b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 30.5%

Time: 36.4s

Alternatives: 6

Speedup: 10.1×

• could not determine a ground truth

  1. a = -3.82982505763553e+214
  2. b = 3.054642616747308e-301
  3. angle = -3.315596310044344e+222
  4. x-scale = -6.168598832121298e+298
  5. y-scale = -2.0787219940878388e-287

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 0.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 30.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ 0.25 \cdot \left(b\_m \cdot \left(y-scale\_m \cdot \sqrt{8 \cdot \left({t\_0}^{2} - \sqrt{{t\_0}^{4}}\right)}\right)\right) \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale y-scale_m)
 :precision binary64
 (let* ((t_0 (cos (* 0.005555555555555556 (* angle PI)))))
   (*
    0.25
    (*
     b_m
     (* y-scale_m (sqrt (* 8.0 (- (pow t_0 2.0) (sqrt (pow t_0 4.0))))))))))

C

b_m = fabs(b);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	return 0.25 * (b_m * (y_45_scale_m * sqrt((8.0 * (pow(t_0, 2.0) - sqrt(pow(t_0, 4.0)))))));
}

Java

b_m = Math.abs(b);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = Math.cos((0.005555555555555556 * (angle * Math.PI)));
	return 0.25 * (b_m * (y_45_scale_m * Math.sqrt((8.0 * (Math.pow(t_0, 2.0) - Math.sqrt(Math.pow(t_0, 4.0)))))));
}

Python

b_m = math.fabs(b)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale, y_45_scale_m):
	t_0 = math.cos((0.005555555555555556 * (angle * math.pi)))
	return 0.25 * (b_m * (y_45_scale_m * math.sqrt((8.0 * (math.pow(t_0, 2.0) - math.sqrt(math.pow(t_0, 4.0)))))))

Julia

b_m = abs(b)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale, y_45_scale_m)
	t_0 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	return Float64(0.25 * Float64(b_m * Float64(y_45_scale_m * sqrt(Float64(8.0 * Float64((t_0 ^ 2.0) - sqrt((t_0 ^ 4.0))))))))
end

MATLAB

b_m = abs(b);
y-scale_m = abs(y_45_scale);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale_m)
	t_0 = cos((0.005555555555555556 * (angle * pi)));
	tmp = 0.25 * (b_m * (y_45_scale_m * sqrt((8.0 * ((t_0 ^ 2.0) - sqrt((t_0 ^ 4.0)))))));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale$95$m_] := Block[{t$95$0 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(0.25 * N[(b$95$m * N[(y$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[Sqrt[N[Power[t$95$0, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. 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 x-scale around 0

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

4. Applied rewrites 0.8%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 4.2%

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

8. Taylor expanded in b around 0

   \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left({y-scale}^{2} \cdot \color{blue}{\sqrt{8 \cdot \frac{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}}{{y-scale}^{2}}}}\right)\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

10. Applied rewrites 13.3%

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

11. Taylor expanded in y-scale around 0

    \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \sqrt{8 \cdot \left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}\right)}\right)\right) \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

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

    2. lower-sqrt.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower--.f64 N/A

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

13. Applied rewrites 30.5%

    \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{8 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} - \sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}\right)\right) \]
14. Add Preprocessing

Alternative 2: 15.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.02 \cdot 10^{-167}:\\ \;\;\;\;\frac{-0.25}{b\_m} \cdot \frac{\left(a \cdot \left(y-scale\_m \cdot y-scale\_m\right)\right) \cdot \sqrt{8 \cdot \left(\left(\left(b\_m \cdot b\_m\right) \cdot \left(b\_m \cdot b\_m\right)\right) \cdot \frac{0.5 - \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot 0.5}{y-scale\_m \cdot y-scale\_m}\right)}}{b\_m}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{a \cdot \left(y-scale\_m \cdot \sqrt{8 \cdot \left({b\_m}^{4} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}\right)}{{b\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale y-scale_m)
 :precision binary64
 (if (<= b_m 1.02e-167)
   (*
    (/ -0.25 b_m)
    (/
     (*
      (* a (* y-scale_m y-scale_m))
      (sqrt
       (*
        8.0
        (*
         (* (* b_m b_m) (* b_m b_m))
         (/
          (- 0.5 (* (cos (* 0.011111111111111112 (* PI angle))) 0.5))
          (* y-scale_m y-scale_m))))))
     b_m))
   (*
    0.25
    (/
     (*
      a
      (*
       y-scale_m
       (sqrt
        (*
         8.0
         (*
          (pow b_m 4.0)
          (- 0.5 (* 0.5 (cos (* 0.011111111111111112 (* angle PI))))))))))
     (pow b_m 2.0)))))

C

b_m = fabs(b);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if (b_m <= 1.02e-167) {
		tmp = (-0.25 / b_m) * (((a * (y_45_scale_m * y_45_scale_m)) * sqrt((8.0 * (((b_m * b_m) * (b_m * b_m)) * ((0.5 - (cos((0.011111111111111112 * (((double) M_PI) * angle))) * 0.5)) / (y_45_scale_m * y_45_scale_m)))))) / b_m);
	} else {
		tmp = 0.25 * ((a * (y_45_scale_m * sqrt((8.0 * (pow(b_m, 4.0) * (0.5 - (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))))))))))) / pow(b_m, 2.0));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if (b_m <= 1.02e-167) {
		tmp = (-0.25 / b_m) * (((a * (y_45_scale_m * y_45_scale_m)) * Math.sqrt((8.0 * (((b_m * b_m) * (b_m * b_m)) * ((0.5 - (Math.cos((0.011111111111111112 * (Math.PI * angle))) * 0.5)) / (y_45_scale_m * y_45_scale_m)))))) / b_m);
	} else {
		tmp = 0.25 * ((a * (y_45_scale_m * Math.sqrt((8.0 * (Math.pow(b_m, 4.0) * (0.5 - (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI)))))))))) / Math.pow(b_m, 2.0));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale, y_45_scale_m):
	tmp = 0
	if b_m <= 1.02e-167:
		tmp = (-0.25 / b_m) * (((a * (y_45_scale_m * y_45_scale_m)) * math.sqrt((8.0 * (((b_m * b_m) * (b_m * b_m)) * ((0.5 - (math.cos((0.011111111111111112 * (math.pi * angle))) * 0.5)) / (y_45_scale_m * y_45_scale_m)))))) / b_m)
	else:
		tmp = 0.25 * ((a * (y_45_scale_m * math.sqrt((8.0 * (math.pow(b_m, 4.0) * (0.5 - (0.5 * math.cos((0.011111111111111112 * (angle * math.pi)))))))))) / math.pow(b_m, 2.0))
	return tmp

Julia

b_m = abs(b)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0
	if (b_m <= 1.02e-167)
		tmp = Float64(Float64(-0.25 / b_m) * Float64(Float64(Float64(a * Float64(y_45_scale_m * y_45_scale_m)) * sqrt(Float64(8.0 * Float64(Float64(Float64(b_m * b_m) * Float64(b_m * b_m)) * Float64(Float64(0.5 - Float64(cos(Float64(0.011111111111111112 * Float64(pi * angle))) * 0.5)) / Float64(y_45_scale_m * y_45_scale_m)))))) / b_m));
	else
		tmp = Float64(0.25 * Float64(Float64(a * Float64(y_45_scale_m * sqrt(Float64(8.0 * Float64((b_m ^ 4.0) * Float64(0.5 - Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi)))))))))) / (b_m ^ 2.0)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0;
	if (b_m <= 1.02e-167)
		tmp = (-0.25 / b_m) * (((a * (y_45_scale_m * y_45_scale_m)) * sqrt((8.0 * (((b_m * b_m) * (b_m * b_m)) * ((0.5 - (cos((0.011111111111111112 * (pi * angle))) * 0.5)) / (y_45_scale_m * y_45_scale_m)))))) / b_m);
	else
		tmp = 0.25 * ((a * (y_45_scale_m * sqrt((8.0 * ((b_m ^ 4.0) * (0.5 - (0.5 * cos((0.011111111111111112 * (angle * pi)))))))))) / (b_m ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale$95$m_] := If[LessEqual[b$95$m, 1.02e-167], N[(N[(-0.25 / b$95$m), $MachinePrecision] * N[(N[(N[(a * N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / b$95$m), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(a * N[(y$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[Power[b$95$m, 4.0], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.0199999999999999e-167

   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 x-scale around 0

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

   4. Applied rewrites 0.8%

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

   6. Applied rewrites 3.2%

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

   7. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 3.9%

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

   11. Applied rewrites 9.6%

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

   if 1.0199999999999999e-167 < b

   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 x-scale around 0

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

   4. Applied rewrites 0.8%

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

   6. Applied rewrites 3.2%

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

   7. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 3.9%

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

   10. Taylor expanded in y-scale around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

   12. Applied rewrites 10.0%

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

Alternative 3: 10.2% accurate, 8.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.02 \cdot 10^{-167}:\\ \;\;\;\;\frac{-0.25}{b\_m} \cdot \frac{\left(a \cdot \left(y-scale\_m \cdot y-scale\_m\right)\right) \cdot \sqrt{8 \cdot \left(\left(\left(b\_m \cdot b\_m\right) \cdot \left(b\_m \cdot b\_m\right)\right) \cdot \frac{0.5 - \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot 0.5}{y-scale\_m \cdot y-scale\_m}\right)}}{b\_m}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{a \cdot \left({y-scale\_m}^{2} \cdot \sqrt{8 \cdot \frac{{b\_m}^{4} \cdot \left(0.5 - 0.5\right)}{{y-scale\_m}^{2}}}\right)}{{b\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale y-scale_m)
 :precision binary64
 (if (<= b_m 1.02e-167)
   (*
    (/ -0.25 b_m)
    (/
     (*
      (* a (* y-scale_m y-scale_m))
      (sqrt
       (*
        8.0
        (*
         (* (* b_m b_m) (* b_m b_m))
         (/
          (- 0.5 (* (cos (* 0.011111111111111112 (* PI angle))) 0.5))
          (* y-scale_m y-scale_m))))))
     b_m))
   (*
    -0.25
    (/
     (*
      a
      (*
       (pow y-scale_m 2.0)
       (sqrt (* 8.0 (/ (* (pow b_m 4.0) (- 0.5 0.5)) (pow y-scale_m 2.0))))))
     (pow b_m 2.0)))))

C

b_m = fabs(b);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if (b_m <= 1.02e-167) {
		tmp = (-0.25 / b_m) * (((a * (y_45_scale_m * y_45_scale_m)) * sqrt((8.0 * (((b_m * b_m) * (b_m * b_m)) * ((0.5 - (cos((0.011111111111111112 * (((double) M_PI) * angle))) * 0.5)) / (y_45_scale_m * y_45_scale_m)))))) / b_m);
	} else {
		tmp = -0.25 * ((a * (pow(y_45_scale_m, 2.0) * sqrt((8.0 * ((pow(b_m, 4.0) * (0.5 - 0.5)) / pow(y_45_scale_m, 2.0)))))) / pow(b_m, 2.0));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if (b_m <= 1.02e-167) {
		tmp = (-0.25 / b_m) * (((a * (y_45_scale_m * y_45_scale_m)) * Math.sqrt((8.0 * (((b_m * b_m) * (b_m * b_m)) * ((0.5 - (Math.cos((0.011111111111111112 * (Math.PI * angle))) * 0.5)) / (y_45_scale_m * y_45_scale_m)))))) / b_m);
	} else {
		tmp = -0.25 * ((a * (Math.pow(y_45_scale_m, 2.0) * Math.sqrt((8.0 * ((Math.pow(b_m, 4.0) * (0.5 - 0.5)) / Math.pow(y_45_scale_m, 2.0)))))) / Math.pow(b_m, 2.0));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale, y_45_scale_m):
	tmp = 0
	if b_m <= 1.02e-167:
		tmp = (-0.25 / b_m) * (((a * (y_45_scale_m * y_45_scale_m)) * math.sqrt((8.0 * (((b_m * b_m) * (b_m * b_m)) * ((0.5 - (math.cos((0.011111111111111112 * (math.pi * angle))) * 0.5)) / (y_45_scale_m * y_45_scale_m)))))) / b_m)
	else:
		tmp = -0.25 * ((a * (math.pow(y_45_scale_m, 2.0) * math.sqrt((8.0 * ((math.pow(b_m, 4.0) * (0.5 - 0.5)) / math.pow(y_45_scale_m, 2.0)))))) / math.pow(b_m, 2.0))
	return tmp

Julia

b_m = abs(b)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0
	if (b_m <= 1.02e-167)
		tmp = Float64(Float64(-0.25 / b_m) * Float64(Float64(Float64(a * Float64(y_45_scale_m * y_45_scale_m)) * sqrt(Float64(8.0 * Float64(Float64(Float64(b_m * b_m) * Float64(b_m * b_m)) * Float64(Float64(0.5 - Float64(cos(Float64(0.011111111111111112 * Float64(pi * angle))) * 0.5)) / Float64(y_45_scale_m * y_45_scale_m)))))) / b_m));
	else
		tmp = Float64(-0.25 * Float64(Float64(a * Float64((y_45_scale_m ^ 2.0) * sqrt(Float64(8.0 * Float64(Float64((b_m ^ 4.0) * Float64(0.5 - 0.5)) / (y_45_scale_m ^ 2.0)))))) / (b_m ^ 2.0)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0;
	if (b_m <= 1.02e-167)
		tmp = (-0.25 / b_m) * (((a * (y_45_scale_m * y_45_scale_m)) * sqrt((8.0 * (((b_m * b_m) * (b_m * b_m)) * ((0.5 - (cos((0.011111111111111112 * (pi * angle))) * 0.5)) / (y_45_scale_m * y_45_scale_m)))))) / b_m);
	else
		tmp = -0.25 * ((a * ((y_45_scale_m ^ 2.0) * sqrt((8.0 * (((b_m ^ 4.0) * (0.5 - 0.5)) / (y_45_scale_m ^ 2.0)))))) / (b_m ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale$95$m_] := If[LessEqual[b$95$m, 1.02e-167], N[(N[(-0.25 / b$95$m), $MachinePrecision] * N[(N[(N[(a * N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / b$95$m), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[(a * N[(N[Power[y$45$scale$95$m, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Power[b$95$m, 4.0], $MachinePrecision] * N[(0.5 - 0.5), $MachinePrecision]), $MachinePrecision] / N[Power[y$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.0199999999999999e-167

   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 x-scale around 0

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

   4. Applied rewrites 0.8%

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

   6. Applied rewrites 3.2%

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

   7. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 3.9%

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

   11. Applied rewrites 9.6%

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

   if 1.0199999999999999e-167 < b

   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 x-scale around 0

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

   4. Applied rewrites 0.8%

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

   6. Applied rewrites 3.2%

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

   7. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   9. Applied rewrites 3.9%

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

   10. Taylor expanded in angle around 0

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

   12. Applied rewrites 4.6%

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

Alternative 4: 9.6% accurate, 8.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ y-scale_m = \left|y-scale\right| \\ \frac{-0.25}{b\_m} \cdot \frac{\left(a \cdot \left(y-scale\_m \cdot y-scale\_m\right)\right) \cdot \sqrt{8 \cdot \left(\left(\left(b\_m \cdot b\_m\right) \cdot \left(b\_m \cdot b\_m\right)\right) \cdot \frac{0.5 - \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot 0.5}{y-scale\_m \cdot y-scale\_m}\right)}}{b\_m} \end{array} \]

FPCore

b_m = (fabs.f64 b)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale y-scale_m)
 :precision binary64
 (*
  (/ -0.25 b_m)
  (/
   (*
    (* a (* y-scale_m y-scale_m))
    (sqrt
     (*
      8.0
      (*
       (* (* b_m b_m) (* b_m b_m))
       (/
        (- 0.5 (* (cos (* 0.011111111111111112 (* PI angle))) 0.5))
        (* y-scale_m y-scale_m))))))
   b_m)))

C

b_m = fabs(b);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale_m) {
	return (-0.25 / b_m) * (((a * (y_45_scale_m * y_45_scale_m)) * sqrt((8.0 * (((b_m * b_m) * (b_m * b_m)) * ((0.5 - (cos((0.011111111111111112 * (((double) M_PI) * angle))) * 0.5)) / (y_45_scale_m * y_45_scale_m)))))) / b_m);
}

Java

b_m = Math.abs(b);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale_m) {
	return (-0.25 / b_m) * (((a * (y_45_scale_m * y_45_scale_m)) * Math.sqrt((8.0 * (((b_m * b_m) * (b_m * b_m)) * ((0.5 - (Math.cos((0.011111111111111112 * (Math.PI * angle))) * 0.5)) / (y_45_scale_m * y_45_scale_m)))))) / b_m);
}

Python

b_m = math.fabs(b)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale, y_45_scale_m):
	return (-0.25 / b_m) * (((a * (y_45_scale_m * y_45_scale_m)) * math.sqrt((8.0 * (((b_m * b_m) * (b_m * b_m)) * ((0.5 - (math.cos((0.011111111111111112 * (math.pi * angle))) * 0.5)) / (y_45_scale_m * y_45_scale_m)))))) / b_m)

Julia

b_m = abs(b)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale, y_45_scale_m)
	return Float64(Float64(-0.25 / b_m) * Float64(Float64(Float64(a * Float64(y_45_scale_m * y_45_scale_m)) * sqrt(Float64(8.0 * Float64(Float64(Float64(b_m * b_m) * Float64(b_m * b_m)) * Float64(Float64(0.5 - Float64(cos(Float64(0.011111111111111112 * Float64(pi * angle))) * 0.5)) / Float64(y_45_scale_m * y_45_scale_m)))))) / b_m))
end

MATLAB

b_m = abs(b);
y-scale_m = abs(y_45_scale);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale_m)
	tmp = (-0.25 / b_m) * (((a * (y_45_scale_m * y_45_scale_m)) * sqrt((8.0 * (((b_m * b_m) * (b_m * b_m)) * ((0.5 - (cos((0.011111111111111112 * (pi * angle))) * 0.5)) / (y_45_scale_m * y_45_scale_m)))))) / b_m);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale$95$m_] := N[(N[(-0.25 / b$95$m), $MachinePrecision] * N[(N[(N[(a * N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / b$95$m), $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 x-scale around 0

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

4. Applied rewrites 0.8%

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

6. Applied rewrites 3.2%

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

7. Taylor expanded in a around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

9. Applied rewrites 3.9%

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

11. Applied rewrites 9.6%

    \[\leadsto \frac{-0.25}{b} \cdot \frac{\left(a \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \sqrt{8 \cdot \left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \frac{0.5 - \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot 0.5}{y-scale \cdot y-scale}\right)}}{\color{blue}{b}} \]
12. Add Preprocessing

Alternative 5: 3.7% accurate, 8.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ y-scale_m = \left|y-scale\right| \\ -0.25 \cdot \left(a \cdot \frac{\sqrt{8 \cdot \left(\left(\left(b\_m \cdot b\_m\right) \cdot \left(b\_m \cdot b\_m\right)\right) \cdot \frac{0.5 - \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot 0.5}{y-scale\_m \cdot y-scale\_m}\right)} \cdot \left(y-scale\_m \cdot y-scale\_m\right)}{b\_m \cdot b\_m}\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale y-scale_m)
 :precision binary64
 (*
  -0.25
  (*
   a
   (/
    (*
     (sqrt
      (*
       8.0
       (*
        (* (* b_m b_m) (* b_m b_m))
        (/
         (- 0.5 (* (cos (* 0.011111111111111112 (* PI angle))) 0.5))
         (* y-scale_m y-scale_m)))))
     (* y-scale_m y-scale_m))
    (* b_m b_m)))))

C

b_m = fabs(b);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale_m) {
	return -0.25 * (a * ((sqrt((8.0 * (((b_m * b_m) * (b_m * b_m)) * ((0.5 - (cos((0.011111111111111112 * (((double) M_PI) * angle))) * 0.5)) / (y_45_scale_m * y_45_scale_m))))) * (y_45_scale_m * y_45_scale_m)) / (b_m * b_m)));
}

Java

b_m = Math.abs(b);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale_m) {
	return -0.25 * (a * ((Math.sqrt((8.0 * (((b_m * b_m) * (b_m * b_m)) * ((0.5 - (Math.cos((0.011111111111111112 * (Math.PI * angle))) * 0.5)) / (y_45_scale_m * y_45_scale_m))))) * (y_45_scale_m * y_45_scale_m)) / (b_m * b_m)));
}

Python

b_m = math.fabs(b)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale, y_45_scale_m):
	return -0.25 * (a * ((math.sqrt((8.0 * (((b_m * b_m) * (b_m * b_m)) * ((0.5 - (math.cos((0.011111111111111112 * (math.pi * angle))) * 0.5)) / (y_45_scale_m * y_45_scale_m))))) * (y_45_scale_m * y_45_scale_m)) / (b_m * b_m)))

Julia

b_m = abs(b)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale, y_45_scale_m)
	return Float64(-0.25 * Float64(a * Float64(Float64(sqrt(Float64(8.0 * Float64(Float64(Float64(b_m * b_m) * Float64(b_m * b_m)) * Float64(Float64(0.5 - Float64(cos(Float64(0.011111111111111112 * Float64(pi * angle))) * 0.5)) / Float64(y_45_scale_m * y_45_scale_m))))) * Float64(y_45_scale_m * y_45_scale_m)) / Float64(b_m * b_m))))
end

MATLAB

b_m = abs(b);
y-scale_m = abs(y_45_scale);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale_m)
	tmp = -0.25 * (a * ((sqrt((8.0 * (((b_m * b_m) * (b_m * b_m)) * ((0.5 - (cos((0.011111111111111112 * (pi * angle))) * 0.5)) / (y_45_scale_m * y_45_scale_m))))) * (y_45_scale_m * y_45_scale_m)) / (b_m * b_m)));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale$95$m_] := N[(-0.25 * N[(a * N[(N[(N[Sqrt[N[(8.0 * N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1%

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

2. Taylor expanded in x-scale around 0

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

4. Applied rewrites 0.8%

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

6. Applied rewrites 3.2%

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

7. Taylor expanded in a around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

9. Applied rewrites 3.9%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-pow.f64 N/A

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

    4. associate-/l* N/A

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

    5. lower-*.f64 N/A

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

    6. lower-/.f64 N/A

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

11. Applied rewrites 3.7%

    \[\leadsto -0.25 \cdot \left(a \cdot \frac{\sqrt{8 \cdot \left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \frac{0.5 - \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot 0.5}{y-scale \cdot y-scale}\right)} \cdot \left(y-scale \cdot y-scale\right)}{\color{blue}{b \cdot b}}\right) \]
12. Add Preprocessing

Alternative 6: 1.6% accurate, 10.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \left(\left(x-scale \cdot x-scale\right) \cdot y-scale\_m\right) \cdot y-scale\_m\\ t_1 := \left(-a\right) \cdot b\_m\\ \frac{-\sqrt{0 \cdot \left(\left(\left(\left(\left(\left(a \cdot b\_m\right) \cdot b\_m\right) \cdot \frac{-a}{t\_0}\right) \cdot 4\right) \cdot 2\right) \cdot \left(t\_1 \cdot \left(a \cdot b\_m\right)\right)\right)}}{t\_1 \cdot \left(\left(a \cdot b\_m\right) \cdot 4\right)} \cdot t\_0 \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale y-scale_m)
 :precision binary64
 (let* ((t_0 (* (* (* x-scale x-scale) y-scale_m) y-scale_m))
        (t_1 (* (- a) b_m)))
   (*
    (/
     (-
      (sqrt
       (*
        0.0
        (*
         (* (* (* (* (* a b_m) b_m) (/ (- a) t_0)) 4.0) 2.0)
         (* t_1 (* a b_m))))))
     (* t_1 (* (* a b_m) 4.0)))
    t_0)))

C

b_m = fabs(b);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = ((x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m;
	double t_1 = -a * b_m;
	return (-sqrt((0.0 * ((((((a * b_m) * b_m) * (-a / t_0)) * 4.0) * 2.0) * (t_1 * (a * b_m))))) / (t_1 * ((a * b_m) * 4.0))) * t_0;
}

Fortran

b_m =     private
y-scale_m =     private
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_m, angle, x_45scale, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    real(8) :: t_1
    t_0 = ((x_45scale * x_45scale) * y_45scale_m) * y_45scale_m
    t_1 = -a * b_m
    code = (-sqrt((0.0d0 * ((((((a * b_m) * b_m) * (-a / t_0)) * 4.0d0) * 2.0d0) * (t_1 * (a * b_m))))) / (t_1 * ((a * b_m) * 4.0d0))) * t_0
end function

Java

b_m = Math.abs(b);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale_m) {
	double t_0 = ((x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m;
	double t_1 = -a * b_m;
	return (-Math.sqrt((0.0 * ((((((a * b_m) * b_m) * (-a / t_0)) * 4.0) * 2.0) * (t_1 * (a * b_m))))) / (t_1 * ((a * b_m) * 4.0))) * t_0;
}

Python

b_m = math.fabs(b)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale, y_45_scale_m):
	t_0 = ((x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m
	t_1 = -a * b_m
	return (-math.sqrt((0.0 * ((((((a * b_m) * b_m) * (-a / t_0)) * 4.0) * 2.0) * (t_1 * (a * b_m))))) / (t_1 * ((a * b_m) * 4.0))) * t_0

Julia

b_m = abs(b)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale, y_45_scale_m)
	t_0 = Float64(Float64(Float64(x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m)
	t_1 = Float64(Float64(-a) * b_m)
	return Float64(Float64(Float64(-sqrt(Float64(0.0 * Float64(Float64(Float64(Float64(Float64(Float64(a * b_m) * b_m) * Float64(Float64(-a) / t_0)) * 4.0) * 2.0) * Float64(t_1 * Float64(a * b_m)))))) / Float64(t_1 * Float64(Float64(a * b_m) * 4.0))) * t_0)
end

MATLAB

b_m = abs(b);
y-scale_m = abs(y_45_scale);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale_m)
	t_0 = ((x_45_scale * x_45_scale) * y_45_scale_m) * y_45_scale_m;
	t_1 = -a * b_m;
	tmp = (-sqrt((0.0 * ((((((a * b_m) * b_m) * (-a / t_0)) * 4.0) * 2.0) * (t_1 * (a * b_m))))) / (t_1 * ((a * b_m) * 4.0))) * t_0;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale$95$m_] := Block[{t$95$0 = N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale$95$m), $MachinePrecision] * y$45$scale$95$m), $MachinePrecision]}, Block[{t$95$1 = N[((-a) * b$95$m), $MachinePrecision]}, N[(N[((-N[Sqrt[N[(0.0 * N[(N[(N[(N[(N[(N[(a * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * N[((-a) / t$95$0), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$1 * N[(a * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(t$95$1 * N[(N[(a * b$95$m), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $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 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. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-pow.f64 1.9

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

7. Applied rewrites 1.9%

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

9. Applied rewrites 0.8%

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

10. Taylor expanded in y-scale around 0

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

12. Applied rewrites 1.6%

    \[\leadsto \frac{-\sqrt{0 \cdot \left(\left(\left(\left(\left(\left(a \cdot b\right) \cdot b\right) \cdot \frac{-a}{\left(\left(x-scale \cdot x-scale\right) \cdot y-scale\right) \cdot y-scale}\right) \cdot 4\right) \cdot 2\right) \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot \left(a \cdot b\right)\right)\right)}}{\left(\left(-a\right) \cdot b\right) \cdot \left(\left(a \cdot b\right) \cdot 4\right)} \cdot \left(\left(\left(x-scale \cdot x-scale\right) \cdot y-scale\right) \cdot y-scale\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(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))))