b from scale-rotated-ellipse

Percentage Accurate: 0.0% → 30.5%

Time: 32.7s

Alternatives: 7

Speedup: 10.1×

• could not determine a ground truth

  1. a = 1.1265381753016791e+212
  2. b = 1.8888624634483204e-307
  3. angle = 3.292235039020281e+44
  4. x-scale = 1.0618756433455896e-250
  5. y-scale = 2.7690943381552144e+276

Specification

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (/ angle 180.0) PI))
       (t_1 (sin t_0))
       (t_2 (cos t_0))
       (t_3
        (/
         (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale)
         y-scale))
       (t_4
        (/
         (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale)
         x-scale))
       (t_5 (* (* b a) (* b (- a))))
       (t_6 (/ (* 4.0 t_5) (pow (* x-scale y-scale) 2.0))))
  (/
   (-
    (sqrt
     (*
      (* (* 2.0 t_6) t_5)
      (-
       (+ t_4 t_3)
       (sqrt
        (+
         (pow (- t_4 t_3) 2.0)
         (pow
          (/
           (/
            (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2)
            x-scale)
           y-scale)
          2.0)))))))
   t_6)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / pow((x_45_scale * y_45_scale), 2.0);
	return -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - sqrt((pow((t_4 - t_3), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / Math.pow((x_45_scale * y_45_scale), 2.0);
	return -Math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - Math.sqrt((Math.pow((t_4 - t_3), 2.0) + Math.pow((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale
	t_4 = ((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale
	t_5 = (b * a) * (b * -a)
	t_6 = (4.0 * t_5) / math.pow((x_45_scale * y_45_scale), 2.0)
	return -math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - math.sqrt((math.pow((t_4 - t_3), 2.0) + math.pow((((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)
	t_5 = Float64(Float64(b * a) * Float64(b * Float64(-a)))
	t_6 = Float64(Float64(4.0 * t_5) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_6) * t_5) * Float64(Float64(t_4 + t_3) - sqrt(Float64((Float64(t_4 - t_3) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0))))))) / t_6)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_4 = ((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale;
	t_5 = (b * a) * (b * -a);
	t_6 = (4.0 * t_5) / ((x_45_scale * y_45_scale) ^ 2.0);
	tmp = -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - sqrt((((t_4 - t_3) ^ 2.0) + ((((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0)))))) / t_6;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(4.0 * t$95$5), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$6), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(t$95$4 + t$95$3), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$4 - t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]]]]]]]]

Initial Program: 0.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (/ angle 180.0) PI))
       (t_1 (sin t_0))
       (t_2 (cos t_0))
       (t_3
        (/
         (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale)
         y-scale))
       (t_4
        (/
         (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale)
         x-scale))
       (t_5 (* (* b a) (* b (- a))))
       (t_6 (/ (* 4.0 t_5) (pow (* x-scale y-scale) 2.0))))
  (/
   (-
    (sqrt
     (*
      (* (* 2.0 t_6) t_5)
      (-
       (+ t_4 t_3)
       (sqrt
        (+
         (pow (- t_4 t_3) 2.0)
         (pow
          (/
           (/
            (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2)
            x-scale)
           y-scale)
          2.0)))))))
   t_6)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / pow((x_45_scale * y_45_scale), 2.0);
	return -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - sqrt((pow((t_4 - t_3), 2.0) + pow((((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale;
	double t_5 = (b * a) * (b * -a);
	double t_6 = (4.0 * t_5) / Math.pow((x_45_scale * y_45_scale), 2.0);
	return -Math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - Math.sqrt((Math.pow((t_4 - t_3), 2.0) + Math.pow((((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale
	t_4 = ((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale
	t_5 = (b * a) * (b * -a)
	t_6 = (4.0 * t_5) / math.pow((x_45_scale * y_45_scale), 2.0)
	return -math.sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - math.sqrt((math.pow((t_4 - t_3), 2.0) + math.pow((((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale), 2.0)))))) / t_6

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_4 = Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)
	t_5 = Float64(Float64(b * a) * Float64(b * Float64(-a)))
	t_6 = Float64(Float64(4.0 * t_5) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_6) * t_5) * Float64(Float64(t_4 + t_3) - sqrt(Float64((Float64(t_4 - t_3) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0))))))) / t_6)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_4 = ((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale;
	t_5 = (b * a) * (b * -a);
	t_6 = (4.0 * t_5) / ((x_45_scale * y_45_scale) ^ 2.0);
	tmp = -sqrt((((2.0 * t_6) * t_5) * ((t_4 + t_3) - sqrt((((t_4 - t_3) ^ 2.0) + ((((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale) ^ 2.0)))))) / t_6;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(4.0 * t$95$5), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$6), $MachinePrecision] * t$95$5), $MachinePrecision] * N[(N[(t$95$4 + t$95$3), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$4 - t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]]]]]]]]

Alternative 1: 30.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;\left|y-scale\right| \leq 2.5 \cdot 10^{-168}:\\ \;\;\;\;0.25 \cdot \frac{\left|a\right| \cdot \left({\left(\left|y-scale\right|\right)}^{2} \cdot \sqrt{8 \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\left|\left|y-scale\right|\right|}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left|a\right| \cdot \left(\left|y-scale\right| \cdot \sqrt{8 \cdot \left({t\_0}^{2} - \sqrt{{t\_0}^{4}}\right)}\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (sin (* 0.005555555555555556 (* angle PI)))))
  (if (<= (fabs y-scale) 2.5e-168)
    (*
     0.25
     (/
      (*
       (fabs a)
       (*
        (pow (fabs y-scale) 2.0)
        (sqrt
         (*
          8.0
          (-
           0.5
           (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))))))
      (fabs (fabs y-scale))))
    (*
     0.25
     (*
      (fabs a)
      (*
       (fabs y-scale)
       (sqrt (* 8.0 (- (pow t_0 2.0) (sqrt (pow t_0 4.0)))))))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (fabs(y_45_scale) <= 2.5e-168) {
		tmp = 0.25 * ((fabs(a) * (pow(fabs(y_45_scale), 2.0) * sqrt((8.0 * (0.5 - (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI)))))))))) / fabs(fabs(y_45_scale)));
	} else {
		tmp = 0.25 * (fabs(a) * (fabs(y_45_scale) * sqrt((8.0 * (pow(t_0, 2.0) - sqrt(pow(t_0, 4.0)))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.sin((0.005555555555555556 * (angle * Math.PI)));
	double tmp;
	if (Math.abs(y_45_scale) <= 2.5e-168) {
		tmp = 0.25 * ((Math.abs(a) * (Math.pow(Math.abs(y_45_scale), 2.0) * Math.sqrt((8.0 * (0.5 - (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI))))))))) / Math.abs(Math.abs(y_45_scale)));
	} else {
		tmp = 0.25 * (Math.abs(a) * (Math.abs(y_45_scale) * Math.sqrt((8.0 * (Math.pow(t_0, 2.0) - Math.sqrt(Math.pow(t_0, 4.0)))))));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.sin((0.005555555555555556 * (angle * math.pi)))
	tmp = 0
	if math.fabs(y_45_scale) <= 2.5e-168:
		tmp = 0.25 * ((math.fabs(a) * (math.pow(math.fabs(y_45_scale), 2.0) * math.sqrt((8.0 * (0.5 - (0.5 * math.cos((0.011111111111111112 * (angle * math.pi))))))))) / math.fabs(math.fabs(y_45_scale)))
	else:
		tmp = 0.25 * (math.fabs(a) * (math.fabs(y_45_scale) * math.sqrt((8.0 * (math.pow(t_0, 2.0) - math.sqrt(math.pow(t_0, 4.0)))))))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (abs(y_45_scale) <= 2.5e-168)
		tmp = Float64(0.25 * Float64(Float64(abs(a) * Float64((abs(y_45_scale) ^ 2.0) * sqrt(Float64(8.0 * Float64(0.5 - Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))))))) / abs(abs(y_45_scale))));
	else
		tmp = Float64(0.25 * Float64(abs(a) * Float64(abs(y_45_scale) * sqrt(Float64(8.0 * Float64((t_0 ^ 2.0) - sqrt((t_0 ^ 4.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = sin((0.005555555555555556 * (angle * pi)));
	tmp = 0.0;
	if (abs(y_45_scale) <= 2.5e-168)
		tmp = 0.25 * ((abs(a) * ((abs(y_45_scale) ^ 2.0) * sqrt((8.0 * (0.5 - (0.5 * cos((0.011111111111111112 * (angle * pi))))))))) / abs(abs(y_45_scale)));
	else
		tmp = 0.25 * (abs(a) * (abs(y_45_scale) * sqrt((8.0 * ((t_0 ^ 2.0) - sqrt((t_0 ^ 4.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 2.5e-168], N[(0.25 * N[(N[(N[Abs[a], $MachinePrecision] * N[(N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(0.5 - N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[N[Abs[y$45$scale], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[Abs[a], $MachinePrecision] * N[(N[Abs[y$45$scale], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[Sqrt[N[Power[t$95$0, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.5e-168

   1. Initial program 0.0%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 0.8%

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

   6. Applied rewrites 3.5%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 4.9%

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

   10. Taylor expanded in b around 0

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

       1. lower-/.f64 N/A

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

   12. Applied rewrites 23.7%

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

   if 2.5e-168 < y-scale

   1. Initial program 0.0%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 4.1%

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

   8. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

   10. Applied rewrites 12.8%

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

   11. Taylor expanded in y-scale around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

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

   13. Applied rewrites 29.5%

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

Alternative 2: 29.3% accurate, 6.0× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|y-scale\right|\right)}^{2}\\ t_1 := \left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\\ t_2 := \frac{4 \cdot t\_1}{{\left(x-scale \cdot \left|y-scale\right|\right)}^{2}}\\ \mathbf{if}\;\left|y-scale\right| \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;0.25 \cdot \frac{a \cdot \left(t\_0 \cdot \sqrt{8 \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{\left|\left|y-scale\right|\right|}\\ \mathbf{elif}\;\left|y-scale\right| \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(t\_0 \cdot \left(angle \cdot \sqrt{8 \cdot \frac{3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - \sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}}}{t\_0}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_2\right) \cdot t\_1\right) \cdot \frac{0}{t\_0}}}{t\_2}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (pow (fabs y-scale) 2.0))
       (t_1 (* (* b a) (* b (- a))))
       (t_2 (/ (* 4.0 t_1) (pow (* x-scale (fabs y-scale)) 2.0))))
  (if (<= (fabs y-scale) 1.55e-162)
    (*
     0.25
     (/
      (*
       a
       (*
        t_0
        (sqrt
         (*
          8.0
          (-
           0.5
           (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))))))
      (fabs (fabs y-scale))))
    (if (<= (fabs y-scale) 1.32e+154)
      (*
       0.25
       (*
        a
        (*
         t_0
         (*
          angle
          (sqrt
           (*
            8.0
            (/
             (-
              (* 3.08641975308642e-5 (pow PI 2.0))
              (sqrt (* 9.525986892242036e-10 (pow PI 4.0))))
             t_0)))))))
      (/ (- (sqrt (* (* (* 2.0 t_2) t_1) (/ 0.0 t_0)))) t_2)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = pow(fabs(y_45_scale), 2.0);
	double t_1 = (b * a) * (b * -a);
	double t_2 = (4.0 * t_1) / pow((x_45_scale * fabs(y_45_scale)), 2.0);
	double tmp;
	if (fabs(y_45_scale) <= 1.55e-162) {
		tmp = 0.25 * ((a * (t_0 * sqrt((8.0 * (0.5 - (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI)))))))))) / fabs(fabs(y_45_scale)));
	} else if (fabs(y_45_scale) <= 1.32e+154) {
		tmp = 0.25 * (a * (t_0 * (angle * sqrt((8.0 * (((3.08641975308642e-5 * pow(((double) M_PI), 2.0)) - sqrt((9.525986892242036e-10 * pow(((double) M_PI), 4.0)))) / t_0))))));
	} else {
		tmp = -sqrt((((2.0 * t_2) * t_1) * (0.0 / t_0))) / t_2;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.pow(Math.abs(y_45_scale), 2.0);
	double t_1 = (b * a) * (b * -a);
	double t_2 = (4.0 * t_1) / Math.pow((x_45_scale * Math.abs(y_45_scale)), 2.0);
	double tmp;
	if (Math.abs(y_45_scale) <= 1.55e-162) {
		tmp = 0.25 * ((a * (t_0 * Math.sqrt((8.0 * (0.5 - (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI))))))))) / Math.abs(Math.abs(y_45_scale)));
	} else if (Math.abs(y_45_scale) <= 1.32e+154) {
		tmp = 0.25 * (a * (t_0 * (angle * Math.sqrt((8.0 * (((3.08641975308642e-5 * Math.pow(Math.PI, 2.0)) - Math.sqrt((9.525986892242036e-10 * Math.pow(Math.PI, 4.0)))) / t_0))))));
	} else {
		tmp = -Math.sqrt((((2.0 * t_2) * t_1) * (0.0 / t_0))) / t_2;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.pow(math.fabs(y_45_scale), 2.0)
	t_1 = (b * a) * (b * -a)
	t_2 = (4.0 * t_1) / math.pow((x_45_scale * math.fabs(y_45_scale)), 2.0)
	tmp = 0
	if math.fabs(y_45_scale) <= 1.55e-162:
		tmp = 0.25 * ((a * (t_0 * math.sqrt((8.0 * (0.5 - (0.5 * math.cos((0.011111111111111112 * (angle * math.pi))))))))) / math.fabs(math.fabs(y_45_scale)))
	elif math.fabs(y_45_scale) <= 1.32e+154:
		tmp = 0.25 * (a * (t_0 * (angle * math.sqrt((8.0 * (((3.08641975308642e-5 * math.pow(math.pi, 2.0)) - math.sqrt((9.525986892242036e-10 * math.pow(math.pi, 4.0)))) / t_0))))))
	else:
		tmp = -math.sqrt((((2.0 * t_2) * t_1) * (0.0 / t_0))) / t_2
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(y_45_scale) ^ 2.0
	t_1 = Float64(Float64(b * a) * Float64(b * Float64(-a)))
	t_2 = Float64(Float64(4.0 * t_1) / (Float64(x_45_scale * abs(y_45_scale)) ^ 2.0))
	tmp = 0.0
	if (abs(y_45_scale) <= 1.55e-162)
		tmp = Float64(0.25 * Float64(Float64(a * Float64(t_0 * sqrt(Float64(8.0 * Float64(0.5 - Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))))))) / abs(abs(y_45_scale))));
	elseif (abs(y_45_scale) <= 1.32e+154)
		tmp = Float64(0.25 * Float64(a * Float64(t_0 * Float64(angle * sqrt(Float64(8.0 * Float64(Float64(Float64(3.08641975308642e-5 * (pi ^ 2.0)) - sqrt(Float64(9.525986892242036e-10 * (pi ^ 4.0)))) / t_0)))))));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_2) * t_1) * Float64(0.0 / t_0)))) / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(y_45_scale) ^ 2.0;
	t_1 = (b * a) * (b * -a);
	t_2 = (4.0 * t_1) / ((x_45_scale * abs(y_45_scale)) ^ 2.0);
	tmp = 0.0;
	if (abs(y_45_scale) <= 1.55e-162)
		tmp = 0.25 * ((a * (t_0 * sqrt((8.0 * (0.5 - (0.5 * cos((0.011111111111111112 * (angle * pi))))))))) / abs(abs(y_45_scale)));
	elseif (abs(y_45_scale) <= 1.32e+154)
		tmp = 0.25 * (a * (t_0 * (angle * sqrt((8.0 * (((3.08641975308642e-5 * (pi ^ 2.0)) - sqrt((9.525986892242036e-10 * (pi ^ 4.0)))) / t_0))))));
	else
		tmp = -sqrt((((2.0 * t_2) * t_1) * (0.0 / t_0))) / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * t$95$1), $MachinePrecision] / N[Power[N[(x$45$scale * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 1.55e-162], N[(0.25 * N[(N[(a * N[(t$95$0 * N[Sqrt[N[(8.0 * N[(0.5 - N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[N[Abs[y$45$scale], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 1.32e+154], N[(0.25 * N[(a * N[(t$95$0 * N[(angle * N[Sqrt[N[(8.0 * N[(N[(N[(3.08641975308642e-5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(9.525986892242036e-10 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(0.0 / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 1.5499999999999999e-162

   1. Initial program 0.0%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 0.8%

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

   6. Applied rewrites 3.5%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 4.9%

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

   10. Taylor expanded in b around 0

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

       1. lower-/.f64 N/A

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

   12. Applied rewrites 23.7%

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

   if 1.5499999999999999e-162 < y-scale < 1.32e154

   1. Initial program 0.0%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 4.1%

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

   8. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

   10. Applied rewrites 12.8%

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

   11. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

   13. Applied rewrites 14.2%

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

   if 1.32e154 < y-scale

   1. Initial program 0.0%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 2.3%

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in a around 0

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{0}{{y-scale}^{2}}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 3.8%

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{0}{{y-scale}^{2}}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 28.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|y-scale\right|\right)}^{2}\\ t_1 := \left|\left|y-scale\right|\right|\\ \mathbf{if}\;\left|y-scale\right| \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;0.25 \cdot \frac{a \cdot \left(t\_0 \cdot \sqrt{8 \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{t\_1}\\ \mathbf{elif}\;\left|y-scale\right| \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(t\_0 \cdot \left(angle \cdot \sqrt{8 \cdot \frac{3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - \sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}}}{t\_0}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left|y-scale\right| \cdot \left(\left|y-scale\right| \cdot \frac{\sqrt{\left(8 \cdot {\left(a \cdot b\right)}^{4}\right) \cdot \left(b \cdot b - \sqrt{{b}^{4}}\right)}}{t\_1 \cdot \left(\left(\left(a \cdot b\right) \cdot a\right) \cdot b\right)}\right)\right) \cdot 0.25\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (pow (fabs y-scale) 2.0)) (t_1 (fabs (fabs y-scale))))
  (if (<= (fabs y-scale) 1.55e-162)
    (*
     0.25
     (/
      (*
       a
       (*
        t_0
        (sqrt
         (*
          8.0
          (-
           0.5
           (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))))))
      t_1))
    (if (<= (fabs y-scale) 1.4e+154)
      (*
       0.25
       (*
        a
        (*
         t_0
         (*
          angle
          (sqrt
           (*
            8.0
            (/
             (-
              (* 3.08641975308642e-5 (pow PI 2.0))
              (sqrt (* 9.525986892242036e-10 (pow PI 4.0))))
             t_0)))))))
      (*
       (*
        (fabs y-scale)
        (*
         (fabs y-scale)
         (/
          (sqrt
           (*
            (* 8.0 (pow (* a b) 4.0))
            (- (* b b) (sqrt (pow b 4.0)))))
          (* t_1 (* (* (* a b) a) b)))))
       0.25)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = pow(fabs(y_45_scale), 2.0);
	double t_1 = fabs(fabs(y_45_scale));
	double tmp;
	if (fabs(y_45_scale) <= 1.55e-162) {
		tmp = 0.25 * ((a * (t_0 * sqrt((8.0 * (0.5 - (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI)))))))))) / t_1);
	} else if (fabs(y_45_scale) <= 1.4e+154) {
		tmp = 0.25 * (a * (t_0 * (angle * sqrt((8.0 * (((3.08641975308642e-5 * pow(((double) M_PI), 2.0)) - sqrt((9.525986892242036e-10 * pow(((double) M_PI), 4.0)))) / t_0))))));
	} else {
		tmp = (fabs(y_45_scale) * (fabs(y_45_scale) * (sqrt(((8.0 * pow((a * b), 4.0)) * ((b * b) - sqrt(pow(b, 4.0))))) / (t_1 * (((a * b) * a) * b))))) * 0.25;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.pow(Math.abs(y_45_scale), 2.0);
	double t_1 = Math.abs(Math.abs(y_45_scale));
	double tmp;
	if (Math.abs(y_45_scale) <= 1.55e-162) {
		tmp = 0.25 * ((a * (t_0 * Math.sqrt((8.0 * (0.5 - (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI))))))))) / t_1);
	} else if (Math.abs(y_45_scale) <= 1.4e+154) {
		tmp = 0.25 * (a * (t_0 * (angle * Math.sqrt((8.0 * (((3.08641975308642e-5 * Math.pow(Math.PI, 2.0)) - Math.sqrt((9.525986892242036e-10 * Math.pow(Math.PI, 4.0)))) / t_0))))));
	} else {
		tmp = (Math.abs(y_45_scale) * (Math.abs(y_45_scale) * (Math.sqrt(((8.0 * Math.pow((a * b), 4.0)) * ((b * b) - Math.sqrt(Math.pow(b, 4.0))))) / (t_1 * (((a * b) * a) * b))))) * 0.25;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.pow(math.fabs(y_45_scale), 2.0)
	t_1 = math.fabs(math.fabs(y_45_scale))
	tmp = 0
	if math.fabs(y_45_scale) <= 1.55e-162:
		tmp = 0.25 * ((a * (t_0 * math.sqrt((8.0 * (0.5 - (0.5 * math.cos((0.011111111111111112 * (angle * math.pi))))))))) / t_1)
	elif math.fabs(y_45_scale) <= 1.4e+154:
		tmp = 0.25 * (a * (t_0 * (angle * math.sqrt((8.0 * (((3.08641975308642e-5 * math.pow(math.pi, 2.0)) - math.sqrt((9.525986892242036e-10 * math.pow(math.pi, 4.0)))) / t_0))))))
	else:
		tmp = (math.fabs(y_45_scale) * (math.fabs(y_45_scale) * (math.sqrt(((8.0 * math.pow((a * b), 4.0)) * ((b * b) - math.sqrt(math.pow(b, 4.0))))) / (t_1 * (((a * b) * a) * b))))) * 0.25
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(y_45_scale) ^ 2.0
	t_1 = abs(abs(y_45_scale))
	tmp = 0.0
	if (abs(y_45_scale) <= 1.55e-162)
		tmp = Float64(0.25 * Float64(Float64(a * Float64(t_0 * sqrt(Float64(8.0 * Float64(0.5 - Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))))))) / t_1));
	elseif (abs(y_45_scale) <= 1.4e+154)
		tmp = Float64(0.25 * Float64(a * Float64(t_0 * Float64(angle * sqrt(Float64(8.0 * Float64(Float64(Float64(3.08641975308642e-5 * (pi ^ 2.0)) - sqrt(Float64(9.525986892242036e-10 * (pi ^ 4.0)))) / t_0)))))));
	else
		tmp = Float64(Float64(abs(y_45_scale) * Float64(abs(y_45_scale) * Float64(sqrt(Float64(Float64(8.0 * (Float64(a * b) ^ 4.0)) * Float64(Float64(b * b) - sqrt((b ^ 4.0))))) / Float64(t_1 * Float64(Float64(Float64(a * b) * a) * b))))) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(y_45_scale) ^ 2.0;
	t_1 = abs(abs(y_45_scale));
	tmp = 0.0;
	if (abs(y_45_scale) <= 1.55e-162)
		tmp = 0.25 * ((a * (t_0 * sqrt((8.0 * (0.5 - (0.5 * cos((0.011111111111111112 * (angle * pi))))))))) / t_1);
	elseif (abs(y_45_scale) <= 1.4e+154)
		tmp = 0.25 * (a * (t_0 * (angle * sqrt((8.0 * (((3.08641975308642e-5 * (pi ^ 2.0)) - sqrt((9.525986892242036e-10 * (pi ^ 4.0)))) / t_0))))));
	else
		tmp = (abs(y_45_scale) * (abs(y_45_scale) * (sqrt(((8.0 * ((a * b) ^ 4.0)) * ((b * b) - sqrt((b ^ 4.0))))) / (t_1 * (((a * b) * a) * b))))) * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[Abs[y$45$scale], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 1.55e-162], N[(0.25 * N[(N[(a * N[(t$95$0 * N[Sqrt[N[(8.0 * N[(0.5 - N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 1.4e+154], N[(0.25 * N[(a * N[(t$95$0 * N[(angle * N[Sqrt[N[(8.0 * N[(N[(N[(3.08641975308642e-5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(9.525986892242036e-10 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[y$45$scale], $MachinePrecision] * N[(N[Abs[y$45$scale], $MachinePrecision] * N[(N[Sqrt[N[(N[(8.0 * N[Power[N[(a * b), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] - N[Sqrt[N[Power[b, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 * N[(N[(N[(a * b), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 1.5499999999999999e-162

   1. Initial program 0.0%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 0.8%

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

   6. Applied rewrites 3.5%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 4.9%

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

   10. Taylor expanded in b around 0

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

       1. lower-/.f64 N/A

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

   12. Applied rewrites 23.7%

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

   if 1.5499999999999999e-162 < y-scale < 1.4e154

   1. Initial program 0.0%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 4.1%

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

   8. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

   10. Applied rewrites 12.8%

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

   11. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

   13. Applied rewrites 14.2%

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

   if 1.4e154 < y-scale

   1. Initial program 0.0%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in angle around 0

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 0.7%

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

   7. Applied rewrites 0.7%

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

   9. Applied rewrites 7.2%

      \[\leadsto \left(\left(y-scale \cdot y-scale\right) \cdot \frac{\frac{\sqrt{8 \cdot \left({\left(a \cdot b\right)}^{4} \cdot \left(b \cdot b - \sqrt{{b}^{4}}\right)\right)}}{\left|y-scale\right|}}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}\right) \cdot \color{blue}{0.25} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

   11. Applied rewrites 5.4%

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

Alternative 4: 24.9% accurate, 8.9× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| \cdot b\\ t_1 := \left|\left|y-scale\right|\right|\\ \mathbf{if}\;\left|y-scale\right| \leq 2.2 \cdot 10^{+170}:\\ \;\;\;\;0.25 \cdot \frac{\left|a\right| \cdot \left({\left(\left|y-scale\right|\right)}^{2} \cdot \sqrt{8 \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left|y-scale\right| \cdot \left(\left|y-scale\right| \cdot \frac{\sqrt{\left(8 \cdot {t\_0}^{4}\right) \cdot \left(b \cdot b - \sqrt{{b}^{4}}\right)}}{t\_1 \cdot \left(\left(t\_0 \cdot \left|a\right|\right) \cdot b\right)}\right)\right) \cdot 0.25\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (fabs a) b)) (t_1 (fabs (fabs y-scale))))
  (if (<= (fabs y-scale) 2.2e+170)
    (*
     0.25
     (/
      (*
       (fabs a)
       (*
        (pow (fabs y-scale) 2.0)
        (sqrt
         (*
          8.0
          (-
           0.5
           (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))))))
      t_1))
    (*
     (*
      (fabs y-scale)
      (*
       (fabs y-scale)
       (/
        (sqrt
         (* (* 8.0 (pow t_0 4.0)) (- (* b b) (sqrt (pow b 4.0)))))
        (* t_1 (* (* t_0 (fabs a)) b)))))
     0.25))))

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(fabs(y_45_scale));
	double tmp;
	if (fabs(y_45_scale) <= 2.2e+170) {
		tmp = 0.25 * ((fabs(a) * (pow(fabs(y_45_scale), 2.0) * sqrt((8.0 * (0.5 - (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI)))))))))) / t_1);
	} else {
		tmp = (fabs(y_45_scale) * (fabs(y_45_scale) * (sqrt(((8.0 * pow(t_0, 4.0)) * ((b * b) - sqrt(pow(b, 4.0))))) / (t_1 * ((t_0 * fabs(a)) * b))))) * 0.25;
	}
	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(a) * b;
	double t_1 = Math.abs(Math.abs(y_45_scale));
	double tmp;
	if (Math.abs(y_45_scale) <= 2.2e+170) {
		tmp = 0.25 * ((Math.abs(a) * (Math.pow(Math.abs(y_45_scale), 2.0) * Math.sqrt((8.0 * (0.5 - (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI))))))))) / t_1);
	} else {
		tmp = (Math.abs(y_45_scale) * (Math.abs(y_45_scale) * (Math.sqrt(((8.0 * Math.pow(t_0, 4.0)) * ((b * b) - Math.sqrt(Math.pow(b, 4.0))))) / (t_1 * ((t_0 * Math.abs(a)) * b))))) * 0.25;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.fabs(a) * b
	t_1 = math.fabs(math.fabs(y_45_scale))
	tmp = 0
	if math.fabs(y_45_scale) <= 2.2e+170:
		tmp = 0.25 * ((math.fabs(a) * (math.pow(math.fabs(y_45_scale), 2.0) * math.sqrt((8.0 * (0.5 - (0.5 * math.cos((0.011111111111111112 * (angle * math.pi))))))))) / t_1)
	else:
		tmp = (math.fabs(y_45_scale) * (math.fabs(y_45_scale) * (math.sqrt(((8.0 * math.pow(t_0, 4.0)) * ((b * b) - math.sqrt(math.pow(b, 4.0))))) / (t_1 * ((t_0 * math.fabs(a)) * b))))) * 0.25
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(a) * b)
	t_1 = abs(abs(y_45_scale))
	tmp = 0.0
	if (abs(y_45_scale) <= 2.2e+170)
		tmp = Float64(0.25 * Float64(Float64(abs(a) * Float64((abs(y_45_scale) ^ 2.0) * sqrt(Float64(8.0 * Float64(0.5 - Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))))))) / t_1));
	else
		tmp = Float64(Float64(abs(y_45_scale) * Float64(abs(y_45_scale) * Float64(sqrt(Float64(Float64(8.0 * (t_0 ^ 4.0)) * Float64(Float64(b * b) - sqrt((b ^ 4.0))))) / Float64(t_1 * Float64(Float64(t_0 * abs(a)) * b))))) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(a) * b;
	t_1 = abs(abs(y_45_scale));
	tmp = 0.0;
	if (abs(y_45_scale) <= 2.2e+170)
		tmp = 0.25 * ((abs(a) * ((abs(y_45_scale) ^ 2.0) * sqrt((8.0 * (0.5 - (0.5 * cos((0.011111111111111112 * (angle * pi))))))))) / t_1);
	else
		tmp = (abs(y_45_scale) * (abs(y_45_scale) * (sqrt(((8.0 * (t_0 ^ 4.0)) * ((b * b) - sqrt((b ^ 4.0))))) / (t_1 * ((t_0 * abs(a)) * b))))) * 0.25;
	end
	tmp_2 = 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[N[Abs[y$45$scale], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 2.2e+170], N[(0.25 * N[(N[(N[Abs[a], $MachinePrecision] * N[(N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(0.5 - N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[y$45$scale], $MachinePrecision] * N[(N[Abs[y$45$scale], $MachinePrecision] * N[(N[Sqrt[N[(N[(8.0 * N[Power[t$95$0, 4.0], $MachinePrecision]), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] - N[Sqrt[N[Power[b, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 * N[(N[(t$95$0 * N[Abs[a], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.1999999999999999e170

   1. Initial program 0.0%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 0.8%

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

   6. Applied rewrites 3.5%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 4.9%

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

   10. Taylor expanded in b around 0

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

       1. lower-/.f64 N/A

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

   12. Applied rewrites 23.7%

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

   if 2.1999999999999999e170 < y-scale

   1. Initial program 0.0%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in angle around 0

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 0.7%

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

   7. Applied rewrites 0.7%

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

   9. Applied rewrites 7.2%

      \[\leadsto \left(\left(y-scale \cdot y-scale\right) \cdot \frac{\frac{\sqrt{8 \cdot \left({\left(a \cdot b\right)}^{4} \cdot \left(b \cdot b - \sqrt{{b}^{4}}\right)\right)}}{\left|y-scale\right|}}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}\right) \cdot \color{blue}{0.25} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

   11. Applied rewrites 5.4%

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

Alternative 5: 14.8% accurate, 9.1× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\left(8 \cdot {\left(a \cdot b\right)}^{4}\right) \cdot \left(b \cdot b - \sqrt{{b}^{4}}\right)}\\ t_1 := \left|\left|y-scale\right|\right|\\ \mathbf{if}\;\left|y-scale\right| \leq 3.8 \cdot 10^{+152}:\\ \;\;\;\;\left(\left(\left|y-scale\right| \cdot \left|y-scale\right|\right) \cdot \frac{\frac{\frac{t\_0}{t\_1}}{a \cdot b}}{a \cdot b}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\left(\left|y-scale\right| \cdot \left(\left|y-scale\right| \cdot \frac{t\_0}{t\_1 \cdot \left(\left(\left(a \cdot b\right) \cdot a\right) \cdot b\right)}\right)\right) \cdot 0.25\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0
        (sqrt
         (*
          (* 8.0 (pow (* a b) 4.0))
          (- (* b b) (sqrt (pow b 4.0))))))
       (t_1 (fabs (fabs y-scale))))
  (if (<= (fabs y-scale) 3.8e+152)
    (*
     (*
      (* (fabs y-scale) (fabs y-scale))
      (/ (/ (/ t_0 t_1) (* a b)) (* a b)))
     0.25)
    (*
     (*
      (fabs y-scale)
      (* (fabs y-scale) (/ t_0 (* t_1 (* (* (* a b) a) b)))))
     0.25))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sqrt(((8.0 * pow((a * b), 4.0)) * ((b * b) - sqrt(pow(b, 4.0)))));
	double t_1 = fabs(fabs(y_45_scale));
	double tmp;
	if (fabs(y_45_scale) <= 3.8e+152) {
		tmp = ((fabs(y_45_scale) * fabs(y_45_scale)) * (((t_0 / t_1) / (a * b)) / (a * b))) * 0.25;
	} else {
		tmp = (fabs(y_45_scale) * (fabs(y_45_scale) * (t_0 / (t_1 * (((a * b) * a) * b))))) * 0.25;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(((8.0d0 * ((a * b) ** 4.0d0)) * ((b * b) - sqrt((b ** 4.0d0)))))
    t_1 = abs(abs(y_45scale))
    if (abs(y_45scale) <= 3.8d+152) then
        tmp = ((abs(y_45scale) * abs(y_45scale)) * (((t_0 / t_1) / (a * b)) / (a * b))) * 0.25d0
    else
        tmp = (abs(y_45scale) * (abs(y_45scale) * (t_0 / (t_1 * (((a * b) * a) * b))))) * 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.sqrt(((8.0 * Math.pow((a * b), 4.0)) * ((b * b) - Math.sqrt(Math.pow(b, 4.0)))));
	double t_1 = Math.abs(Math.abs(y_45_scale));
	double tmp;
	if (Math.abs(y_45_scale) <= 3.8e+152) {
		tmp = ((Math.abs(y_45_scale) * Math.abs(y_45_scale)) * (((t_0 / t_1) / (a * b)) / (a * b))) * 0.25;
	} else {
		tmp = (Math.abs(y_45_scale) * (Math.abs(y_45_scale) * (t_0 / (t_1 * (((a * b) * a) * b))))) * 0.25;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.sqrt(((8.0 * math.pow((a * b), 4.0)) * ((b * b) - math.sqrt(math.pow(b, 4.0)))))
	t_1 = math.fabs(math.fabs(y_45_scale))
	tmp = 0
	if math.fabs(y_45_scale) <= 3.8e+152:
		tmp = ((math.fabs(y_45_scale) * math.fabs(y_45_scale)) * (((t_0 / t_1) / (a * b)) / (a * b))) * 0.25
	else:
		tmp = (math.fabs(y_45_scale) * (math.fabs(y_45_scale) * (t_0 / (t_1 * (((a * b) * a) * b))))) * 0.25
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = sqrt(Float64(Float64(8.0 * (Float64(a * b) ^ 4.0)) * Float64(Float64(b * b) - sqrt((b ^ 4.0)))))
	t_1 = abs(abs(y_45_scale))
	tmp = 0.0
	if (abs(y_45_scale) <= 3.8e+152)
		tmp = Float64(Float64(Float64(abs(y_45_scale) * abs(y_45_scale)) * Float64(Float64(Float64(t_0 / t_1) / Float64(a * b)) / Float64(a * b))) * 0.25);
	else
		tmp = Float64(Float64(abs(y_45_scale) * Float64(abs(y_45_scale) * Float64(t_0 / Float64(t_1 * Float64(Float64(Float64(a * b) * a) * b))))) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = sqrt(((8.0 * ((a * b) ^ 4.0)) * ((b * b) - sqrt((b ^ 4.0)))));
	t_1 = abs(abs(y_45_scale));
	tmp = 0.0;
	if (abs(y_45_scale) <= 3.8e+152)
		tmp = ((abs(y_45_scale) * abs(y_45_scale)) * (((t_0 / t_1) / (a * b)) / (a * b))) * 0.25;
	else
		tmp = (abs(y_45_scale) * (abs(y_45_scale) * (t_0 / (t_1 * (((a * b) * a) * b))))) * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sqrt[N[(N[(8.0 * N[Power[N[(a * b), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] - N[Sqrt[N[Power[b, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[Abs[y$45$scale], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 3.8e+152], N[(N[(N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 / t$95$1), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision], N[(N[(N[Abs[y$45$scale], $MachinePrecision] * N[(N[Abs[y$45$scale], $MachinePrecision] * N[(t$95$0 / N[(t$95$1 * N[(N[(N[(a * b), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 3.8e152

   1. Initial program 0.0%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in angle around 0

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 0.7%

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

   7. Applied rewrites 0.7%

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

   9. Applied rewrites 7.2%

      \[\leadsto \left(\left(y-scale \cdot y-scale\right) \cdot \frac{\frac{\sqrt{8 \cdot \left({\left(a \cdot b\right)}^{4} \cdot \left(b \cdot b - \sqrt{{b}^{4}}\right)\right)}}{\left|y-scale\right|}}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}\right) \cdot \color{blue}{0.25} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

   11. Applied rewrites 13.8%

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

   if 3.8e152 < y-scale

   1. Initial program 0.0%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in angle around 0

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 0.7%

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

   7. Applied rewrites 0.7%

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

   9. Applied rewrites 7.2%

      \[\leadsto \left(\left(y-scale \cdot y-scale\right) \cdot \frac{\frac{\sqrt{8 \cdot \left({\left(a \cdot b\right)}^{4} \cdot \left(b \cdot b - \sqrt{{b}^{4}}\right)\right)}}{\left|y-scale\right|}}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}\right) \cdot \color{blue}{0.25} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

   11. Applied rewrites 5.4%

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

Alternative 6: 9.0% accurate, 9.2× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| \cdot b\\ \mathbf{if}\;\left|a\right| \leq 9.5 \cdot 10^{-93}:\\ \;\;\;\;0.25 \cdot \frac{\left|a\right| \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(0.5 - 0.5\right)\right)}\right)}{{b}^{2} \cdot \left|y-scale\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(y-scale \cdot \left(y-scale \cdot \frac{\sqrt{\left(8 \cdot {t\_0}^{4}\right) \cdot \left(b \cdot b - \sqrt{{b}^{4}}\right)}}{\left|y-scale\right| \cdot \left(\left(t\_0 \cdot \left|a\right|\right) \cdot b\right)}\right)\right) \cdot 0.25\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (fabs a) b)))
  (if (<= (fabs a) 9.5e-93)
    (*
     0.25
     (/
      (*
       (fabs a)
       (*
        (pow y-scale 2.0)
        (sqrt (* 8.0 (* (pow b 4.0) (- 0.5 0.5))))))
      (* (pow b 2.0) (fabs y-scale))))
    (*
     (*
      y-scale
      (*
       y-scale
       (/
        (sqrt
         (* (* 8.0 (pow t_0 4.0)) (- (* b b) (sqrt (pow b 4.0)))))
        (* (fabs y-scale) (* (* t_0 (fabs a)) b)))))
     0.25))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(a) * b;
	double tmp;
	if (fabs(a) <= 9.5e-93) {
		tmp = 0.25 * ((fabs(a) * (pow(y_45_scale, 2.0) * sqrt((8.0 * (pow(b, 4.0) * (0.5 - 0.5)))))) / (pow(b, 2.0) * fabs(y_45_scale)));
	} else {
		tmp = (y_45_scale * (y_45_scale * (sqrt(((8.0 * pow(t_0, 4.0)) * ((b * b) - sqrt(pow(b, 4.0))))) / (fabs(y_45_scale) * ((t_0 * fabs(a)) * b))))) * 0.25;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(a) * b
    if (abs(a) <= 9.5d-93) then
        tmp = 0.25d0 * ((abs(a) * ((y_45scale ** 2.0d0) * sqrt((8.0d0 * ((b ** 4.0d0) * (0.5d0 - 0.5d0)))))) / ((b ** 2.0d0) * abs(y_45scale)))
    else
        tmp = (y_45scale * (y_45scale * (sqrt(((8.0d0 * (t_0 ** 4.0d0)) * ((b * b) - sqrt((b ** 4.0d0))))) / (abs(y_45scale) * ((t_0 * abs(a)) * b))))) * 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.abs(a) * b;
	double tmp;
	if (Math.abs(a) <= 9.5e-93) {
		tmp = 0.25 * ((Math.abs(a) * (Math.pow(y_45_scale, 2.0) * Math.sqrt((8.0 * (Math.pow(b, 4.0) * (0.5 - 0.5)))))) / (Math.pow(b, 2.0) * Math.abs(y_45_scale)));
	} else {
		tmp = (y_45_scale * (y_45_scale * (Math.sqrt(((8.0 * Math.pow(t_0, 4.0)) * ((b * b) - Math.sqrt(Math.pow(b, 4.0))))) / (Math.abs(y_45_scale) * ((t_0 * Math.abs(a)) * b))))) * 0.25;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.fabs(a) * b
	tmp = 0
	if math.fabs(a) <= 9.5e-93:
		tmp = 0.25 * ((math.fabs(a) * (math.pow(y_45_scale, 2.0) * math.sqrt((8.0 * (math.pow(b, 4.0) * (0.5 - 0.5)))))) / (math.pow(b, 2.0) * math.fabs(y_45_scale)))
	else:
		tmp = (y_45_scale * (y_45_scale * (math.sqrt(((8.0 * math.pow(t_0, 4.0)) * ((b * b) - math.sqrt(math.pow(b, 4.0))))) / (math.fabs(y_45_scale) * ((t_0 * math.fabs(a)) * b))))) * 0.25
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(a) * b)
	tmp = 0.0
	if (abs(a) <= 9.5e-93)
		tmp = Float64(0.25 * Float64(Float64(abs(a) * Float64((y_45_scale ^ 2.0) * sqrt(Float64(8.0 * Float64((b ^ 4.0) * Float64(0.5 - 0.5)))))) / Float64((b ^ 2.0) * abs(y_45_scale))));
	else
		tmp = Float64(Float64(y_45_scale * Float64(y_45_scale * Float64(sqrt(Float64(Float64(8.0 * (t_0 ^ 4.0)) * Float64(Float64(b * b) - sqrt((b ^ 4.0))))) / Float64(abs(y_45_scale) * Float64(Float64(t_0 * abs(a)) * b))))) * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(a) * b;
	tmp = 0.0;
	if (abs(a) <= 9.5e-93)
		tmp = 0.25 * ((abs(a) * ((y_45_scale ^ 2.0) * sqrt((8.0 * ((b ^ 4.0) * (0.5 - 0.5)))))) / ((b ^ 2.0) * abs(y_45_scale)));
	else
		tmp = (y_45_scale * (y_45_scale * (sqrt(((8.0 * (t_0 ^ 4.0)) * ((b * b) - sqrt((b ^ 4.0))))) / (abs(y_45_scale) * ((t_0 * abs(a)) * b))))) * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[a], $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 9.5e-93], N[(0.25 * N[(N[(N[Abs[a], $MachinePrecision] * N[(N[Power[y$45$scale, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[Power[b, 4.0], $MachinePrecision] * N[(0.5 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 2.0], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$45$scale * N[(y$45$scale * N[(N[Sqrt[N[(N[(8.0 * N[Power[t$95$0, 4.0], $MachinePrecision]), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] - N[Sqrt[N[Power[b, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Abs[y$45$scale], $MachinePrecision] * N[(N[(t$95$0 * N[Abs[a], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 9.5000000000000001e-93

   1. Initial program 0.0%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 0.8%

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

   6. Applied rewrites 3.5%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 4.9%

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

   10. Taylor expanded in angle around 0

       \[\leadsto 0.25 \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(0.5 - \frac{1}{2}\right)\right)}\right)}{{b}^{2} \cdot \left|y-scale\right|} \]
   11. Step-by-step derivation

   12. Applied rewrites 5.4%

       \[\leadsto 0.25 \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(0.5 - 0.5\right)\right)}\right)}{{b}^{2} \cdot \left|y-scale\right|} \]

   if 9.5000000000000001e-93 < a

   1. Initial program 0.0%

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

   2. Taylor expanded in x-scale around 0

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in angle around 0

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 0.7%

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

   7. Applied rewrites 0.7%

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

   9. Applied rewrites 7.2%

      \[\leadsto \left(\left(y-scale \cdot y-scale\right) \cdot \frac{\frac{\sqrt{8 \cdot \left({\left(a \cdot b\right)}^{4} \cdot \left(b \cdot b - \sqrt{{b}^{4}}\right)\right)}}{\left|y-scale\right|}}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}\right) \cdot \color{blue}{0.25} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

   11. Applied rewrites 5.4%

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

Alternative 7: 5.4% accurate, 10.1× speedup

Math

\[0.25 \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(0.5 - 0.5\right)\right)}\right)}{{b}^{2} \cdot \left|y-scale\right|} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (*
 0.25
 (/
  (*
   a
   (* (pow y-scale 2.0) (sqrt (* 8.0 (* (pow b 4.0) (- 0.5 0.5))))))
  (* (pow b 2.0) (fabs y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * ((a * (pow(y_45_scale, 2.0) * sqrt((8.0 * (pow(b, 4.0) * (0.5 - 0.5)))))) / (pow(b, 2.0) * fabs(y_45_scale)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = 0.25d0 * ((a * ((y_45scale ** 2.0d0) * sqrt((8.0d0 * ((b ** 4.0d0) * (0.5d0 - 0.5d0)))))) / ((b ** 2.0d0) * abs(y_45scale)))
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * ((a * (Math.pow(y_45_scale, 2.0) * Math.sqrt((8.0 * (Math.pow(b, 4.0) * (0.5 - 0.5)))))) / (Math.pow(b, 2.0) * Math.abs(y_45_scale)));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 0.25 * ((a * (math.pow(y_45_scale, 2.0) * math.sqrt((8.0 * (math.pow(b, 4.0) * (0.5 - 0.5)))))) / (math.pow(b, 2.0) * math.fabs(y_45_scale)))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(0.25 * Float64(Float64(a * Float64((y_45_scale ^ 2.0) * sqrt(Float64(8.0 * Float64((b ^ 4.0) * Float64(0.5 - 0.5)))))) / Float64((b ^ 2.0) * abs(y_45_scale))))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.25 * ((a * ((y_45_scale ^ 2.0) * sqrt((8.0 * ((b ^ 4.0) * (0.5 - 0.5)))))) / ((b ^ 2.0) * abs(y_45_scale)));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(0.25 * N[(N[(a * N[(N[Power[y$45$scale, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[Power[b, 4.0], $MachinePrecision] * N[(0.5 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 2.0], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0%

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

2. Taylor expanded in x-scale around 0

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

4. Applied rewrites 0.8%

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

6. Applied rewrites 3.5%

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

7. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

9. Applied rewrites 4.9%

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

10. Taylor expanded in angle around 0

    \[\leadsto 0.25 \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(0.5 - \frac{1}{2}\right)\right)}\right)}{{b}^{2} \cdot \left|y-scale\right|} \]
11. Step-by-step derivation

12. Applied rewrites 5.4%

    \[\leadsto 0.25 \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(0.5 - 0.5\right)\right)}\right)}{{b}^{2} \cdot \left|y-scale\right|} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (a b angle x-scale y-scale)
  :name "b from scale-rotated-ellipse"
  :precision binary64
  (/ (- (sqrt (* (* (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))) (* (* b a) (* b (- a)))) (- (+ (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) (sqrt (+ (pow (- (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) 2.0) (pow (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale) 2.0))))))) (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))