b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 22.2%

Time: 39.8s

Alternatives: 11

Speedup: 8.7×

• could not determine a ground truth

  1. a = -8.157159620228829e+306
  2. b = -1.4268795491354986e-287
  3. angle = -2.6641037508599736e-124
  4. x-scale = 1.526352054153162e+298
  5. y-scale = 5.15113052474051e-295

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 0.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 22.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} t_0 := \left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\\ t_1 := \frac{4 \cdot t\_0}{{\left(\left|x-scale\right| \cdot y-scale\right)}^{2}}\\ \mathbf{if}\;\left|x-scale\right| \leq 5.2 \cdot 10^{+161}:\\ \;\;\;\;\frac{-0.25}{a} \cdot \frac{\left(b \cdot \left(\left|x-scale\right| \cdot \left|x-scale\right|\right)\right) \cdot \frac{{a}^{2} \cdot \sqrt{8 \cdot \left(0.5 - \left(\sqrt{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{\left|\left|x-scale\right|\right|}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_1\right) \cdot t\_0\right) \cdot 0}}{t\_1}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* b a) (* b (- a))))
        (t_1 (/ (* 4.0 t_0) (pow (* (fabs x-scale) y-scale) 2.0))))
   (if (<= (fabs x-scale) 5.2e+161)
     (*
      (/ -0.25 a)
      (/
       (*
        (* b (* (fabs x-scale) (fabs x-scale)))
        (/
         (*
          (pow a 2.0)
          (sqrt
           (*
            8.0
            (-
             0.5
             (+
              (sqrt (pow (sin (* 0.005555555555555556 (* angle PI))) 4.0))
              (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))))))
         (fabs (fabs x-scale))))
       a))
     (/ (- (sqrt (* (* (* 2.0 t_1) t_0) 0.0))) t_1))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * a) * (b * -a);
	double t_1 = (4.0 * t_0) / pow((fabs(x_45_scale) * y_45_scale), 2.0);
	double tmp;
	if (fabs(x_45_scale) <= 5.2e+161) {
		tmp = (-0.25 / a) * (((b * (fabs(x_45_scale) * fabs(x_45_scale))) * ((pow(a, 2.0) * sqrt((8.0 * (0.5 - (sqrt(pow(sin((0.005555555555555556 * (angle * ((double) M_PI)))), 4.0)) + (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI)))))))))) / fabs(fabs(x_45_scale)))) / a);
	} else {
		tmp = -sqrt((((2.0 * t_1) * t_0) * 0.0)) / t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * a) * (b * -a);
	double t_1 = (4.0 * t_0) / Math.pow((Math.abs(x_45_scale) * y_45_scale), 2.0);
	double tmp;
	if (Math.abs(x_45_scale) <= 5.2e+161) {
		tmp = (-0.25 / a) * (((b * (Math.abs(x_45_scale) * Math.abs(x_45_scale))) * ((Math.pow(a, 2.0) * Math.sqrt((8.0 * (0.5 - (Math.sqrt(Math.pow(Math.sin((0.005555555555555556 * (angle * Math.PI))), 4.0)) + (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI))))))))) / Math.abs(Math.abs(x_45_scale)))) / a);
	} else {
		tmp = -Math.sqrt((((2.0 * t_1) * t_0) * 0.0)) / t_1;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (b * a) * (b * -a)
	t_1 = (4.0 * t_0) / math.pow((math.fabs(x_45_scale) * y_45_scale), 2.0)
	tmp = 0
	if math.fabs(x_45_scale) <= 5.2e+161:
		tmp = (-0.25 / a) * (((b * (math.fabs(x_45_scale) * math.fabs(x_45_scale))) * ((math.pow(a, 2.0) * math.sqrt((8.0 * (0.5 - (math.sqrt(math.pow(math.sin((0.005555555555555556 * (angle * math.pi))), 4.0)) + (0.5 * math.cos((0.011111111111111112 * (angle * math.pi))))))))) / math.fabs(math.fabs(x_45_scale)))) / a)
	else:
		tmp = -math.sqrt((((2.0 * t_1) * t_0) * 0.0)) / t_1
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b * a) * Float64(b * Float64(-a)))
	t_1 = Float64(Float64(4.0 * t_0) / (Float64(abs(x_45_scale) * y_45_scale) ^ 2.0))
	tmp = 0.0
	if (abs(x_45_scale) <= 5.2e+161)
		tmp = Float64(Float64(-0.25 / a) * Float64(Float64(Float64(b * Float64(abs(x_45_scale) * abs(x_45_scale))) * Float64(Float64((a ^ 2.0) * sqrt(Float64(8.0 * Float64(0.5 - Float64(sqrt((sin(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 4.0)) + Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))))))) / abs(abs(x_45_scale)))) / a));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_1) * t_0) * 0.0))) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (b * a) * (b * -a);
	t_1 = (4.0 * t_0) / ((abs(x_45_scale) * y_45_scale) ^ 2.0);
	tmp = 0.0;
	if (abs(x_45_scale) <= 5.2e+161)
		tmp = (-0.25 / a) * (((b * (abs(x_45_scale) * abs(x_45_scale))) * (((a ^ 2.0) * sqrt((8.0 * (0.5 - (sqrt((sin((0.005555555555555556 * (angle * pi))) ^ 4.0)) + (0.5 * cos((0.011111111111111112 * (angle * pi))))))))) / abs(abs(x_45_scale)))) / a);
	else
		tmp = -sqrt((((2.0 * t_1) * t_0) * 0.0)) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * t$95$0), $MachinePrecision] / N[Power[N[(N[Abs[x$45$scale], $MachinePrecision] * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 5.2e+161], N[(N[(-0.25 / a), $MachinePrecision] * N[(N[(N[(b * N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(0.5 - N[(N[Sqrt[N[Power[N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[N[Abs[x$45$scale], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] * 0.0), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 5.1999999999999996e161

   1. Initial program 0.1%

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

   2. Taylor expanded in b around -inf

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 3.7%

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

   9. Applied rewrites 19.1%

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

   10. Taylor expanded in a around 0

       \[\leadsto \frac{-0.25}{a} \cdot \frac{\left(b \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{{a}^{2} \cdot \sqrt{8 \cdot \left(\frac{1}{2} - \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{\left|x-scale\right|}}{a} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{4}}{a} \cdot \frac{\left(b \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{{a}^{2} \cdot \sqrt{8 \cdot \left(\frac{1}{2} - \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}}{\left|x-scale\right|}}{a} \]

       2. lower-pow.f64 N/A

          \[\leadsto \frac{\frac{-1}{4}}{a} \cdot \frac{\left(b \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{{a}^{2} \cdot \sqrt{8 \cdot \left(\frac{1}{2} - \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}}{\left|x-scale\right|}}{a} \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \frac{\frac{-1}{4}}{a} \cdot \frac{\left(b \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{{a}^{2} \cdot \sqrt{8 \cdot \left(\frac{1}{2} - \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}}{\left|x-scale\right|}}{a} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{-1}{4}}{a} \cdot \frac{\left(b \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{{a}^{2} \cdot \sqrt{8 \cdot \left(\frac{1}{2} - \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}}{\left|x-scale\right|}}{a} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{\frac{-1}{4}}{a} \cdot \frac{\left(b \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{{a}^{2} \cdot \sqrt{8 \cdot \left(\frac{1}{2} - \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}}{\left|x-scale\right|}}{a} \]

       6. lower-+.f64 N/A

          \[\leadsto \frac{\frac{-1}{4}}{a} \cdot \frac{\left(b \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{{a}^{2} \cdot \sqrt{8 \cdot \left(\frac{1}{2} - \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}}{\left|x-scale\right|}}{a} \]

   12. Applied rewrites 20.6%

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

   if 5.1999999999999996e161 < x-scale

   1. Initial program 0.1%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 0.2%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 0.1%

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

      16. lift-pow.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 0.1%

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

      19. lift-pow.f64 N/A

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

      20. pow2 N/A

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

      21. lift-*.f64 0.1%

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

      22. lift-*.f64 N/A

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

   6. Applied rewrites 0.1%

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

   7. Taylor expanded in y-scale around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 2.0%

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

   9. Applied rewrites 2.0%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 4.9%

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

Alternative 2: 21.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} t_0 := 8 \cdot \left(0.5 - \mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5, \sqrt{{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{4}}\right)\right)\\ t_1 := \frac{-0.25}{\left|a\right|}\\ \mathbf{if}\;\left|a\right| \leq 4.4 \cdot 10^{-100}:\\ \;\;\;\;t\_1 \cdot \frac{b \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \frac{\sqrt{t\_0 \cdot {\left(\left|a\right|\right)}^{4}}}{\left|x-scale\right|}\right)}{\left|a\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\left(\left(b \cdot x-scale\right) \cdot x-scale\right) \cdot \left(\frac{\sqrt{t\_0}}{\left|x-scale\right|} \cdot \left(\left|a\right| \cdot \left|a\right|\right)\right)}{\left|a\right|}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (*
          8.0
          (-
           0.5
           (fma
            (cos (* (* PI angle) 0.011111111111111112))
            0.5
            (sqrt (pow (sin (* PI (* angle 0.005555555555555556))) 4.0))))))
        (t_1 (/ -0.25 (fabs a))))
   (if (<= (fabs a) 4.4e-100)
     (*
      t_1
      (/
       (*
        b
        (*
         (* x-scale x-scale)
         (/ (sqrt (* t_0 (pow (fabs a) 4.0))) (fabs x-scale))))
       (fabs a)))
     (*
      t_1
      (/
       (*
        (* (* b x-scale) x-scale)
        (* (/ (sqrt t_0) (fabs x-scale)) (* (fabs a) (fabs a))))
       (fabs a))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 8.0 * (0.5 - fma(cos(((((double) M_PI) * angle) * 0.011111111111111112)), 0.5, sqrt(pow(sin((((double) M_PI) * (angle * 0.005555555555555556))), 4.0))));
	double t_1 = -0.25 / fabs(a);
	double tmp;
	if (fabs(a) <= 4.4e-100) {
		tmp = t_1 * ((b * ((x_45_scale * x_45_scale) * (sqrt((t_0 * pow(fabs(a), 4.0))) / fabs(x_45_scale)))) / fabs(a));
	} else {
		tmp = t_1 * ((((b * x_45_scale) * x_45_scale) * ((sqrt(t_0) / fabs(x_45_scale)) * (fabs(a) * fabs(a)))) / fabs(a));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(8.0 * Float64(0.5 - fma(cos(Float64(Float64(pi * angle) * 0.011111111111111112)), 0.5, sqrt((sin(Float64(pi * Float64(angle * 0.005555555555555556))) ^ 4.0)))))
	t_1 = Float64(-0.25 / abs(a))
	tmp = 0.0
	if (abs(a) <= 4.4e-100)
		tmp = Float64(t_1 * Float64(Float64(b * Float64(Float64(x_45_scale * x_45_scale) * Float64(sqrt(Float64(t_0 * (abs(a) ^ 4.0))) / abs(x_45_scale)))) / abs(a)));
	else
		tmp = Float64(t_1 * Float64(Float64(Float64(Float64(b * x_45_scale) * x_45_scale) * Float64(Float64(sqrt(t_0) / abs(x_45_scale)) * Float64(abs(a) * abs(a)))) / abs(a)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(8.0 * N[(0.5 - N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * 0.5 + N[Sqrt[N[Power[N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.25 / N[Abs[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 4.4e-100], N[(t$95$1 * N[(N[(b * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(N[Sqrt[N[(t$95$0 * N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(N[(N[(b * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * N[(N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 4.3999999999999998e-100

   1. Initial program 0.1%

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

   2. Taylor expanded in b around -inf

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 3.7%

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

   9. Applied rewrites 19.1%

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

   11. Applied rewrites 20.1%

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

   if 4.3999999999999998e-100 < a

   1. Initial program 0.1%

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

   2. Taylor expanded in b around -inf

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

   10. Applied rewrites 4.5%

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

   12. Applied rewrites 19.9%

       \[\leadsto \frac{-0.25}{a} \cdot \color{blue}{\frac{\left(\left(b \cdot x-scale\right) \cdot x-scale\right) \cdot \left(\frac{\sqrt{8 \cdot \left(0.5 - \mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5, \sqrt{{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{4}}\right)\right)}}{\left|x-scale\right|} \cdot \left(a \cdot a\right)\right)}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 21.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|a\right| \leq 7 \cdot 10^{+76}:\\ \;\;\;\;\frac{-0.25}{\left|a\right|} \cdot \frac{\left(b \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{angle \cdot \sqrt{-8 \cdot \left({\left(\left|a\right|\right)}^{4} \cdot \left(\sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}} + -3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2}\right)\right)}}{\left|x-scale\right|}}{\left|a\right|}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\frac{\left(\frac{\sqrt{8 \cdot \left(0.5 - \mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5, \sqrt{{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{4}}\right)\right)}}{\left|x-scale\right|} \cdot \left(\left|a\right| \cdot \left|a\right|\right)\right) \cdot \left(x-scale \cdot x-scale\right)}{\left|a\right|} \cdot \frac{b}{\left|a\right|}\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= (fabs a) 7e+76)
   (*
    (/ -0.25 (fabs a))
    (/
     (*
      (* b (* x-scale x-scale))
      (/
       (*
        angle
        (sqrt
         (*
          -8.0
          (*
           (pow (fabs a) 4.0)
           (+
            (sqrt (* 9.525986892242036e-10 (pow PI 4.0)))
            (* -3.08641975308642e-5 (pow PI 2.0)))))))
       (fabs x-scale)))
     (fabs a)))
   (*
    -0.25
    (*
     (/
      (*
       (*
        (/
         (sqrt
          (*
           8.0
           (-
            0.5
            (fma
             (cos (* (* PI angle) 0.011111111111111112))
             0.5
             (sqrt (pow (sin (* PI (* angle 0.005555555555555556))) 4.0))))))
         (fabs x-scale))
        (* (fabs a) (fabs a)))
       (* x-scale x-scale))
      (fabs a))
     (/ b (fabs a))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (fabs(a) <= 7e+76) {
		tmp = (-0.25 / fabs(a)) * (((b * (x_45_scale * x_45_scale)) * ((angle * sqrt((-8.0 * (pow(fabs(a), 4.0) * (sqrt((9.525986892242036e-10 * pow(((double) M_PI), 4.0))) + (-3.08641975308642e-5 * pow(((double) M_PI), 2.0))))))) / fabs(x_45_scale))) / fabs(a));
	} else {
		tmp = -0.25 * (((((sqrt((8.0 * (0.5 - fma(cos(((((double) M_PI) * angle) * 0.011111111111111112)), 0.5, sqrt(pow(sin((((double) M_PI) * (angle * 0.005555555555555556))), 4.0)))))) / fabs(x_45_scale)) * (fabs(a) * fabs(a))) * (x_45_scale * x_45_scale)) / fabs(a)) * (b / fabs(a)));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (abs(a) <= 7e+76)
		tmp = Float64(Float64(-0.25 / abs(a)) * Float64(Float64(Float64(b * Float64(x_45_scale * x_45_scale)) * Float64(Float64(angle * sqrt(Float64(-8.0 * Float64((abs(a) ^ 4.0) * Float64(sqrt(Float64(9.525986892242036e-10 * (pi ^ 4.0))) + Float64(-3.08641975308642e-5 * (pi ^ 2.0))))))) / abs(x_45_scale))) / abs(a)));
	else
		tmp = Float64(-0.25 * Float64(Float64(Float64(Float64(Float64(sqrt(Float64(8.0 * Float64(0.5 - fma(cos(Float64(Float64(pi * angle) * 0.011111111111111112)), 0.5, sqrt((sin(Float64(pi * Float64(angle * 0.005555555555555556))) ^ 4.0)))))) / abs(x_45_scale)) * Float64(abs(a) * abs(a))) * Float64(x_45_scale * x_45_scale)) / abs(a)) * Float64(b / abs(a))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[N[Abs[a], $MachinePrecision], 7e+76], N[(N[(-0.25 / N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(b * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(angle * N[Sqrt[N[(-8.0 * N[(N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision] * N[(N[Sqrt[N[(9.525986892242036e-10 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-3.08641975308642e-5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[(N[(N[(N[(N[Sqrt[N[(8.0 * N[(0.5 - N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * 0.5 + N[Sqrt[N[Power[N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] / N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(b / N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7e76

   1. Initial program 0.1%

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

   2. Taylor expanded in b around -inf

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 3.7%

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

   9. Applied rewrites 19.1%

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

   10. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-+.f64 N/A

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

   12. Applied rewrites 20.4%

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

   if 7e76 < a

   1. Initial program 0.1%

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

   2. Taylor expanded in b around -inf

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

   10. Applied rewrites 4.5%

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

   12. Applied rewrites 14.6%

       \[\leadsto -0.25 \cdot \left(\frac{\left(\frac{\sqrt{8 \cdot \left(0.5 - \mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5, \sqrt{{\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{4}}\right)\right)}}{\left|x-scale\right|} \cdot \left(a \cdot a\right)\right) \cdot \left(x-scale \cdot x-scale\right)}{a} \cdot \color{blue}{\frac{b}{a}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 21.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}}\\ t_1 := {\left(\left|a\right|\right)}^{2}\\ \mathbf{if}\;\left|a\right| \leq 8.6 \cdot 10^{+73}:\\ \;\;\;\;\frac{-0.25}{\left|a\right|} \cdot \frac{\left(b \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{angle \cdot \sqrt{-8 \cdot \left({\left(\left|a\right|\right)}^{4} \cdot \left(t\_0 + -3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2}\right)\right)}}{\left|x-scale\right|}}{\left|a\right|}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \left(t\_1 \cdot \left(angle \cdot \sqrt{8 \cdot \frac{3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - t\_0}{{x-scale}^{2}}}\right)\right)\right)}{t\_1}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sqrt (* 9.525986892242036e-10 (pow PI 4.0))))
        (t_1 (pow (fabs a) 2.0)))
   (if (<= (fabs a) 8.6e+73)
     (*
      (/ -0.25 (fabs a))
      (/
       (*
        (* b (* x-scale x-scale))
        (/
         (*
          angle
          (sqrt
           (*
            -8.0
            (*
             (pow (fabs a) 4.0)
             (+ t_0 (* -3.08641975308642e-5 (pow PI 2.0)))))))
         (fabs x-scale)))
       (fabs a)))
     (*
      -0.25
      (/
       (*
        b
        (*
         (pow x-scale 2.0)
         (*
          t_1
          (*
           angle
           (sqrt
            (*
             8.0
             (/
              (- (* 3.08641975308642e-5 (pow PI 2.0)) t_0)
              (pow x-scale 2.0))))))))
       t_1)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = sqrt((9.525986892242036e-10 * pow(((double) M_PI), 4.0)));
	double t_1 = pow(fabs(a), 2.0);
	double tmp;
	if (fabs(a) <= 8.6e+73) {
		tmp = (-0.25 / fabs(a)) * (((b * (x_45_scale * x_45_scale)) * ((angle * sqrt((-8.0 * (pow(fabs(a), 4.0) * (t_0 + (-3.08641975308642e-5 * pow(((double) M_PI), 2.0))))))) / fabs(x_45_scale))) / fabs(a));
	} else {
		tmp = -0.25 * ((b * (pow(x_45_scale, 2.0) * (t_1 * (angle * sqrt((8.0 * (((3.08641975308642e-5 * pow(((double) M_PI), 2.0)) - t_0) / pow(x_45_scale, 2.0)))))))) / t_1);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.sqrt((9.525986892242036e-10 * Math.pow(Math.PI, 4.0)));
	double t_1 = Math.pow(Math.abs(a), 2.0);
	double tmp;
	if (Math.abs(a) <= 8.6e+73) {
		tmp = (-0.25 / Math.abs(a)) * (((b * (x_45_scale * x_45_scale)) * ((angle * Math.sqrt((-8.0 * (Math.pow(Math.abs(a), 4.0) * (t_0 + (-3.08641975308642e-5 * Math.pow(Math.PI, 2.0))))))) / Math.abs(x_45_scale))) / Math.abs(a));
	} else {
		tmp = -0.25 * ((b * (Math.pow(x_45_scale, 2.0) * (t_1 * (angle * Math.sqrt((8.0 * (((3.08641975308642e-5 * Math.pow(Math.PI, 2.0)) - t_0) / Math.pow(x_45_scale, 2.0)))))))) / t_1);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.sqrt((9.525986892242036e-10 * math.pow(math.pi, 4.0)))
	t_1 = math.pow(math.fabs(a), 2.0)
	tmp = 0
	if math.fabs(a) <= 8.6e+73:
		tmp = (-0.25 / math.fabs(a)) * (((b * (x_45_scale * x_45_scale)) * ((angle * math.sqrt((-8.0 * (math.pow(math.fabs(a), 4.0) * (t_0 + (-3.08641975308642e-5 * math.pow(math.pi, 2.0))))))) / math.fabs(x_45_scale))) / math.fabs(a))
	else:
		tmp = -0.25 * ((b * (math.pow(x_45_scale, 2.0) * (t_1 * (angle * math.sqrt((8.0 * (((3.08641975308642e-5 * math.pow(math.pi, 2.0)) - t_0) / math.pow(x_45_scale, 2.0)))))))) / t_1)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = sqrt(Float64(9.525986892242036e-10 * (pi ^ 4.0)))
	t_1 = abs(a) ^ 2.0
	tmp = 0.0
	if (abs(a) <= 8.6e+73)
		tmp = Float64(Float64(-0.25 / abs(a)) * Float64(Float64(Float64(b * Float64(x_45_scale * x_45_scale)) * Float64(Float64(angle * sqrt(Float64(-8.0 * Float64((abs(a) ^ 4.0) * Float64(t_0 + Float64(-3.08641975308642e-5 * (pi ^ 2.0))))))) / abs(x_45_scale))) / abs(a)));
	else
		tmp = Float64(-0.25 * Float64(Float64(b * Float64((x_45_scale ^ 2.0) * Float64(t_1 * Float64(angle * sqrt(Float64(8.0 * Float64(Float64(Float64(3.08641975308642e-5 * (pi ^ 2.0)) - t_0) / (x_45_scale ^ 2.0)))))))) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = sqrt((9.525986892242036e-10 * (pi ^ 4.0)));
	t_1 = abs(a) ^ 2.0;
	tmp = 0.0;
	if (abs(a) <= 8.6e+73)
		tmp = (-0.25 / abs(a)) * (((b * (x_45_scale * x_45_scale)) * ((angle * sqrt((-8.0 * ((abs(a) ^ 4.0) * (t_0 + (-3.08641975308642e-5 * (pi ^ 2.0))))))) / abs(x_45_scale))) / abs(a));
	else
		tmp = -0.25 * ((b * ((x_45_scale ^ 2.0) * (t_1 * (angle * sqrt((8.0 * (((3.08641975308642e-5 * (pi ^ 2.0)) - t_0) / (x_45_scale ^ 2.0)))))))) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Sqrt[N[(9.525986892242036e-10 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 8.6e+73], N[(N[(-0.25 / N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(b * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(angle * N[Sqrt[N[(-8.0 * N[(N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision] * N[(t$95$0 + N[(-3.08641975308642e-5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[(b * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[(t$95$1 * N[(angle * N[Sqrt[N[(8.0 * N[(N[(N[(3.08641975308642e-5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 8.6000000000000003e73

   1. Initial program 0.1%

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

   2. Taylor expanded in b around -inf

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 3.7%

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

   9. Applied rewrites 19.1%

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

   10. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-+.f64 N/A

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

   12. Applied rewrites 20.4%

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

   if 8.6000000000000003e73 < a

   1. Initial program 0.1%

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

   2. Taylor expanded in b around -inf

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

   10. Applied rewrites 4.5%

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

   11. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

   13. Applied rewrites 4.8%

       \[\leadsto -0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \left({a}^{2} \cdot \left(angle \cdot \sqrt{8 \cdot \frac{3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2} - \sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}}}{{x-scale}^{2}}}\right)\right)\right)}{{a}^{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 21.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} t_0 := \left(b \cdot \left|a\right|\right) \cdot \left(b \cdot \left(-\left|a\right|\right)\right)\\ t_1 := \frac{4 \cdot t\_0}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \mathbf{if}\;\left|a\right| \leq 7 \cdot 10^{+76}:\\ \;\;\;\;\frac{-0.25}{\left|a\right|} \cdot \frac{\left(b \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{angle \cdot \sqrt{-8 \cdot \left({\left(\left|a\right|\right)}^{4} \cdot \left(\sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}} + -3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2}\right)\right)}}{\left|x-scale\right|}}{\left|a\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_1\right) \cdot t\_0\right) \cdot 0}}{t\_1}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* b (fabs a)) (* b (- (fabs a)))))
        (t_1 (/ (* 4.0 t_0) (pow (* x-scale y-scale) 2.0))))
   (if (<= (fabs a) 7e+76)
     (*
      (/ -0.25 (fabs a))
      (/
       (*
        (* b (* x-scale x-scale))
        (/
         (*
          angle
          (sqrt
           (*
            -8.0
            (*
             (pow (fabs a) 4.0)
             (+
              (sqrt (* 9.525986892242036e-10 (pow PI 4.0)))
              (* -3.08641975308642e-5 (pow PI 2.0)))))))
         (fabs x-scale)))
       (fabs a)))
     (/ (- (sqrt (* (* (* 2.0 t_1) t_0) 0.0))) t_1))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * fabs(a)) * (b * -fabs(a));
	double t_1 = (4.0 * t_0) / pow((x_45_scale * y_45_scale), 2.0);
	double tmp;
	if (fabs(a) <= 7e+76) {
		tmp = (-0.25 / fabs(a)) * (((b * (x_45_scale * x_45_scale)) * ((angle * sqrt((-8.0 * (pow(fabs(a), 4.0) * (sqrt((9.525986892242036e-10 * pow(((double) M_PI), 4.0))) + (-3.08641975308642e-5 * pow(((double) M_PI), 2.0))))))) / fabs(x_45_scale))) / fabs(a));
	} else {
		tmp = -sqrt((((2.0 * t_1) * t_0) * 0.0)) / t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * Math.abs(a)) * (b * -Math.abs(a));
	double t_1 = (4.0 * t_0) / Math.pow((x_45_scale * y_45_scale), 2.0);
	double tmp;
	if (Math.abs(a) <= 7e+76) {
		tmp = (-0.25 / Math.abs(a)) * (((b * (x_45_scale * x_45_scale)) * ((angle * Math.sqrt((-8.0 * (Math.pow(Math.abs(a), 4.0) * (Math.sqrt((9.525986892242036e-10 * Math.pow(Math.PI, 4.0))) + (-3.08641975308642e-5 * Math.pow(Math.PI, 2.0))))))) / Math.abs(x_45_scale))) / Math.abs(a));
	} else {
		tmp = -Math.sqrt((((2.0 * t_1) * t_0) * 0.0)) / t_1;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (b * math.fabs(a)) * (b * -math.fabs(a))
	t_1 = (4.0 * t_0) / math.pow((x_45_scale * y_45_scale), 2.0)
	tmp = 0
	if math.fabs(a) <= 7e+76:
		tmp = (-0.25 / math.fabs(a)) * (((b * (x_45_scale * x_45_scale)) * ((angle * math.sqrt((-8.0 * (math.pow(math.fabs(a), 4.0) * (math.sqrt((9.525986892242036e-10 * math.pow(math.pi, 4.0))) + (-3.08641975308642e-5 * math.pow(math.pi, 2.0))))))) / math.fabs(x_45_scale))) / math.fabs(a))
	else:
		tmp = -math.sqrt((((2.0 * t_1) * t_0) * 0.0)) / t_1
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b * abs(a)) * Float64(b * Float64(-abs(a))))
	t_1 = Float64(Float64(4.0 * t_0) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	tmp = 0.0
	if (abs(a) <= 7e+76)
		tmp = Float64(Float64(-0.25 / abs(a)) * Float64(Float64(Float64(b * Float64(x_45_scale * x_45_scale)) * Float64(Float64(angle * sqrt(Float64(-8.0 * Float64((abs(a) ^ 4.0) * Float64(sqrt(Float64(9.525986892242036e-10 * (pi ^ 4.0))) + Float64(-3.08641975308642e-5 * (pi ^ 2.0))))))) / abs(x_45_scale))) / abs(a)));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_1) * t_0) * 0.0))) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (b * abs(a)) * (b * -abs(a));
	t_1 = (4.0 * t_0) / ((x_45_scale * y_45_scale) ^ 2.0);
	tmp = 0.0;
	if (abs(a) <= 7e+76)
		tmp = (-0.25 / abs(a)) * (((b * (x_45_scale * x_45_scale)) * ((angle * sqrt((-8.0 * ((abs(a) ^ 4.0) * (sqrt((9.525986892242036e-10 * (pi ^ 4.0))) + (-3.08641975308642e-5 * (pi ^ 2.0))))))) / abs(x_45_scale))) / abs(a));
	else
		tmp = -sqrt((((2.0 * t_1) * t_0) * 0.0)) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(b * (-N[Abs[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * t$95$0), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 7e+76], N[(N[(-0.25 / N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(b * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(angle * N[Sqrt[N[(-8.0 * N[(N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision] * N[(N[Sqrt[N[(9.525986892242036e-10 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-3.08641975308642e-5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] * 0.0), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 7e76

   1. Initial program 0.1%

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

   2. Taylor expanded in b around -inf

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 3.7%

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

   9. Applied rewrites 19.1%

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

   10. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-+.f64 N/A

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

   12. Applied rewrites 20.4%

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

   if 7e76 < a

   1. Initial program 0.1%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 0.2%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-sub N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 0.1%

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

      16. lift-pow.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 0.1%

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

      19. lift-pow.f64 N/A

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

      20. pow2 N/A

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

      21. lift-*.f64 0.1%

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

      22. lift-*.f64 N/A

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

   6. Applied rewrites 0.1%

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

   7. Taylor expanded in y-scale around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 2.0%

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

   9. Applied rewrites 2.0%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 4.9%

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

Alternative 6: 20.4% accurate, 8.2× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  (/ -0.25 a)
  (/
   (*
    (* b (* x-scale x-scale))
    (/
     (*
      angle
      (sqrt
       (*
        -8.0
        (*
         (pow a 4.0)
         (+
          (sqrt (* 9.525986892242036e-10 (pow PI 4.0)))
          (* -3.08641975308642e-5 (pow PI 2.0)))))))
     (fabs x-scale)))
   a)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (-0.25 / a) * (((b * (x_45_scale * x_45_scale)) * ((angle * sqrt((-8.0 * (pow(a, 4.0) * (sqrt((9.525986892242036e-10 * pow(((double) M_PI), 4.0))) + (-3.08641975308642e-5 * pow(((double) M_PI), 2.0))))))) / fabs(x_45_scale))) / a);
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (-0.25 / a) * (((b * (x_45_scale * x_45_scale)) * ((angle * Math.sqrt((-8.0 * (Math.pow(a, 4.0) * (Math.sqrt((9.525986892242036e-10 * Math.pow(Math.PI, 4.0))) + (-3.08641975308642e-5 * Math.pow(Math.PI, 2.0))))))) / Math.abs(x_45_scale))) / a);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (-0.25 / a) * (((b * (x_45_scale * x_45_scale)) * ((angle * math.sqrt((-8.0 * (math.pow(a, 4.0) * (math.sqrt((9.525986892242036e-10 * math.pow(math.pi, 4.0))) + (-3.08641975308642e-5 * math.pow(math.pi, 2.0))))))) / math.fabs(x_45_scale))) / a)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(-0.25 / a) * Float64(Float64(Float64(b * Float64(x_45_scale * x_45_scale)) * Float64(Float64(angle * sqrt(Float64(-8.0 * Float64((a ^ 4.0) * Float64(sqrt(Float64(9.525986892242036e-10 * (pi ^ 4.0))) + Float64(-3.08641975308642e-5 * (pi ^ 2.0))))))) / abs(x_45_scale))) / a))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = (-0.25 / a) * (((b * (x_45_scale * x_45_scale)) * ((angle * sqrt((-8.0 * ((a ^ 4.0) * (sqrt((9.525986892242036e-10 * (pi ^ 4.0))) + (-3.08641975308642e-5 * (pi ^ 2.0))))))) / abs(x_45_scale))) / a);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(-0.25 / a), $MachinePrecision] * N[(N[(N[(b * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(angle * N[Sqrt[N[(-8.0 * N[(N[Power[a, 4.0], $MachinePrecision] * N[(N[Sqrt[N[(9.525986892242036e-10 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-3.08641975308642e-5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1%

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

2. Taylor expanded in b around -inf

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

4. Applied rewrites 0.5%

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

5. Taylor expanded in y-scale around 0

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

   1. lower-sqrt.f64 N/A

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

7. Applied rewrites 3.7%

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

9. Applied rewrites 19.1%

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

10. Taylor expanded in angle around 0

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

    1. lower-*.f64 N/A

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

    2. lower-sqrt.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-*.f64 N/A

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

    5. lower-pow.f64 N/A

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

    6. lower-+.f64 N/A

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

12. Applied rewrites 20.4%

    \[\leadsto \frac{-0.25}{a} \cdot \frac{\left(b \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{angle \cdot \sqrt{-8 \cdot \left({a}^{4} \cdot \left(\sqrt{9.525986892242036 \cdot 10^{-10} \cdot {\pi}^{4}} + -3.08641975308642 \cdot 10^{-5} \cdot {\pi}^{2}\right)\right)}}{\left|x-scale\right|}}{a} \]
13. Add Preprocessing

Alternative 7: 3.1% accurate, 8.7× speedup

Math

\[\left(b \cdot \frac{\frac{\left(\sqrt{\left(\frac{\frac{1}{x-scale \cdot x-scale} - \sqrt{{x-scale}^{-4}}}{\left(y-scale \cdot \left(x-scale \cdot x-scale\right)\right) \cdot y-scale} \cdot {a}^{4}\right) \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)}{a}}{a}\right) \cdot -0.25 \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (b * ((((sqrt((((((1.0d0 / (x_45scale * x_45scale)) - sqrt((x_45scale ** (-4.0d0)))) / ((y_45scale * (x_45scale * x_45scale)) * y_45scale)) * (a ** 4.0d0)) * 8.0d0)) * (y_45scale * y_45scale)) * (x_45scale * x_45scale)) / a) / a)) * (-0.25d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 0.1%

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

2. Taylor expanded in b around -inf

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

4. Applied rewrites 0.5%

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

5. Taylor expanded in angle around 0

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

   1. lower--.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 0.7%

      \[\leadsto -0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\frac{1}{{x-scale}^{2}} - \sqrt{\frac{1}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{a}^{2}} \]

7. Applied rewrites 0.7%

   \[\leadsto -0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\frac{1}{{x-scale}^{2}} - \sqrt{\frac{1}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{a}^{2}} \]

9. Applied rewrites 1.2%

   \[\leadsto \color{blue}{\left(b \cdot \frac{\left(\sqrt{\left({a}^{4} \cdot \frac{\frac{1}{x-scale \cdot x-scale} - \sqrt{{x-scale}^{-4}}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}\right) \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)}{a \cdot a}\right) \cdot -0.25} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

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

11. Applied rewrites 3.1%

    \[\leadsto \left(b \cdot \frac{\frac{\left(\sqrt{\left(\frac{\frac{1}{x-scale \cdot x-scale} - \sqrt{{x-scale}^{-4}}}{\left(y-scale \cdot \left(x-scale \cdot x-scale\right)\right) \cdot y-scale} \cdot {a}^{4}\right) \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)}{a}}{a}\right) \cdot -0.25 \]
12. Add Preprocessing

Alternative 8: 2.9% accurate, 8.7× speedup

Math

\[\frac{-0.25}{a} \cdot \frac{\left(b \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\sqrt{\left({a}^{4} \cdot \frac{\frac{1}{x-scale \cdot x-scale} - \sqrt{{x-scale}^{-4}}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}\right) \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right)}{a} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = ((-0.25d0) / a) * (((b * (x_45scale * x_45scale)) * (sqrt((((a ** 4.0d0) * (((1.0d0 / (x_45scale * x_45scale)) - sqrt((x_45scale ** (-4.0d0)))) / (((y_45scale * x_45scale) * x_45scale) * y_45scale))) * 8.0d0)) * (y_45scale * y_45scale))) / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 0.1%

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

2. Taylor expanded in b around -inf

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

4. Applied rewrites 0.5%

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

5. Taylor expanded in angle around 0

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

   1. lower--.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 0.7%

      \[\leadsto -0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\frac{1}{{x-scale}^{2}} - \sqrt{\frac{1}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{a}^{2}} \]

7. Applied rewrites 0.7%

   \[\leadsto -0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\frac{1}{{x-scale}^{2}} - \sqrt{\frac{1}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{a}^{2}} \]

9. Applied rewrites 2.7%

   \[\leadsto \frac{-0.25}{a} \cdot \color{blue}{\frac{\left(b \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\sqrt{\left({a}^{4} \cdot \frac{\frac{1}{x-scale \cdot x-scale} - \sqrt{{x-scale}^{-4}}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}\right) \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right)}{a}} \]
10. Add Preprocessing

Alternative 9: 2.7% accurate, 8.7× speedup

Math

\[\left(b \cdot \frac{\left(\left(\sqrt{\left(\frac{\frac{1}{x-scale \cdot x-scale} - \sqrt{{x-scale}^{-4}}}{\left(y-scale \cdot \left(x-scale \cdot x-scale\right)\right) \cdot y-scale} \cdot {a}^{4}\right) \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot x-scale\right) \cdot x-scale}{a \cdot a}\right) \cdot -0.25 \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (b * ((((sqrt((((((1.0d0 / (x_45scale * x_45scale)) - sqrt((x_45scale ** (-4.0d0)))) / ((y_45scale * (x_45scale * x_45scale)) * y_45scale)) * (a ** 4.0d0)) * 8.0d0)) * (y_45scale * y_45scale)) * x_45scale) * x_45scale) / (a * a))) * (-0.25d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 0.1%

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

2. Taylor expanded in b around -inf

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

4. Applied rewrites 0.5%

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

5. Taylor expanded in angle around 0

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

   1. lower--.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 0.7%

      \[\leadsto -0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\frac{1}{{x-scale}^{2}} - \sqrt{\frac{1}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{a}^{2}} \]

7. Applied rewrites 0.7%

   \[\leadsto -0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\frac{1}{{x-scale}^{2}} - \sqrt{\frac{1}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{a}^{2}} \]

9. Applied rewrites 1.2%

   \[\leadsto \color{blue}{\left(b \cdot \frac{\left(\sqrt{\left({a}^{4} \cdot \frac{\frac{1}{x-scale \cdot x-scale} - \sqrt{{x-scale}^{-4}}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}\right) \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)}{a \cdot a}\right) \cdot -0.25} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

11. Applied rewrites 2.9%

    \[\leadsto \left(b \cdot \frac{\left(\left(\sqrt{\left(\frac{\frac{1}{x-scale \cdot x-scale} - \sqrt{{x-scale}^{-4}}}{\left(y-scale \cdot \left(x-scale \cdot x-scale\right)\right) \cdot y-scale} \cdot {a}^{4}\right) \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot x-scale\right) \cdot x-scale}{a \cdot a}\right) \cdot -0.25 \]
12. Add Preprocessing

Alternative 10: 1.4% accurate, 8.7× speedup

Math

\[\left(b \cdot \frac{\left(\left(\sqrt{\left(\frac{\frac{1}{x-scale \cdot x-scale} - \sqrt{{x-scale}^{-4}}}{\left(y-scale \cdot \left(x-scale \cdot x-scale\right)\right) \cdot y-scale} \cdot {a}^{4}\right) \cdot 8} \cdot y-scale\right) \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}{a \cdot a}\right) \cdot -0.25 \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (b * ((((sqrt((((((1.0d0 / (x_45scale * x_45scale)) - sqrt((x_45scale ** (-4.0d0)))) / ((y_45scale * (x_45scale * x_45scale)) * y_45scale)) * (a ** 4.0d0)) * 8.0d0)) * y_45scale) * y_45scale) * (x_45scale * x_45scale)) / (a * a))) * (-0.25d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 0.1%

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

2. Taylor expanded in b around -inf

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

4. Applied rewrites 0.5%

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

5. Taylor expanded in angle around 0

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

   1. lower--.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 0.7%

      \[\leadsto -0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\frac{1}{{x-scale}^{2}} - \sqrt{\frac{1}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{a}^{2}} \]

7. Applied rewrites 0.7%

   \[\leadsto -0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\frac{1}{{x-scale}^{2}} - \sqrt{\frac{1}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{a}^{2}} \]

9. Applied rewrites 1.2%

   \[\leadsto \color{blue}{\left(b \cdot \frac{\left(\sqrt{\left({a}^{4} \cdot \frac{\frac{1}{x-scale \cdot x-scale} - \sqrt{{x-scale}^{-4}}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}\right) \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)}{a \cdot a}\right) \cdot -0.25} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

11. Applied rewrites 1.4%

    \[\leadsto \left(b \cdot \frac{\left(\left(\sqrt{\left(\frac{\frac{1}{x-scale \cdot x-scale} - \sqrt{{x-scale}^{-4}}}{\left(y-scale \cdot \left(x-scale \cdot x-scale\right)\right) \cdot y-scale} \cdot {a}^{4}\right) \cdot 8} \cdot y-scale\right) \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}{a \cdot a}\right) \cdot -0.25 \]
12. Add Preprocessing

Alternative 11: 1.2% accurate, 8.7× speedup

Math

\[\left(b \cdot \frac{\left(\sqrt{\left({a}^{4} \cdot \frac{\frac{1}{x-scale \cdot x-scale} - \sqrt{{x-scale}^{-4}}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}\right) \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)}{a \cdot a}\right) \cdot -0.25 \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (b * (((sqrt((((a ** 4.0d0) * (((1.0d0 / (x_45scale * x_45scale)) - sqrt((x_45scale ** (-4.0d0)))) / (((y_45scale * x_45scale) * x_45scale) * y_45scale))) * 8.0d0)) * (y_45scale * y_45scale)) * (x_45scale * x_45scale)) / (a * a))) * (-0.25d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 0.1%

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

2. Taylor expanded in b around -inf

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

4. Applied rewrites 0.5%

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

5. Taylor expanded in angle around 0

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

   1. lower--.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 0.7%

      \[\leadsto -0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\frac{1}{{x-scale}^{2}} - \sqrt{\frac{1}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{a}^{2}} \]

7. Applied rewrites 0.7%

   \[\leadsto -0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\frac{1}{{x-scale}^{2}} - \sqrt{\frac{1}{{x-scale}^{4}}}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{a}^{2}} \]

9. Applied rewrites 1.2%

   \[\leadsto \color{blue}{\left(b \cdot \frac{\left(\sqrt{\left({a}^{4} \cdot \frac{\frac{1}{x-scale \cdot x-scale} - \sqrt{{x-scale}^{-4}}}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}\right) \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)}{a \cdot a}\right) \cdot -0.25} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025189 
(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))))