around 0

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} - \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Applied rewrites 7.2%

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(0.5 \cdot \frac{1}{{x-scale}^{2}} - \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   10. Taylor expanded in y-scale around 0

       \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\frac{1}{2} + \left(\sqrt{{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{y-scale \cdot \color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{1}{4} \cdot \frac{a \cdot \sqrt{8 \cdot \left(\frac{1}{2} + \left(\sqrt{{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}}{y-scale \cdot \left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   12. Applied rewrites 8.6%

       \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{y-scale \cdot \color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   if 8.4999999999999998e73 < y-scale

   1. Initial program 2.7%

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

   3. Applied rewrites 6.7%

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

   4. Taylor expanded in b around 0

      \[\leadsto \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   6. Applied rewrites 2.5%

      \[\leadsto \left(\color{blue}{\left(0.25 \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   7. Taylor expanded in a around 0

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} - \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Applied rewrites 7.2%

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(0.5 \cdot \frac{1}{{x-scale}^{2}} - \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   10. Taylor expanded in x-scale around 0

       \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\frac{1}{2} + \sqrt{{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}\right) - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}}{x-scale \cdot \color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{1}{4} \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\frac{1}{2} + \sqrt{{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}}\right) - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}{x-scale \cdot \left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   12. Applied rewrites 4.2%

       \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(0.5 + \sqrt{{\left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}\right) - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}}{x-scale \cdot \color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 18.5% accurate, 5.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{b}{x-scale \cdot x-scale}\\ t_1 := 4 \cdot \left(\left|a\right| \cdot b\right)\\ t_2 := x-scale \cdot \left|y-scale\right|\\ t_3 := \left(t\_2 \cdot x-scale\right) \cdot \left|y-scale\right|\\ t_4 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ t_5 := \frac{\left|a\right|}{\left|y-scale\right| \cdot \left|y-scale\right|}\\ t_6 := \left|y-scale\right| \cdot x-scale\\ t_7 := -\left|a\right|\\ t_8 := t\_7 \cdot b\\ \mathbf{if}\;\left|y-scale\right| \leq 4.8 \cdot 10^{+89}:\\ \;\;\;\;\left(\left(0.25 \cdot \frac{\left|a\right| \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + t\_4\right)}^{2}} + t\_4\right)\right)}}{\left|y-scale\right| \cdot \left|t\_2\right|}\right) \cdot t\_6\right) \cdot t\_6\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\sqrt{\left(\left(\frac{\left(t\_1 \cdot b\right) \cdot t\_7}{t\_3} \cdot 2\right) \cdot \left(\left(t\_8 \cdot b\right) \cdot \left|a\right|\right)\right) \cdot \mathsf{fma}\left(\left|a\right|, t\_5, \mathsf{fma}\left(b, t\_0, \left|\left|a\right| \cdot t\_5 - b \cdot t\_0\right|\right)\right)}}{t\_1}}{t\_8} \cdot t\_3\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (/ b (* x-scale x-scale)))
       (t_1 (* 4.0 (* (fabs a) b)))
       (t_2 (* x-scale (fabs y-scale)))
       (t_3 (* (* t_2 x-scale) (fabs y-scale)))
       (t_4 (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
       (t_5 (/ (fabs a) (* (fabs y-scale) (fabs y-scale))))
       (t_6 (* (fabs y-scale) x-scale))
       (t_7 (- (fabs a)))
       (t_8 (* t_7 b)))
  (if (<= (fabs y-scale) 4.8e+89)
    (*
     (*
      (*
       0.25
       (/
        (*
         (fabs a)
         (sqrt (* 8.0 (+ 0.5 (+ (sqrt (pow (+ 0.5 t_4) 2.0)) t_4)))))
        (* (fabs y-scale) (fabs t_2))))
      t_6)
     t_6)
    (*
     (/
      (/
       (-
        (sqrt
         (*
          (* (* (/ (* (* t_1 b) t_7) t_3) 2.0) (* (* t_8 b) (fabs a)))
          (fma
           (fabs a)
           t_5
           (fma b t_0 (fabs (- (* (fabs a) t_5) (* b t_0))))))))
       t_1)
      t_8)
     t_3))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (x_45_scale * x_45_scale);
	double t_1 = 4.0 * (fabs(a) * b);
	double t_2 = x_45_scale * fabs(y_45_scale);
	double t_3 = (t_2 * x_45_scale) * fabs(y_45_scale);
	double t_4 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double t_5 = fabs(a) / (fabs(y_45_scale) * fabs(y_45_scale));
	double t_6 = fabs(y_45_scale) * x_45_scale;
	double t_7 = -fabs(a);
	double t_8 = t_7 * b;
	double tmp;
	if (fabs(y_45_scale) <= 4.8e+89) {
		tmp = ((0.25 * ((fabs(a) * sqrt((8.0 * (0.5 + (sqrt(pow((0.5 + t_4), 2.0)) + t_4))))) / (fabs(y_45_scale) * fabs(t_2)))) * t_6) * t_6;
	} else {
		tmp = ((-sqrt(((((((t_1 * b) * t_7) / t_3) * 2.0) * ((t_8 * b) * fabs(a))) * fma(fabs(a), t_5, fma(b, t_0, fabs(((fabs(a) * t_5) - (b * t_0))))))) / t_1) / t_8) * t_3;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(x_45_scale * x_45_scale))
	t_1 = Float64(4.0 * Float64(abs(a) * b))
	t_2 = Float64(x_45_scale * abs(y_45_scale))
	t_3 = Float64(Float64(t_2 * x_45_scale) * abs(y_45_scale))
	t_4 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	t_5 = Float64(abs(a) / Float64(abs(y_45_scale) * abs(y_45_scale)))
	t_6 = Float64(abs(y_45_scale) * x_45_scale)
	t_7 = Float64(-abs(a))
	t_8 = Float64(t_7 * b)
	tmp = 0.0
	if (abs(y_45_scale) <= 4.8e+89)
		tmp = Float64(Float64(Float64(0.25 * Float64(Float64(abs(a) * sqrt(Float64(8.0 * Float64(0.5 + Float64(sqrt((Float64(0.5 + t_4) ^ 2.0)) + t_4))))) / Float64(abs(y_45_scale) * abs(t_2)))) * t_6) * t_6);
	else
		tmp = Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(Float64(Float64(Float64(Float64(t_1 * b) * t_7) / t_3) * 2.0) * Float64(Float64(t_8 * b) * abs(a))) * fma(abs(a), t_5, fma(b, t_0, abs(Float64(Float64(abs(a) * t_5) - Float64(b * t_0)))))))) / t_1) / t_8) * t_3);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(4.0 * N[(N[Abs[a], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x$45$scale * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 * x$45$scale), $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Abs[a], $MachinePrecision] / N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Abs[y$45$scale], $MachinePrecision] * x$45$scale), $MachinePrecision]}, Block[{t$95$7 = (-N[Abs[a], $MachinePrecision])}, Block[{t$95$8 = N[(t$95$7 * b), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 4.8e+89], N[(N[(N[(0.25 * N[(N[(N[Abs[a], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(0.5 + N[(N[Sqrt[N[Power[N[(0.5 + t$95$4), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$6), $MachinePrecision] * t$95$6), $MachinePrecision], N[(N[(N[((-N[Sqrt[N[(N[(N[(N[(N[(N[(t$95$1 * b), $MachinePrecision] * t$95$7), $MachinePrecision] / t$95$3), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(t$95$8 * b), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] * t$95$5 + N[(b * t$95$0 + N[Abs[N[(N[(N[Abs[a], $MachinePrecision] * t$95$5), $MachinePrecision] - N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision] / t$95$8), $MachinePrecision] * t$95$3), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 4.8000000000000001e89

   1. Initial program 2.7%

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

   3. Applied rewrites 6.7%

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

   4. Taylor expanded in b around 0

      \[\leadsto \left(\color{blue}{\left(\frac{1}{4} \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   6. Applied rewrites 2.5%

      \[\leadsto \left(\color{blue}{\left(0.25 \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   7. Taylor expanded in a around 0

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} - \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Applied rewrites 7.2%

      \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(0.5 \cdot \frac{1}{{x-scale}^{2}} - \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   10. Taylor expanded in y-scale around 0

       \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(\frac{1}{2} + \left(\sqrt{{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{y-scale \cdot \color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{1}{4} \cdot \frac{a \cdot \sqrt{8 \cdot \left(\frac{1}{2} + \left(\sqrt{{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{2}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}}{y-scale \cdot \left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   12. Applied rewrites 8.6%

       \[\leadsto \left(\left(0.25 \cdot \frac{a \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{y-scale \cdot \color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   if 4.8000000000000001e89 < y-scale

   1. Initial program 2.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 4.1%

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

   6. Applied rewrites 4.1%

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

   8. Applied rewrites 8.1%

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

Alternative 8: 11.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| \cdot b\\ t_1 := -\left|a\right|\\ t_2 := \frac{b}{x-scale \cdot x-scale}\\ t_3 := x-scale \cdot \left|y-scale\right|\\ t_4 := \left(t\_3 \cdot x-scale\right) \cdot \left|y-scale\right|\\ t_5 := \frac{\left|a\right|}{\left|y-scale\right| \cdot \left|y-scale\right|}\\ t_6 := \left|y-scale\right| \cdot x-scale\\ \mathbf{if}\;\left|a\right| \leq 2.7 \cdot 10^{-81}:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{fma}\left(\left|a\right|, t\_5, \mathsf{fma}\left(b, t\_2, \left|\left|a\right| \cdot t\_5 - b \cdot t\_2\right|\right)\right) \cdot \left(\frac{\left(\left(4 \cdot t\_0\right) \cdot b\right) \cdot t\_1}{t\_4} \cdot 2\right)\right) \cdot \left(\left(\left(t\_1 \cdot b\right) \cdot b\right) \cdot \left|a\right|\right)}}{\left(4 \cdot \left(b \cdot \left|a\right|\right)\right) \cdot \left(b \cdot t\_1\right)} \cdot t\_4\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\left({t\_0}^{4} \cdot \mathsf{fma}\left(\left|a\right|, \left|a\right|, \sqrt{{\left(\left|a\right|\right)}^{4}}\right)\right) \cdot 8}}{\left|t\_3\right| \cdot \left|y-scale\right|}}{t\_0 \cdot 4}}{t\_0} \cdot t\_6\right) \cdot t\_6\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (fabs a) b))
       (t_1 (- (fabs a)))
       (t_2 (/ b (* x-scale x-scale)))
       (t_3 (* x-scale (fabs y-scale)))
       (t_4 (* (* t_3 x-scale) (fabs y-scale)))
       (t_5 (/ (fabs a) (* (fabs y-scale) (fabs y-scale))))
       (t_6 (* (fabs y-scale) x-scale)))
  (if (<= (fabs a) 2.7e-81)
    (*
     (/
      (-
       (sqrt
        (*
         (*
          (fma
           (fabs a)
           t_5
           (fma b t_2 (fabs (- (* (fabs a) t_5) (* b t_2)))))
          (* (/ (* (* (* 4.0 t_0) b) t_1) t_4) 2.0))
         (* (* (* t_1 b) b) (fabs a)))))
      (* (* 4.0 (* b (fabs a))) (* b t_1)))
     t_4)
    (*
     (*
      (/
       (/
        (/
         (sqrt
          (*
           (*
            (pow t_0 4.0)
            (fma (fabs a) (fabs a) (sqrt (pow (fabs a) 4.0))))
           8.0))
         (* (fabs t_3) (fabs y-scale)))
        (* t_0 4.0))
       t_0)
      t_6)
     t_6))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(a) * b;
	double t_1 = -fabs(a);
	double t_2 = b / (x_45_scale * x_45_scale);
	double t_3 = x_45_scale * fabs(y_45_scale);
	double t_4 = (t_3 * x_45_scale) * fabs(y_45_scale);
	double t_5 = fabs(a) / (fabs(y_45_scale) * fabs(y_45_scale));
	double t_6 = fabs(y_45_scale) * x_45_scale;
	double tmp;
	if (fabs(a) <= 2.7e-81) {
		tmp = (-sqrt(((fma(fabs(a), t_5, fma(b, t_2, fabs(((fabs(a) * t_5) - (b * t_2))))) * (((((4.0 * t_0) * b) * t_1) / t_4) * 2.0)) * (((t_1 * b) * b) * fabs(a)))) / ((4.0 * (b * fabs(a))) * (b * t_1))) * t_4;
	} else {
		tmp = ((((sqrt(((pow(t_0, 4.0) * fma(fabs(a), fabs(a), sqrt(pow(fabs(a), 4.0)))) * 8.0)) / (fabs(t_3) * fabs(y_45_scale))) / (t_0 * 4.0)) / t_0) * t_6) * t_6;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(a) * b)
	t_1 = Float64(-abs(a))
	t_2 = Float64(b / Float64(x_45_scale * x_45_scale))
	t_3 = Float64(x_45_scale * abs(y_45_scale))
	t_4 = Float64(Float64(t_3 * x_45_scale) * abs(y_45_scale))
	t_5 = Float64(abs(a) / Float64(abs(y_45_scale) * abs(y_45_scale)))
	t_6 = Float64(abs(y_45_scale) * x_45_scale)
	tmp = 0.0
	if (abs(a) <= 2.7e-81)
		tmp = Float64(Float64(Float64(-sqrt(Float64(Float64(fma(abs(a), t_5, fma(b, t_2, abs(Float64(Float64(abs(a) * t_5) - Float64(b * t_2))))) * Float64(Float64(Float64(Float64(Float64(4.0 * t_0) * b) * t_1) / t_4) * 2.0)) * Float64(Float64(Float64(t_1 * b) * b) * abs(a))))) / Float64(Float64(4.0 * Float64(b * abs(a))) * Float64(b * t_1))) * t_4);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64((t_0 ^ 4.0) * fma(abs(a), abs(a), sqrt((abs(a) ^ 4.0)))) * 8.0)) / Float64(abs(t_3) * abs(y_45_scale))) / Float64(t_0 * 4.0)) / t_0) * t_6) * t_6);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[a], $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$1 = (-N[Abs[a], $MachinePrecision])}, Block[{t$95$2 = N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x$45$scale * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$3 * x$45$scale), $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Abs[a], $MachinePrecision] / N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Abs[y$45$scale], $MachinePrecision] * x$45$scale), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 2.7e-81], N[(N[((-N[Sqrt[N[(N[(N[(N[Abs[a], $MachinePrecision] * t$95$5 + N[(b * t$95$2 + N[Abs[N[(N[(N[Abs[a], $MachinePrecision] * t$95$5), $MachinePrecision] - N[(b * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(4.0 * t$95$0), $MachinePrecision] * b), $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$4), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$1 * b), $MachinePrecision] * b), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(4.0 * N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[Power[t$95$0, 4.0], $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision] + N[Sqrt[N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] / N[(N[Abs[t$95$3], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * 4.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$6), $MachinePrecision] * t$95$6), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.6999999999999999e-81

   1. Initial program 2.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 4.1%

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

   6. Applied rewrites 4.1%

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

   8. Applied rewrites 5.3%

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

   if 2.6999999999999999e-81 < a

   1. Initial program 2.7%

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

   3. Applied rewrites 6.7%

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

   4. Taylor expanded in y-scale around 0

      \[\leadsto \left(\frac{\color{blue}{\frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left({a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2}} + \left({a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\right)}}{y-scale \cdot \left|x-scale \cdot y-scale\right|}}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   6. Applied rewrites 1.0%

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

   7. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 1.0%

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

   9. Applied rewrites 1.0%

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

   11. Applied rewrites 5.0%

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

Alternative 9: 9.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} t_0 := \left|y-scale\right| \cdot x-scale\\ \left(\frac{\frac{\frac{\sqrt{\left({\left(a \cdot b\right)}^{4} \cdot \mathsf{fma}\left(a, a, \sqrt{{a}^{4}}\right)\right) \cdot 8}}{\left|x-scale \cdot \left|y-scale\right|\right| \cdot \left|y-scale\right|}}{\left(a \cdot b\right) \cdot 4}}{a \cdot b} \cdot t\_0\right) \cdot t\_0 \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (fabs y-scale) x-scale)))
  (*
   (*
    (/
     (/
      (/
       (sqrt
        (* (* (pow (* a b) 4.0) (fma a a (sqrt (pow a 4.0)))) 8.0))
       (* (fabs (* x-scale (fabs y-scale))) (fabs y-scale)))
      (* (* a b) 4.0))
     (* a b))
    t_0)
   t_0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(y_45_scale) * x_45_scale;
	return ((((sqrt(((pow((a * b), 4.0) * fma(a, a, sqrt(pow(a, 4.0)))) * 8.0)) / (fabs((x_45_scale * fabs(y_45_scale))) * fabs(y_45_scale))) / ((a * b) * 4.0)) / (a * b)) * t_0) * t_0;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(y_45_scale) * x_45_scale)
	return Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64((Float64(a * b) ^ 4.0) * fma(a, a, sqrt((a ^ 4.0)))) * 8.0)) / Float64(abs(Float64(x_45_scale * abs(y_45_scale))) * abs(y_45_scale))) / Float64(Float64(a * b) * 4.0)) / Float64(a * b)) * t_0) * t_0)
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[y$45$scale], $MachinePrecision] * x$45$scale), $MachinePrecision]}, N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[Power[N[(a * b), $MachinePrecision], 4.0], $MachinePrecision] * N[(a * a + N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] / N[(N[Abs[N[(x$45$scale * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a * b), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 2.7%

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

3. Applied rewrites 6.7%

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

4. Taylor expanded in y-scale around 0

   \[\leadsto \left(\frac{\color{blue}{\frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left({a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2}} + \left({a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\right)}}{y-scale \cdot \left|x-scale \cdot y-scale\right|}}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

6. Applied rewrites 1.0%

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

7. Taylor expanded in angle around 0

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

   1. lower-+.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-pow.f64 1.0%

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

9. Applied rewrites 1.0%

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

11. Applied rewrites 5.0%

    \[\leadsto \left(\color{blue}{\frac{\frac{\frac{\sqrt{\left({\left(a \cdot b\right)}^{4} \cdot \mathsf{fma}\left(a, a, \sqrt{{a}^{4}}\right)\right) \cdot 8}}{\left|x-scale \cdot y-scale\right| \cdot y-scale}}{\left(a \cdot b\right) \cdot 4}}{a \cdot b}} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
12. Add Preprocessing

Alternative 10: 7.3% accurate, 8.3× speedup

Math

\[\begin{array}{l} t_0 := \left|y-scale\right| \cdot x-scale\\ \left(\frac{\frac{\frac{\sqrt{\left({\left(a \cdot b\right)}^{4} \cdot \mathsf{fma}\left(a, a, \sqrt{{a}^{4}}\right)\right) \cdot 8}}{\left|y-scale\right|}}{\left|x-scale \cdot \left|y-scale\right|\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot t\_0\right) \cdot t\_0 \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (fabs y-scale) x-scale)))
  (*
   (*
    (/
     (/
      (/
       (sqrt
        (* (* (pow (* a b) 4.0) (fma a a (sqrt (pow a 4.0)))) 8.0))
       (fabs y-scale))
      (fabs (* x-scale (fabs y-scale))))
     (* (* (* a b) 4.0) (* a b)))
    t_0)
   t_0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(y_45_scale) * x_45_scale;
	return ((((sqrt(((pow((a * b), 4.0) * fma(a, a, sqrt(pow(a, 4.0)))) * 8.0)) / fabs(y_45_scale)) / fabs((x_45_scale * fabs(y_45_scale)))) / (((a * b) * 4.0) * (a * b))) * t_0) * t_0;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(y_45_scale) * x_45_scale)
	return Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64((Float64(a * b) ^ 4.0) * fma(a, a, sqrt((a ^ 4.0)))) * 8.0)) / abs(y_45_scale)) / abs(Float64(x_45_scale * abs(y_45_scale)))) / Float64(Float64(Float64(a * b) * 4.0) * Float64(a * b))) * t_0) * t_0)
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[y$45$scale], $MachinePrecision] * x$45$scale), $MachinePrecision]}, N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[Power[N[(a * b), $MachinePrecision], 4.0], $MachinePrecision] * N[(a * a + N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] / N[Abs[N[(x$45$scale * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(a * b), $MachinePrecision] * 4.0), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 2.7%

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

3. Applied rewrites 6.7%

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

4. Taylor expanded in y-scale around 0

   \[\leadsto \left(\frac{\color{blue}{\frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left({a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2}} + \left({a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\right)}}{y-scale \cdot \left|x-scale \cdot y-scale\right|}}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

6. Applied rewrites 1.0%

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

7. Taylor expanded in angle around 0

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

   1. lower-+.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-pow.f64 1.0%

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

9. Applied rewrites 1.0%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

11. Applied rewrites 3.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left({\left(a \cdot b\right)}^{4} \cdot \mathsf{fma}\left(a, a, \sqrt{{a}^{4}}\right)\right) \cdot 8}}{y-scale}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
12. Add Preprocessing

Alternative 11: 5.4% accurate, 8.4× speedup

Math

\[\begin{array}{l} t_0 := \left|y-scale\right| \cdot x-scale\\ \left(\frac{\frac{\sqrt{\left({\left(a \cdot b\right)}^{4} \cdot \mathsf{fma}\left(a, a, \sqrt{{a}^{4}}\right)\right) \cdot 8}}{\left|x-scale \cdot \left|y-scale\right|\right| \cdot \left|y-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot t\_0\right) \cdot t\_0 \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (fabs y-scale) x-scale)))
  (*
   (*
    (/
     (/
      (sqrt
       (* (* (pow (* a b) 4.0) (fma a a (sqrt (pow a 4.0)))) 8.0))
      (* (fabs (* x-scale (fabs y-scale))) (fabs y-scale)))
     (* (* (* a b) 4.0) (* a b)))
    t_0)
   t_0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(y_45_scale) * x_45_scale;
	return (((sqrt(((pow((a * b), 4.0) * fma(a, a, sqrt(pow(a, 4.0)))) * 8.0)) / (fabs((x_45_scale * fabs(y_45_scale))) * fabs(y_45_scale))) / (((a * b) * 4.0) * (a * b))) * t_0) * t_0;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(y_45_scale) * x_45_scale)
	return Float64(Float64(Float64(Float64(sqrt(Float64(Float64((Float64(a * b) ^ 4.0) * fma(a, a, sqrt((a ^ 4.0)))) * 8.0)) / Float64(abs(Float64(x_45_scale * abs(y_45_scale))) * abs(y_45_scale))) / Float64(Float64(Float64(a * b) * 4.0) * Float64(a * b))) * t_0) * t_0)
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[y$45$scale], $MachinePrecision] * x$45$scale), $MachinePrecision]}, N[(N[(N[(N[(N[Sqrt[N[(N[(N[Power[N[(a * b), $MachinePrecision], 4.0], $MachinePrecision] * N[(a * a + N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] / N[(N[Abs[N[(x$45$scale * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(a * b), $MachinePrecision] * 4.0), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 2.7%

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

3. Applied rewrites 6.7%

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

4. Taylor expanded in y-scale around 0

   \[\leadsto \left(\frac{\color{blue}{\frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left({a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2}} + \left({a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\right)}}{y-scale \cdot \left|x-scale \cdot y-scale\right|}}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

6. Applied rewrites 1.0%

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

7. Taylor expanded in angle around 0

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

   1. lower-+.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-pow.f64 1.0%

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

9. Applied rewrites 1.0%

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

11. Applied rewrites 2.7%

    \[\leadsto \left(\frac{\frac{\sqrt{\left({\left(a \cdot b\right)}^{4} \cdot \mathsf{fma}\left(a, a, \sqrt{{a}^{4}}\right)\right) \cdot 8}}{\color{blue}{\left|x-scale \cdot y-scale\right| \cdot y-scale}}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
12. Add Preprocessing

Alternative 12: 4.3% accurate, 8.4× speedup

Math

\[\begin{array}{l} t_0 := x-scale \cdot \left|y-scale\right|\\ \left(\left(\frac{\frac{\sqrt{\left({\left(a \cdot b\right)}^{4} \cdot \mathsf{fma}\left(a, a, \sqrt{{a}^{4}}\right)\right) \cdot 8}}{\left|t\_0\right| \cdot \left|y-scale\right|}}{\left(\left(\left(a \cdot b\right) \cdot 4\right) \cdot a\right) \cdot b} \cdot t\_0\right) \cdot \left|y-scale\right|\right) \cdot x-scale \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* x-scale (fabs y-scale))))
  (*
   (*
    (*
     (/
      (/
       (sqrt
        (* (* (pow (* a b) 4.0) (fma a a (sqrt (pow a 4.0)))) 8.0))
       (* (fabs t_0) (fabs y-scale)))
      (* (* (* (* a b) 4.0) a) b))
     t_0)
    (fabs y-scale))
   x-scale)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = x_45_scale * fabs(y_45_scale);
	return ((((sqrt(((pow((a * b), 4.0) * fma(a, a, sqrt(pow(a, 4.0)))) * 8.0)) / (fabs(t_0) * fabs(y_45_scale))) / ((((a * b) * 4.0) * a) * b)) * t_0) * fabs(y_45_scale)) * x_45_scale;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(x_45_scale * abs(y_45_scale))
	return Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64((Float64(a * b) ^ 4.0) * fma(a, a, sqrt((a ^ 4.0)))) * 8.0)) / Float64(abs(t_0) * abs(y_45_scale))) / Float64(Float64(Float64(Float64(a * b) * 4.0) * a) * b)) * t_0) * abs(y_45_scale)) * x_45_scale)
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(x$45$scale * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[Power[N[(a * b), $MachinePrecision], 4.0], $MachinePrecision] * N[(a * a + N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] / N[(N[Abs[t$95$0], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(a * b), $MachinePrecision] * 4.0), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision]]

Derivation

1. Initial program 2.7%

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

3. Applied rewrites 6.7%

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

4. Taylor expanded in y-scale around 0

   \[\leadsto \left(\frac{\color{blue}{\frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left({a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2}} + \left({a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\right)}}{y-scale \cdot \left|x-scale \cdot y-scale\right|}}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

6. Applied rewrites 1.0%

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

7. Taylor expanded in angle around 0

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

   1. lower-+.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-pow.f64 1.0%

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

9. Applied rewrites 1.0%

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

11. Applied rewrites 2.2%

    \[\leadsto \color{blue}{\left(\left(\frac{\frac{\sqrt{\left({\left(a \cdot b\right)}^{4} \cdot \mathsf{fma}\left(a, a, \sqrt{{a}^{4}}\right)\right) \cdot 8}}{\left|x-scale \cdot y-scale\right| \cdot y-scale}}{\left(\left(\left(a \cdot b\right) \cdot 4\right) \cdot a\right) \cdot b} \cdot \left(x-scale \cdot y-scale\right)\right) \cdot y-scale\right) \cdot x-scale} \]
12. Add Preprocessing

Alternative 13: 4.3% accurate, 8.4× speedup

Math

\[\left(\left(\frac{\frac{\sqrt{\left({\left(a \cdot b\right)}^{4} \cdot \mathsf{fma}\left(a, a, \sqrt{{a}^{4}}\right)\right) \cdot 8}}{\left|x-scale \cdot \left|y-scale\right|\right| \cdot \left|y-scale\right|}}{\left(\left(\left(a \cdot b\right) \cdot 4\right) \cdot a\right) \cdot b} \cdot \left|y-scale\right|\right) \cdot x-scale\right) \cdot \left(\left|y-scale\right| \cdot x-scale\right) \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (*
 (*
  (*
   (/
    (/
     (sqrt (* (* (pow (* a b) 4.0) (fma a a (sqrt (pow a 4.0)))) 8.0))
     (* (fabs (* x-scale (fabs y-scale))) (fabs y-scale)))
    (* (* (* (* a b) 4.0) a) b))
   (fabs y-scale))
  x-scale)
 (* (fabs y-scale) x-scale)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((((sqrt(((pow((a * b), 4.0) * fma(a, a, sqrt(pow(a, 4.0)))) * 8.0)) / (fabs((x_45_scale * fabs(y_45_scale))) * fabs(y_45_scale))) / ((((a * b) * 4.0) * a) * b)) * fabs(y_45_scale)) * x_45_scale) * (fabs(y_45_scale) * x_45_scale);
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64((Float64(a * b) ^ 4.0) * fma(a, a, sqrt((a ^ 4.0)))) * 8.0)) / Float64(abs(Float64(x_45_scale * abs(y_45_scale))) * abs(y_45_scale))) / Float64(Float64(Float64(Float64(a * b) * 4.0) * a) * b)) * abs(y_45_scale)) * x_45_scale) * Float64(abs(y_45_scale) * x_45_scale))
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[Power[N[(a * b), $MachinePrecision], 4.0], $MachinePrecision] * N[(a * a + N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] / N[(N[Abs[N[(x$45$scale * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(a * b), $MachinePrecision] * 4.0), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision] * N[(N[Abs[y$45$scale], $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.7%

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

3. Applied rewrites 6.7%

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

4. Taylor expanded in y-scale around 0

   \[\leadsto \left(\frac{\color{blue}{\frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left({a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2}} + \left({a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\right)}}{y-scale \cdot \left|x-scale \cdot y-scale\right|}}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

6. Applied rewrites 1.0%

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

7. Taylor expanded in angle around 0

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

   1. lower-+.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-pow.f64 1.0%

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

9. Applied rewrites 1.0%

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

11. Applied rewrites 2.2%

    \[\leadsto \color{blue}{\left(\left(\frac{\frac{\sqrt{\left({\left(a \cdot b\right)}^{4} \cdot \mathsf{fma}\left(a, a, \sqrt{{a}^{4}}\right)\right) \cdot 8}}{\left|x-scale \cdot y-scale\right| \cdot y-scale}}{\left(\left(\left(a \cdot b\right) \cdot 4\right) \cdot a\right) \cdot b} \cdot y-scale\right) \cdot x-scale\right)} \cdot \left(y-scale \cdot x-scale\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (a b angle x-scale y-scale)
  :name "a 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))))