a from scale-rotated-ellipse

Percentage Accurate: 2.9% → 16.3%

Time: 31.7s

Alternatives: 10

Speedup: 8.0×

• 30.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 2.9% 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: 16.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{a}{y-scale \cdot y-scale}\\ t_1 := \frac{b}{x-scale \cdot x-scale}\\ \left(\frac{\frac{\frac{\sqrt{\left(\mathsf{fma}\left(b, t\_1, \mathsf{fma}\left(a, t\_0, \left|\mathsf{fma}\left(t\_0, a, \left(-b\right) \cdot t\_1\right)\right|\right)\right) \cdot {\left(b \cdot a\right)}^{4}\right) \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ a (* y-scale y-scale))) (t_1 (/ b (* x-scale x-scale))))
   (*
    (*
     (/
      (/
       (/
        (sqrt
         (*
          (*
           (fma b t_1 (fma a t_0 (fabs (fma t_0 a (* (- b) t_1)))))
           (pow (* b a) 4.0))
          8.0))
        (fabs (* x-scale y-scale)))
       (* 4.0 (* b a)))
      (* b a))
     (* y-scale x-scale))
    (* y-scale x-scale))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a / (y_45_scale * y_45_scale);
	double t_1 = b / (x_45_scale * x_45_scale);
	return ((((sqrt(((fma(b, t_1, fma(a, t_0, fabs(fma(t_0, a, (-b * t_1))))) * pow((b * a), 4.0)) * 8.0)) / fabs((x_45_scale * y_45_scale))) / (4.0 * (b * a))) / (b * a)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_1 = Float64(b / Float64(x_45_scale * x_45_scale))
	return Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(fma(b, t_1, fma(a, t_0, abs(fma(t_0, a, Float64(Float64(-b) * t_1))))) * (Float64(b * a) ^ 4.0)) * 8.0)) / abs(Float64(x_45_scale * y_45_scale))) / Float64(4.0 * Float64(b * a))) / Float64(b * a)) * Float64(y_45_scale * x_45_scale)) * Float64(y_45_scale * x_45_scale))
end

Wolfram

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

Derivation

1. Initial program 2.9%

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

3. Applied rewrites 6.1%

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

4. Taylor expanded in angle around 0

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

6. Applied rewrites 2.5%

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

8. Applied rewrites 16.1%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 16.3

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

10. Applied rewrites 16.3%

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

Alternative 2: 16.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{a}{y-scale \cdot y-scale}\\ t_1 := \frac{b}{x-scale \cdot x-scale}\\ \left(\frac{\frac{\frac{\sqrt{\left(\mathsf{fma}\left(b, t\_1, \mathsf{fma}\left(a, t\_0, \left|a \cdot t\_0 - b \cdot t\_1\right|\right)\right) \cdot {\left(b \cdot a\right)}^{4}\right) \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ a (* y-scale y-scale))) (t_1 (/ b (* x-scale x-scale))))
   (*
    (*
     (/
      (/
       (/
        (sqrt
         (*
          (*
           (fma b t_1 (fma a t_0 (fabs (- (* a t_0) (* b t_1)))))
           (pow (* b a) 4.0))
          8.0))
        (fabs (* x-scale y-scale)))
       (* 4.0 (* b a)))
      (* b a))
     (* y-scale x-scale))
    (* y-scale x-scale))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a / (y_45_scale * y_45_scale);
	double t_1 = b / (x_45_scale * x_45_scale);
	return ((((sqrt(((fma(b, t_1, fma(a, t_0, fabs(((a * t_0) - (b * t_1))))) * pow((b * a), 4.0)) * 8.0)) / fabs((x_45_scale * y_45_scale))) / (4.0 * (b * a))) / (b * a)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_1 = Float64(b / Float64(x_45_scale * x_45_scale))
	return Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(fma(b, t_1, fma(a, t_0, abs(Float64(Float64(a * t_0) - Float64(b * t_1))))) * (Float64(b * a) ^ 4.0)) * 8.0)) / abs(Float64(x_45_scale * y_45_scale))) / Float64(4.0 * Float64(b * a))) / Float64(b * a)) * Float64(y_45_scale * x_45_scale)) * Float64(y_45_scale * x_45_scale))
end

Wolfram

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

Derivation

1. Initial program 2.9%

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

3. Applied rewrites 6.1%

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

4. Taylor expanded in angle around 0

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

6. Applied rewrites 2.5%

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

8. Applied rewrites 16.1%

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

Alternative 3: 10.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|b\right|}{x-scale \cdot x-scale}\\ t_1 := a \cdot \left|b\right|\\ t_2 := \frac{a}{y-scale \cdot y-scale}\\ t_3 := \left|b\right| \cdot a\\ t_4 := \left(\frac{\frac{\sqrt{\left(\mathsf{fma}\left(\left|b\right|, t\_0, \mathsf{fma}\left(a, t\_2, \left|a \cdot t\_2 - \left|b\right| \cdot t\_0\right|\right)\right) \cdot {t\_3}^{4}\right) \cdot 8}}{\left|y-scale \cdot x-scale\right|}}{\left(t\_1 \cdot 4\right) \cdot t\_1} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)\\ \mathbf{if}\;\left|b\right| \leq 4.7 \cdot 10^{-110}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;\left|b\right| \leq 4 \cdot 10^{+201}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {\left(\left|b\right|\right)}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot t\_3}}{t\_3} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (fabs b) (* x-scale x-scale)))
        (t_1 (* a (fabs b)))
        (t_2 (/ a (* y-scale y-scale)))
        (t_3 (* (fabs b) a))
        (t_4
         (*
          (*
           (/
            (/
             (sqrt
              (*
               (*
                (fma
                 (fabs b)
                 t_0
                 (fma a t_2 (fabs (- (* a t_2) (* (fabs b) t_0)))))
                (pow t_3 4.0))
               8.0))
             (fabs (* y-scale x-scale)))
            (* (* t_1 4.0) t_1))
           (* y-scale x-scale))
          (* y-scale x-scale))))
   (if (<= (fabs b) 4.7e-110)
     t_4
     (if (<= (fabs b) 4e+201)
       (*
        (*
         (/
          (/
           (/
            (sqrt
             (* (/ (* (pow a 6.0) (pow (fabs b) 4.0)) (pow y-scale 2.0)) 8.0))
            (fabs (* x-scale y-scale)))
           (* 4.0 t_3))
          t_3)
         (* y-scale x-scale))
        (* y-scale x-scale))
       t_4))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(b) / (x_45_scale * x_45_scale);
	double t_1 = a * fabs(b);
	double t_2 = a / (y_45_scale * y_45_scale);
	double t_3 = fabs(b) * a;
	double t_4 = (((sqrt(((fma(fabs(b), t_0, fma(a, t_2, fabs(((a * t_2) - (fabs(b) * t_0))))) * pow(t_3, 4.0)) * 8.0)) / fabs((y_45_scale * x_45_scale))) / ((t_1 * 4.0) * t_1)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
	double tmp;
	if (fabs(b) <= 4.7e-110) {
		tmp = t_4;
	} else if (fabs(b) <= 4e+201) {
		tmp = ((((sqrt((((pow(a, 6.0) * pow(fabs(b), 4.0)) / pow(y_45_scale, 2.0)) * 8.0)) / fabs((x_45_scale * y_45_scale))) / (4.0 * t_3)) / t_3) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(b) / Float64(x_45_scale * x_45_scale))
	t_1 = Float64(a * abs(b))
	t_2 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_3 = Float64(abs(b) * a)
	t_4 = Float64(Float64(Float64(Float64(sqrt(Float64(Float64(fma(abs(b), t_0, fma(a, t_2, abs(Float64(Float64(a * t_2) - Float64(abs(b) * t_0))))) * (t_3 ^ 4.0)) * 8.0)) / abs(Float64(y_45_scale * x_45_scale))) / Float64(Float64(t_1 * 4.0) * t_1)) * Float64(y_45_scale * x_45_scale)) * Float64(y_45_scale * x_45_scale))
	tmp = 0.0
	if (abs(b) <= 4.7e-110)
		tmp = t_4;
	elseif (abs(b) <= 4e+201)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64((a ^ 6.0) * (abs(b) ^ 4.0)) / (y_45_scale ^ 2.0)) * 8.0)) / abs(Float64(x_45_scale * y_45_scale))) / Float64(4.0 * t_3)) / t_3) * Float64(y_45_scale * x_45_scale)) * Float64(y_45_scale * x_45_scale));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[b], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(N[Sqrt[N[(N[(N[(N[Abs[b], $MachinePrecision] * t$95$0 + N[(a * t$95$2 + N[Abs[N[(N[(a * t$95$2), $MachinePrecision] - N[(N[Abs[b], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[t$95$3, 4.0], $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(y$45$scale * x$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * 4.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 4.7e-110], t$95$4, If[LessEqual[N[Abs[b], $MachinePrecision], 4e+201], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[(N[Power[a, 6.0], $MachinePrecision] * N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(x$45$scale * y$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(4.0 * t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.69999999999999992e-110 or 4.00000000000000015e201 < b

   1. Initial program 2.9%

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

   3. Applied rewrites 6.1%

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

   4. Taylor expanded in angle around 0

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

   6. Applied rewrites 2.5%

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

   8. Applied rewrites 9.8%

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

   if 4.69999999999999992e-110 < b < 4.00000000000000015e201

   1. Initial program 2.9%

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

   3. Applied rewrites 6.1%

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

   4. Taylor expanded in angle around 0

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

   6. Applied rewrites 2.5%

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

   8. Applied rewrites 16.1%

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

   9. Taylor expanded in a around inf

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

       1. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       5. lower-pow.f64 4.9

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   11. Applied rewrites 4.9%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 9.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|b\right|}{x-scale \cdot x-scale}\\ t_1 := \left|x-scale \cdot y-scale\right|\\ t_2 := \left|b\right| \cdot a\\ t_3 := {t\_2}^{4}\\ t_4 := \frac{a}{y-scale \cdot y-scale}\\ t_5 := 4 \cdot t\_2\\ t_6 := \left(t\_5 \cdot a\right) \cdot \left|b\right|\\ \mathbf{if}\;\left|b\right| \leq 4.7 \cdot 10^{-110}:\\ \;\;\;\;\left(\frac{\sqrt{\left(\mathsf{fma}\left(\left|b\right|, t\_0, \mathsf{fma}\left(a, t\_4, \left|a \cdot t\_4 - \left|b\right| \cdot t\_0\right|\right)\right) \cdot t\_3\right) \cdot 8}}{t\_1 \cdot t\_6} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)\\ \mathbf{elif}\;\left|b\right| \leq 4 \cdot 10^{+201}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {\left(\left|b\right|\right)}^{4}}{{y-scale}^{2}} \cdot 8}}{t\_1}}{t\_5}}{t\_2} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left|\frac{\sqrt{\mathsf{fma}\left(t\_0, \left|b\right|, \mathsf{fma}\left(t\_4, a, \left|\frac{\left|b\right| \cdot \left|b\right|}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)\right) \cdot \left(t\_3 \cdot 8\right)}}{x-scale \cdot y-scale}\right|}{t\_6} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (fabs b) (* x-scale x-scale)))
        (t_1 (fabs (* x-scale y-scale)))
        (t_2 (* (fabs b) a))
        (t_3 (pow t_2 4.0))
        (t_4 (/ a (* y-scale y-scale)))
        (t_5 (* 4.0 t_2))
        (t_6 (* (* t_5 a) (fabs b))))
   (if (<= (fabs b) 4.7e-110)
     (*
      (*
       (/
        (sqrt
         (*
          (*
           (fma (fabs b) t_0 (fma a t_4 (fabs (- (* a t_4) (* (fabs b) t_0)))))
           t_3)
          8.0))
        (* t_1 t_6))
       (* y-scale x-scale))
      (* y-scale x-scale))
     (if (<= (fabs b) 4e+201)
       (*
        (*
         (/
          (/
           (/
            (sqrt
             (* (/ (* (pow a 6.0) (pow (fabs b) 4.0)) (pow y-scale 2.0)) 8.0))
            t_1)
           t_5)
          t_2)
         (* y-scale x-scale))
        (* y-scale x-scale))
       (*
        (*
         (/
          (fabs
           (/
            (sqrt
             (*
              (fma
               t_0
               (fabs b)
               (fma
                t_4
                a
                (fabs
                 (-
                  (/ (* (fabs b) (fabs b)) (* x-scale x-scale))
                  (/ (* a a) (* y-scale y-scale))))))
              (* t_3 8.0)))
            (* x-scale y-scale)))
          t_6)
         (* y-scale x-scale))
        (* y-scale x-scale))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(b) / (x_45_scale * x_45_scale);
	double t_1 = fabs((x_45_scale * y_45_scale));
	double t_2 = fabs(b) * a;
	double t_3 = pow(t_2, 4.0);
	double t_4 = a / (y_45_scale * y_45_scale);
	double t_5 = 4.0 * t_2;
	double t_6 = (t_5 * a) * fabs(b);
	double tmp;
	if (fabs(b) <= 4.7e-110) {
		tmp = ((sqrt(((fma(fabs(b), t_0, fma(a, t_4, fabs(((a * t_4) - (fabs(b) * t_0))))) * t_3) * 8.0)) / (t_1 * t_6)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
	} else if (fabs(b) <= 4e+201) {
		tmp = ((((sqrt((((pow(a, 6.0) * pow(fabs(b), 4.0)) / pow(y_45_scale, 2.0)) * 8.0)) / t_1) / t_5) / t_2) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
	} else {
		tmp = ((fabs((sqrt((fma(t_0, fabs(b), fma(t_4, a, fabs((((fabs(b) * fabs(b)) / (x_45_scale * x_45_scale)) - ((a * a) / (y_45_scale * y_45_scale)))))) * (t_3 * 8.0))) / (x_45_scale * y_45_scale))) / t_6) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(b) / Float64(x_45_scale * x_45_scale))
	t_1 = abs(Float64(x_45_scale * y_45_scale))
	t_2 = Float64(abs(b) * a)
	t_3 = t_2 ^ 4.0
	t_4 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_5 = Float64(4.0 * t_2)
	t_6 = Float64(Float64(t_5 * a) * abs(b))
	tmp = 0.0
	if (abs(b) <= 4.7e-110)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(fma(abs(b), t_0, fma(a, t_4, abs(Float64(Float64(a * t_4) - Float64(abs(b) * t_0))))) * t_3) * 8.0)) / Float64(t_1 * t_6)) * Float64(y_45_scale * x_45_scale)) * Float64(y_45_scale * x_45_scale));
	elseif (abs(b) <= 4e+201)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64((a ^ 6.0) * (abs(b) ^ 4.0)) / (y_45_scale ^ 2.0)) * 8.0)) / t_1) / t_5) / t_2) * Float64(y_45_scale * x_45_scale)) * Float64(y_45_scale * x_45_scale));
	else
		tmp = Float64(Float64(Float64(abs(Float64(sqrt(Float64(fma(t_0, abs(b), fma(t_4, a, abs(Float64(Float64(Float64(abs(b) * abs(b)) / Float64(x_45_scale * x_45_scale)) - Float64(Float64(a * a) / Float64(y_45_scale * y_45_scale)))))) * Float64(t_3 * 8.0))) / Float64(x_45_scale * y_45_scale))) / t_6) * Float64(y_45_scale * x_45_scale)) * Float64(y_45_scale * x_45_scale));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x$45$scale * y$45$scale), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[b], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 4.0], $MachinePrecision]}, Block[{t$95$4 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(4.0 * t$95$2), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$5 * a), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 4.7e-110], N[(N[(N[(N[Sqrt[N[(N[(N[(N[Abs[b], $MachinePrecision] * t$95$0 + N[(a * t$95$4 + N[Abs[N[(N[(a * t$95$4), $MachinePrecision] - N[(N[Abs[b], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 * t$95$6), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 4e+201], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[(N[Power[a, 6.0], $MachinePrecision] * N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$5), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[N[(N[Sqrt[N[(N[(t$95$0 * N[Abs[b], $MachinePrecision] + N[(t$95$4 * a + N[Abs[N[(N[(N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * 8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.69999999999999992e-110

   1. Initial program 2.9%

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

   3. Applied rewrites 6.1%

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

   4. Taylor expanded in angle around 0

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

   6. Applied rewrites 2.5%

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

   8. Applied rewrites 7.0%

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

   if 4.69999999999999992e-110 < b < 4.00000000000000015e201

   1. Initial program 2.9%

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

   3. Applied rewrites 6.1%

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

   4. Taylor expanded in angle around 0

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

   6. Applied rewrites 2.5%

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

   8. Applied rewrites 16.1%

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

   9. Taylor expanded in a around inf

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

       1. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       5. lower-pow.f64 4.9

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   11. Applied rewrites 4.9%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   if 4.00000000000000015e201 < b

   1. Initial program 2.9%

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

   3. Applied rewrites 6.1%

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

   4. Taylor expanded in angle around 0

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

   6. Applied rewrites 2.5%

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

   8. Applied rewrites 16.1%

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

   10. Applied rewrites 7.3%

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

Alternative 5: 7.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|b\right|}{x-scale \cdot x-scale}\\ t_1 := \left|x-scale \cdot y-scale\right|\\ t_2 := \left|b\right| \cdot a\\ t_3 := \frac{a}{y-scale \cdot y-scale}\\ t_4 := 4 \cdot t\_2\\ t_5 := \left(\frac{\sqrt{\left(\mathsf{fma}\left(\left|b\right|, t\_0, \mathsf{fma}\left(a, t\_3, \left|a \cdot t\_3 - \left|b\right| \cdot t\_0\right|\right)\right) \cdot {t\_2}^{4}\right) \cdot 8}}{t\_1 \cdot \left(\left(t\_4 \cdot a\right) \cdot \left|b\right|\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)\\ \mathbf{if}\;\left|b\right| \leq 4.7 \cdot 10^{-110}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;\left|b\right| \leq 4.3 \cdot 10^{+201}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {\left(\left|b\right|\right)}^{4}}{{y-scale}^{2}} \cdot 8}}{t\_1}}{t\_4}}{t\_2} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (fabs b) (* x-scale x-scale)))
        (t_1 (fabs (* x-scale y-scale)))
        (t_2 (* (fabs b) a))
        (t_3 (/ a (* y-scale y-scale)))
        (t_4 (* 4.0 t_2))
        (t_5
         (*
          (*
           (/
            (sqrt
             (*
              (*
               (fma
                (fabs b)
                t_0
                (fma a t_3 (fabs (- (* a t_3) (* (fabs b) t_0)))))
               (pow t_2 4.0))
              8.0))
            (* t_1 (* (* t_4 a) (fabs b))))
           (* y-scale x-scale))
          (* y-scale x-scale))))
   (if (<= (fabs b) 4.7e-110)
     t_5
     (if (<= (fabs b) 4.3e+201)
       (*
        (*
         (/
          (/
           (/
            (sqrt
             (* (/ (* (pow a 6.0) (pow (fabs b) 4.0)) (pow y-scale 2.0)) 8.0))
            t_1)
           t_4)
          t_2)
         (* y-scale x-scale))
        (* y-scale x-scale))
       t_5))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(b) / (x_45_scale * x_45_scale);
	double t_1 = fabs((x_45_scale * y_45_scale));
	double t_2 = fabs(b) * a;
	double t_3 = a / (y_45_scale * y_45_scale);
	double t_4 = 4.0 * t_2;
	double t_5 = ((sqrt(((fma(fabs(b), t_0, fma(a, t_3, fabs(((a * t_3) - (fabs(b) * t_0))))) * pow(t_2, 4.0)) * 8.0)) / (t_1 * ((t_4 * a) * fabs(b)))) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
	double tmp;
	if (fabs(b) <= 4.7e-110) {
		tmp = t_5;
	} else if (fabs(b) <= 4.3e+201) {
		tmp = ((((sqrt((((pow(a, 6.0) * pow(fabs(b), 4.0)) / pow(y_45_scale, 2.0)) * 8.0)) / t_1) / t_4) / t_2) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
	} else {
		tmp = t_5;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(b) / Float64(x_45_scale * x_45_scale))
	t_1 = abs(Float64(x_45_scale * y_45_scale))
	t_2 = Float64(abs(b) * a)
	t_3 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_4 = Float64(4.0 * t_2)
	t_5 = Float64(Float64(Float64(sqrt(Float64(Float64(fma(abs(b), t_0, fma(a, t_3, abs(Float64(Float64(a * t_3) - Float64(abs(b) * t_0))))) * (t_2 ^ 4.0)) * 8.0)) / Float64(t_1 * Float64(Float64(t_4 * a) * abs(b)))) * Float64(y_45_scale * x_45_scale)) * Float64(y_45_scale * x_45_scale))
	tmp = 0.0
	if (abs(b) <= 4.7e-110)
		tmp = t_5;
	elseif (abs(b) <= 4.3e+201)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64((a ^ 6.0) * (abs(b) ^ 4.0)) / (y_45_scale ^ 2.0)) * 8.0)) / t_1) / t_4) / t_2) * Float64(y_45_scale * x_45_scale)) * Float64(y_45_scale * x_45_scale));
	else
		tmp = t_5;
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x$45$scale * y$45$scale), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[b], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$3 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(4.0 * t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[Sqrt[N[(N[(N[(N[Abs[b], $MachinePrecision] * t$95$0 + N[(a * t$95$3 + N[Abs[N[(N[(a * t$95$3), $MachinePrecision] - N[(N[Abs[b], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[t$95$2, 4.0], $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 * N[(N[(t$95$4 * a), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 4.7e-110], t$95$5, If[LessEqual[N[Abs[b], $MachinePrecision], 4.3e+201], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[(N[Power[a, 6.0], $MachinePrecision] * N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$4), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision], t$95$5]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.69999999999999992e-110 or 4.2999999999999999e201 < b

   1. Initial program 2.9%

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

   3. Applied rewrites 6.1%

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

   4. Taylor expanded in angle around 0

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

   6. Applied rewrites 2.5%

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

   8. Applied rewrites 7.0%

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

   if 4.69999999999999992e-110 < b < 4.2999999999999999e201

   1. Initial program 2.9%

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

   3. Applied rewrites 6.1%

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

   4. Taylor expanded in angle around 0

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

   6. Applied rewrites 2.5%

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

   8. Applied rewrites 16.1%

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

   9. Taylor expanded in a around inf

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

       1. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       5. lower-pow.f64 4.9

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   11. Applied rewrites 4.9%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 7.5% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.9%

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

3. Applied rewrites 6.1%

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

4. Taylor expanded in angle around 0

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

6. Applied rewrites 2.5%

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

8. Applied rewrites 16.1%

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

10. Applied rewrites 10.1%

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

Alternative 7: 5.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot a\\ t_1 := 4 \cdot t\_0\\ t_2 := \left(t\_0 \cdot \left|b\right|\right) \cdot \left(-a\right)\\ t_3 := \left(\left(x-scale \cdot y-scale\right) \cdot x-scale\right) \cdot y-scale\\ \mathbf{if}\;\left|b\right| \leq 8.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{-\sqrt{\frac{\mathsf{fma}\left(a, a, \sqrt{{a}^{4}}\right)}{y-scale \cdot y-scale} \cdot \left(\left(\left(4 \cdot \frac{t\_2}{t\_3}\right) \cdot 2\right) \cdot t\_2\right)}}{t\_1 \cdot \left(\left(-a\right) \cdot \left|b\right|\right)} \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {\left(\left|b\right|\right)}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{t\_1}}{t\_0} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (fabs b) a))
        (t_1 (* 4.0 t_0))
        (t_2 (* (* t_0 (fabs b)) (- a)))
        (t_3 (* (* (* x-scale y-scale) x-scale) y-scale)))
   (if (<= (fabs b) 8.5e-112)
     (*
      (/
       (-
        (sqrt
         (*
          (/ (fma a a (sqrt (pow a 4.0))) (* y-scale y-scale))
          (* (* (* 4.0 (/ t_2 t_3)) 2.0) t_2))))
       (* t_1 (* (- a) (fabs b))))
      t_3)
     (*
      (*
       (/
        (/
         (/
          (sqrt
           (* (/ (* (pow a 6.0) (pow (fabs b) 4.0)) (pow y-scale 2.0)) 8.0))
          (fabs (* x-scale y-scale)))
         t_1)
        t_0)
       (* y-scale x-scale))
      (* y-scale x-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(b) * a;
	double t_1 = 4.0 * t_0;
	double t_2 = (t_0 * fabs(b)) * -a;
	double t_3 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale;
	double tmp;
	if (fabs(b) <= 8.5e-112) {
		tmp = (-sqrt(((fma(a, a, sqrt(pow(a, 4.0))) / (y_45_scale * y_45_scale)) * (((4.0 * (t_2 / t_3)) * 2.0) * t_2))) / (t_1 * (-a * fabs(b)))) * t_3;
	} else {
		tmp = ((((sqrt((((pow(a, 6.0) * pow(fabs(b), 4.0)) / pow(y_45_scale, 2.0)) * 8.0)) / fabs((x_45_scale * y_45_scale))) / t_1) / t_0) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(b) * a)
	t_1 = Float64(4.0 * t_0)
	t_2 = Float64(Float64(t_0 * abs(b)) * Float64(-a))
	t_3 = Float64(Float64(Float64(x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)
	tmp = 0.0
	if (abs(b) <= 8.5e-112)
		tmp = Float64(Float64(Float64(-sqrt(Float64(Float64(fma(a, a, sqrt((a ^ 4.0))) / Float64(y_45_scale * y_45_scale)) * Float64(Float64(Float64(4.0 * Float64(t_2 / t_3)) * 2.0) * t_2)))) / Float64(t_1 * Float64(Float64(-a) * abs(b)))) * t_3);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64((a ^ 6.0) * (abs(b) ^ 4.0)) / (y_45_scale ^ 2.0)) * 8.0)) / abs(Float64(x_45_scale * y_45_scale))) / t_1) / t_0) * Float64(y_45_scale * x_45_scale)) * Float64(y_45_scale * x_45_scale));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$1 = N[(4.0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[Abs[b], $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 8.5e-112], N[(N[((-N[Sqrt[N[(N[(N[(a * a + N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(4.0 * N[(t$95$2 / t$95$3), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(t$95$1 * N[((-a) * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[(N[Power[a, 6.0], $MachinePrecision] * N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(x$45$scale * y$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 8.49999999999999992e-112

   1. Initial program 2.9%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 4.2%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 2.6

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

   7. Applied rewrites 2.6%

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

   9. Applied rewrites 2.1%

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

   if 8.49999999999999992e-112 < b

   1. Initial program 2.9%

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

   3. Applied rewrites 6.1%

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

   4. Taylor expanded in angle around 0

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

   6. Applied rewrites 2.5%

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

   8. Applied rewrites 16.1%

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

   9. Taylor expanded in a around inf

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

       1. lower-/.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

       5. lower-pow.f64 4.9

          \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   11. Applied rewrites 4.9%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 4.9% accurate, 8.0× speedup

Math

\[\left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  (*
   (/
    (/
     (/
      (sqrt (* (/ (* (pow a 6.0) (pow b 4.0)) (pow y-scale 2.0)) 8.0))
      (fabs (* x-scale y-scale)))
     (* 4.0 (* b a)))
    (* b a))
   (* y-scale x-scale))
  (* y-scale x-scale)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((((sqrt((((pow(a, 6.0) * pow(b, 4.0)) / pow(y_45_scale, 2.0)) * 8.0)) / fabs((x_45_scale * y_45_scale))) / (4.0 * (b * a))) / (b * a)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_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 = ((((sqrt(((((a ** 6.0d0) * (b ** 4.0d0)) / (y_45scale ** 2.0d0)) * 8.0d0)) / abs((x_45scale * y_45scale))) / (4.0d0 * (b * a))) / (b * a)) * (y_45scale * x_45scale)) * (y_45scale * x_45scale)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((((Math.sqrt((((Math.pow(a, 6.0) * Math.pow(b, 4.0)) / Math.pow(y_45_scale, 2.0)) * 8.0)) / Math.abs((x_45_scale * y_45_scale))) / (4.0 * (b * a))) / (b * a)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return ((((math.sqrt((((math.pow(a, 6.0) * math.pow(b, 4.0)) / math.pow(y_45_scale, 2.0)) * 8.0)) / math.fabs((x_45_scale * y_45_scale))) / (4.0 * (b * a))) / (b * a)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64((a ^ 6.0) * (b ^ 4.0)) / (y_45_scale ^ 2.0)) * 8.0)) / abs(Float64(x_45_scale * y_45_scale))) / Float64(4.0 * Float64(b * a))) / Float64(b * a)) * Float64(y_45_scale * x_45_scale)) * Float64(y_45_scale * x_45_scale))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = ((((sqrt(((((a ^ 6.0) * (b ^ 4.0)) / (y_45_scale ^ 2.0)) * 8.0)) / abs((x_45_scale * y_45_scale))) / (4.0 * (b * a))) / (b * a)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[(N[Power[a, 6.0], $MachinePrecision] * N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(x$45$scale * y$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.9%

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

3. Applied rewrites 6.1%

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

4. Taylor expanded in angle around 0

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

6. Applied rewrites 2.5%

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

8. Applied rewrites 16.1%

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

9. Taylor expanded in a around inf

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

    1. lower-/.f64 N/A

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

    3. lower-pow.f64 N/A

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

    4. lower-pow.f64 N/A

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

    5. lower-pow.f64 4.9

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

11. Applied rewrites 4.9%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\frac{{a}^{6} \cdot {b}^{4}}{{y-scale}^{2}} \cdot 8}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
12. Add Preprocessing

Alternative 9: 2.8% accurate, 8.6× speedup

Math

\[\frac{\frac{-\sqrt{\frac{\frac{{b}^{6} \cdot {a}^{4}}{x-scale \cdot x-scale}}{\left(\left(x-scale \cdot y-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8}}{4 \cdot \left(b \cdot a\right)}}{\left(-a\right) \cdot b} \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right) \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  (/
   (/
    (-
     (sqrt
      (*
       (/
        (/ (* (pow b 6.0) (pow a 4.0)) (* x-scale x-scale))
        (* (* (* x-scale y-scale) x-scale) y-scale))
       8.0)))
    (* 4.0 (* b a)))
   (* (- a) b))
  (* (* (* y-scale x-scale) x-scale) y-scale)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((-sqrt(((((pow(b, 6.0) * pow(a, 4.0)) / (x_45_scale * x_45_scale)) / (((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)) * 8.0)) / (4.0 * (b * a))) / (-a * b)) * (((y_45_scale * x_45_scale) * x_45_scale) * 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 = ((-sqrt((((((b ** 6.0d0) * (a ** 4.0d0)) / (x_45scale * x_45scale)) / (((x_45scale * y_45scale) * x_45scale) * y_45scale)) * 8.0d0)) / (4.0d0 * (b * a))) / (-a * b)) * (((y_45scale * x_45scale) * x_45scale) * y_45scale)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((-Math.sqrt(((((Math.pow(b, 6.0) * Math.pow(a, 4.0)) / (x_45_scale * x_45_scale)) / (((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)) * 8.0)) / (4.0 * (b * a))) / (-a * b)) * (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return ((-math.sqrt(((((math.pow(b, 6.0) * math.pow(a, 4.0)) / (x_45_scale * x_45_scale)) / (((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)) * 8.0)) / (4.0 * (b * a))) / (-a * b)) * (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(Float64(Float64((b ^ 6.0) * (a ^ 4.0)) / Float64(x_45_scale * x_45_scale)) / Float64(Float64(Float64(x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)) * 8.0))) / Float64(4.0 * Float64(b * a))) / Float64(Float64(-a) * b)) * Float64(Float64(Float64(y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = ((-sqrt((((((b ^ 6.0) * (a ^ 4.0)) / (x_45_scale * x_45_scale)) / (((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)) * 8.0)) / (4.0 * (b * a))) / (-a * b)) * (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[((-N[Sqrt[N[(N[(N[(N[(N[Power[b, 6.0], $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision]) / N[(4.0 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-a) * b), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.9%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 0.4%

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

6. Applied rewrites 4.2%

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

7. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{-\sqrt{\frac{\frac{{a}^{4} \cdot {b}^{6}}{{x-scale}^{2}}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(\left(-a\right) \cdot b\right)} \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{-\sqrt{\frac{\frac{{a}^{4} \cdot {b}^{6}}{{x-scale}^{2}}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(\left(-a\right) \cdot b\right)} \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{-\sqrt{\frac{\frac{{a}^{4} \cdot {b}^{6}}{{x-scale}^{2}}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(\left(-a\right) \cdot b\right)} \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right) \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{-\sqrt{\frac{\frac{{a}^{4} \cdot {b}^{6}}{{x-scale}^{2}}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(\left(-a\right) \cdot b\right)} \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right) \]

   5. lower-pow.f64 0.7

      \[\leadsto \frac{-\sqrt{\frac{\frac{{a}^{4} \cdot {b}^{6}}{{x-scale}^{2}}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(\left(-a\right) \cdot b\right)} \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right) \]

9. Applied rewrites 0.7%

   \[\leadsto \frac{-\sqrt{\frac{\frac{{a}^{4} \cdot {b}^{6}}{{x-scale}^{2}}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(\left(-a\right) \cdot b\right)} \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right) \]

11. Applied rewrites 2.8%

    \[\leadsto \color{blue}{\frac{\frac{-\sqrt{\frac{\frac{{b}^{6} \cdot {a}^{4}}{x-scale \cdot x-scale}}{\left(\left(x-scale \cdot y-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8}}{4 \cdot \left(b \cdot a\right)}}{\left(-a\right) \cdot b}} \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right) \]
12. Add Preprocessing

Alternative 10: 0.7% accurate, 8.6× speedup

Math

\[\begin{array}{l} t_0 := \left(x-scale \cdot y-scale\right) \cdot x-scale\\ \left(\frac{-\sqrt{\frac{\frac{{b}^{6} \cdot {a}^{4}}{x-scale \cdot x-scale}}{t\_0 \cdot y-scale} \cdot 8}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(\left(-a\right) \cdot b\right)} \cdot t\_0\right) \cdot y-scale \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* x-scale y-scale) x-scale)))
   (*
    (*
     (/
      (-
       (sqrt
        (*
         (/
          (/ (* (pow b 6.0) (pow a 4.0)) (* x-scale x-scale))
          (* t_0 y-scale))
         8.0)))
      (* (* 4.0 (* b a)) (* (- a) b)))
     t_0)
    y-scale)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (x_45_scale * y_45_scale) * x_45_scale;
	return ((-sqrt(((((pow(b, 6.0) * pow(a, 4.0)) / (x_45_scale * x_45_scale)) / (t_0 * y_45_scale)) * 8.0)) / ((4.0 * (b * a)) * (-a * b))) * t_0) * 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
    real(8) :: t_0
    t_0 = (x_45scale * y_45scale) * x_45scale
    code = ((-sqrt((((((b ** 6.0d0) * (a ** 4.0d0)) / (x_45scale * x_45scale)) / (t_0 * y_45scale)) * 8.0d0)) / ((4.0d0 * (b * a)) * (-a * b))) * t_0) * y_45scale
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (x_45_scale * y_45_scale) * x_45_scale;
	return ((-Math.sqrt(((((Math.pow(b, 6.0) * Math.pow(a, 4.0)) / (x_45_scale * x_45_scale)) / (t_0 * y_45_scale)) * 8.0)) / ((4.0 * (b * a)) * (-a * b))) * t_0) * y_45_scale;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (x_45_scale * y_45_scale) * x_45_scale
	return ((-math.sqrt(((((math.pow(b, 6.0) * math.pow(a, 4.0)) / (x_45_scale * x_45_scale)) / (t_0 * y_45_scale)) * 8.0)) / ((4.0 * (b * a)) * (-a * b))) * t_0) * y_45_scale

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(x_45_scale * y_45_scale) * x_45_scale)
	return Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(Float64(Float64((b ^ 6.0) * (a ^ 4.0)) / Float64(x_45_scale * x_45_scale)) / Float64(t_0 * y_45_scale)) * 8.0))) / Float64(Float64(4.0 * Float64(b * a)) * Float64(Float64(-a) * b))) * t_0) * y_45_scale)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (x_45_scale * y_45_scale) * x_45_scale;
	tmp = ((-sqrt((((((b ^ 6.0) * (a ^ 4.0)) / (x_45_scale * x_45_scale)) / (t_0 * y_45_scale)) * 8.0)) / ((4.0 * (b * a)) * (-a * b))) * t_0) * y_45_scale;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]}, N[(N[(N[((-N[Sqrt[N[(N[(N[(N[(N[Power[b, 6.0], $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * y$45$scale), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision]) / N[(N[(4.0 * N[(b * a), $MachinePrecision]), $MachinePrecision] * N[((-a) * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * y$45$scale), $MachinePrecision]]

Derivation

1. Initial program 2.9%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 0.4%

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

6. Applied rewrites 4.2%

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

7. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{-\sqrt{\frac{\frac{{a}^{4} \cdot {b}^{6}}{{x-scale}^{2}}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(\left(-a\right) \cdot b\right)} \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{-\sqrt{\frac{\frac{{a}^{4} \cdot {b}^{6}}{{x-scale}^{2}}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(\left(-a\right) \cdot b\right)} \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{-\sqrt{\frac{\frac{{a}^{4} \cdot {b}^{6}}{{x-scale}^{2}}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(\left(-a\right) \cdot b\right)} \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right) \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{-\sqrt{\frac{\frac{{a}^{4} \cdot {b}^{6}}{{x-scale}^{2}}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(\left(-a\right) \cdot b\right)} \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right) \]

   5. lower-pow.f64 0.7

      \[\leadsto \frac{-\sqrt{\frac{\frac{{a}^{4} \cdot {b}^{6}}{{x-scale}^{2}}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(\left(-a\right) \cdot b\right)} \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right) \]

9. Applied rewrites 0.7%

   \[\leadsto \frac{-\sqrt{\frac{\frac{{a}^{4} \cdot {b}^{6}}{{x-scale}^{2}}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(\left(-a\right) \cdot b\right)} \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right) \]

11. Applied rewrites 0.7%

    \[\leadsto \color{blue}{\left(\frac{-\sqrt{\frac{\frac{{b}^{6} \cdot {a}^{4}}{x-scale \cdot x-scale}}{\left(\left(x-scale \cdot y-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(\left(-a\right) \cdot b\right)} \cdot \left(\left(x-scale \cdot y-scale\right) \cdot x-scale\right)\right) \cdot y-scale} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025171 
(FPCore (a b angle x-scale y-scale)
  :name "a from scale-rotated-ellipse"
  :precision binary64
  (/ (- (sqrt (* (* (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))) (* (* b a) (* b (- a)))) (+ (+ (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) (sqrt (+ (pow (- (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) 2.0) (pow (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale) 2.0))))))) (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))