b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 2.9%

Time: 57.1s

Alternatives: 8

Speedup: 1.1×

• could not determine a ground truth

  1. a = 1.674272551015081e+171
  2. b = -5.617956271809195e-277
  3. angle = -2.5649879081328116e+137
  4. x-scale = -4.725931842922895e-210
  5. y-scale = -1.2610548541904089e+308

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 0.1% accurate, 1.0× speedup

Math

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

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: 2.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 0.1%

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

3. Applied rewrites 0.1%

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

5. Applied rewrites 0.5%

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

7. Applied rewrites 1.7%

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

9. Applied rewrites 2.0%

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

11. Applied rewrites 2.0%

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

13. Applied rewrites 2.9%

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

Alternative 2: 2.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 0.1%

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

3. Applied rewrites 0.1%

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

5. Applied rewrites 0.5%

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

7. Applied rewrites 0.4%

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

9. Applied rewrites 1.7%

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

11. Applied rewrites 2.0%

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

Alternative 3: 1.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 0.1%

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

3. Applied rewrites 0.1%

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

5. Applied rewrites 0.5%

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

7. Applied rewrites 0.4%

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

9. Applied rewrites 1.7%

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

Alternative 4: 0.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 0.1%

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

3. Applied rewrites 0.1%

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

5. Applied rewrites 0.5%

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

7. Applied rewrites 0.4%

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

9. Applied rewrites 0.8%

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

Alternative 5: 0.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 0.1%

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

3. Applied rewrites 0.1%

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

5. Applied rewrites 0.5%

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

7. Applied rewrites 0.4%

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

9. Applied rewrites 0.9%

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

Alternative 6: 0.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 0.1%

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

3. Applied rewrites 0.1%

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

5. Applied rewrites 0.5%

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

7. Applied rewrites 0.4%

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

9. Applied rewrites 0.4%

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

Alternative 7: 0.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 0.1%

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

3. Applied rewrites 0.1%

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

5. Applied rewrites 0.5%

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

7. Applied rewrites 0.4%

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

9. Applied rewrites 0.4%

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

Alternative 8: 0.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 0.1%

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

3. Applied rewrites 0.1%

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

5. Applied rewrites 0.5%

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

7. Applied rewrites 0.4%

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

9. Applied rewrites 0.4%

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

11. Applied rewrites 0.4%

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

13. Applied rewrites 0.4%

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

15. Applied rewrites 0.4%

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

17. Applied rewrites 0.4%

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

19. Applied rewrites 0.4%

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

21. Applied rewrites 0.4%

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

23. Applied rewrites 0.4%

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

25. Applied rewrites 0.4%

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

27. Applied rewrites 0.4%

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

29. Applied rewrites 0.4%

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

Reproduce

herbie shell --seed 2025121 
(FPCore (a b angle x-scale y-scale)
  :name "b from scale-rotated-ellipse"
  :precision binary64
  (/ (- (sqrt (* (* (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))) (* (* b a) (* b (- a)))) (- (+ (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) (sqrt (+ (pow (- (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) 2.0) (pow (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale) 2.0))))))) (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))