b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 27.6%

Time: 51.4s

Alternatives: 7

Speedup: 17.4×

• could not determine a ground truth

  1. a = 9.352828484200807e+284
  2. b = -2.815985373355598e-283
  3. angle = -1.4980788717638468e-295
  4. x-scale = 6.957013205961828e+304
  5. y-scale = -3.0236101855690335e-164

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: 27.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := 0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;a\_m \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\frac{0.25}{a\_m} \cdot \frac{b \cdot \left(y-scale \cdot \sqrt{8 \cdot \left({a\_m}^{4} \cdot \left(t\_0 - \sqrt{{t\_0}^{2}}\right)\right)}\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{b} \cdot \frac{\left(\left(\left(b \cdot b\right) \cdot \sqrt{\frac{0 \cdot \left(\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\right)}{y-scale \cdot y-scale}}\right) \cdot \sqrt{8}\right) \cdot \left(y-scale \cdot y-scale\right)}{b}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))))
   (if (<= a_m 1.15e+77)
     (*
      (/ 0.25 a_m)
      (/
       (*
        b
        (*
         y-scale
         (sqrt (* 8.0 (* (pow a_m 4.0) (- t_0 (sqrt (pow t_0 2.0))))))))
       a_m))
     (*
      (/ 0.25 b)
      (/
       (*
        (*
         (*
          (* b b)
          (sqrt
           (/
            (*
             0.0
             (*
              (fma (cos (* (* PI angle) 0.011111111111111112)) 0.5 0.5)
              (* b b)))
            (* y-scale y-scale))))
         (sqrt 8.0))
        (* y-scale y-scale))
       b)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.5 + (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI)))));
	double tmp;
	if (a_m <= 1.15e+77) {
		tmp = (0.25 / a_m) * ((b * (y_45_scale * sqrt((8.0 * (pow(a_m, 4.0) * (t_0 - sqrt(pow(t_0, 2.0)))))))) / a_m);
	} else {
		tmp = (0.25 / b) * (((((b * b) * sqrt(((0.0 * (fma(cos(((((double) M_PI) * angle) * 0.011111111111111112)), 0.5, 0.5) * (b * b))) / (y_45_scale * y_45_scale)))) * sqrt(8.0)) * (y_45_scale * y_45_scale)) / b);
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.5 + Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi)))))
	tmp = 0.0
	if (a_m <= 1.15e+77)
		tmp = Float64(Float64(0.25 / a_m) * Float64(Float64(b * Float64(y_45_scale * sqrt(Float64(8.0 * Float64((a_m ^ 4.0) * Float64(t_0 - sqrt((t_0 ^ 2.0)))))))) / a_m));
	else
		tmp = Float64(Float64(0.25 / b) * Float64(Float64(Float64(Float64(Float64(b * b) * sqrt(Float64(Float64(0.0 * Float64(fma(cos(Float64(Float64(pi * angle) * 0.011111111111111112)), 0.5, 0.5) * Float64(b * b))) / Float64(y_45_scale * y_45_scale)))) * sqrt(8.0)) * Float64(y_45_scale * y_45_scale)) / b));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.5 + N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 1.15e+77], N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(b * N[(y$45$scale * N[Sqrt[N[(8.0 * N[(N[Power[a$95$m, 4.0], $MachinePrecision] * N[(t$95$0 - N[Sqrt[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / b), $MachinePrecision] * N[(N[(N[(N[(N[(b * b), $MachinePrecision] * N[Sqrt[N[(N[(0.0 * N[(N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.14999999999999997e77

   1. Initial program 0.1%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 0.5%

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

   6. Applied rewrites 1.0%

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

   7. Taylor expanded in x-scale around 0

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

   9. Applied rewrites 10.5%

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

   10. Taylor expanded in y-scale around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

   12. Applied rewrites 24.9%

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

   if 1.14999999999999997e77 < a

   1. Initial program 0.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 0.7%

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

   5. Taylor expanded in x-scale 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

   7. Applied rewrites 3.7%

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

   9. Applied rewrites 2.5%

      \[\leadsto 0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(e^{\log \left({\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{4}\right) \cdot 0.5} - \sqrt{{b}^{4} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{y-scale}^{2}}}}{{b}^{2}} \]

   11. Applied rewrites 11.2%

       \[\leadsto \frac{0.25}{b} \cdot \color{blue}{\frac{\left(\left(\left(b \cdot b\right) \cdot \sqrt{\frac{0 \cdot \left(\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\right)}{y-scale \cdot y-scale}}\right) \cdot \sqrt{8}\right) \cdot \left(y-scale \cdot y-scale\right)}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 23.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5, 0.5\right)\\ \mathbf{if}\;a\_m \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;\frac{0.25}{a\_m} \cdot \frac{b \cdot \left({y-scale}^{2} \cdot \frac{\sqrt{8 \cdot \left(\left(\left(\left(a\_m \cdot a\_m\right) \cdot a\_m\right) \cdot a\_m\right) \cdot \left(t\_0 - t\_0\right)\right)}}{\left|y-scale\right|}\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{b} \cdot \frac{\left(\left(\left(b \cdot b\right) \cdot \sqrt{\frac{0 \cdot \left(t\_0 \cdot \left(b \cdot b\right)\right)}{y-scale \cdot y-scale}}\right) \cdot \sqrt{8}\right) \cdot \left(y-scale \cdot y-scale\right)}{b}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (fma (cos (* (* PI angle) 0.011111111111111112)) 0.5 0.5)))
   (if (<= a_m 1.15e+77)
     (*
      (/ 0.25 a_m)
      (/
       (*
        b
        (*
         (pow y-scale 2.0)
         (/
          (sqrt (* 8.0 (* (* (* (* a_m a_m) a_m) a_m) (- t_0 t_0))))
          (fabs y-scale))))
       a_m))
     (*
      (/ 0.25 b)
      (/
       (*
        (*
         (* (* b b) (sqrt (/ (* 0.0 (* t_0 (* b b))) (* y-scale y-scale))))
         (sqrt 8.0))
        (* y-scale y-scale))
       b)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fma(cos(((((double) M_PI) * angle) * 0.011111111111111112)), 0.5, 0.5);
	double tmp;
	if (a_m <= 1.15e+77) {
		tmp = (0.25 / a_m) * ((b * (pow(y_45_scale, 2.0) * (sqrt((8.0 * ((((a_m * a_m) * a_m) * a_m) * (t_0 - t_0)))) / fabs(y_45_scale)))) / a_m);
	} else {
		tmp = (0.25 / b) * (((((b * b) * sqrt(((0.0 * (t_0 * (b * b))) / (y_45_scale * y_45_scale)))) * sqrt(8.0)) * (y_45_scale * y_45_scale)) / b);
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = fma(cos(Float64(Float64(pi * angle) * 0.011111111111111112)), 0.5, 0.5)
	tmp = 0.0
	if (a_m <= 1.15e+77)
		tmp = Float64(Float64(0.25 / a_m) * Float64(Float64(b * Float64((y_45_scale ^ 2.0) * Float64(sqrt(Float64(8.0 * Float64(Float64(Float64(Float64(a_m * a_m) * a_m) * a_m) * Float64(t_0 - t_0)))) / abs(y_45_scale)))) / a_m));
	else
		tmp = Float64(Float64(0.25 / b) * Float64(Float64(Float64(Float64(Float64(b * b) * sqrt(Float64(Float64(0.0 * Float64(t_0 * Float64(b * b))) / Float64(y_45_scale * y_45_scale)))) * sqrt(8.0)) * Float64(y_45_scale * y_45_scale)) / b));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]}, If[LessEqual[a$95$m, 1.15e+77], N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(b * N[(N[Power[y$45$scale, 2.0], $MachinePrecision] * N[(N[Sqrt[N[(8.0 * N[(N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] * a$95$m), $MachinePrecision] * a$95$m), $MachinePrecision] * N[(t$95$0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / b), $MachinePrecision] * N[(N[(N[(N[(N[(b * b), $MachinePrecision] * N[Sqrt[N[(N[(0.0 * N[(t$95$0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.14999999999999997e77

   1. Initial program 0.1%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 0.5%

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

   6. Applied rewrites 1.0%

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

   7. Taylor expanded in x-scale around 0

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

   9. Applied rewrites 10.5%

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

   11. Applied rewrites 20.7%

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

   if 1.14999999999999997e77 < a

   1. Initial program 0.1%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 0.7%

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

   5. Taylor expanded in x-scale 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

   7. Applied rewrites 3.7%

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

   9. Applied rewrites 2.5%

      \[\leadsto 0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(e^{\log \left({\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{4}\right) \cdot 0.5} - \sqrt{{b}^{4} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{y-scale}^{2}}}}{{b}^{2}} \]

   11. Applied rewrites 11.2%

       \[\leadsto \frac{0.25}{b} \cdot \color{blue}{\frac{\left(\left(\left(b \cdot b\right) \cdot \sqrt{\frac{0 \cdot \left(\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\right)}{y-scale \cdot y-scale}}\right) \cdot \sqrt{8}\right) \cdot \left(y-scale \cdot y-scale\right)}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 16.4% accurate, 8.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{0.25}{b} \cdot \frac{\left(\left(\left(b \cdot b\right) \cdot \sqrt{\frac{0 \cdot \left(\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\right)}{y-scale \cdot y-scale}}\right) \cdot \sqrt{8}\right) \cdot \left(y-scale \cdot y-scale\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{0}}{{b}^{2}}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (if (<= b 1.55e-162)
   (*
    (/ 0.25 b)
    (/
     (*
      (*
       (*
        (* b b)
        (sqrt
         (/
          (*
           0.0
           (*
            (fma (cos (* (* PI angle) 0.011111111111111112)) 0.5 0.5)
            (* b b)))
          (* y-scale y-scale))))
       (sqrt 8.0))
      (* y-scale y-scale))
     b))
   (* 0.25 (/ (* (pow y-scale 2.0) (sqrt 0.0)) (pow b 2.0)))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b <= 1.55e-162) {
		tmp = (0.25 / b) * (((((b * b) * sqrt(((0.0 * (fma(cos(((((double) M_PI) * angle) * 0.011111111111111112)), 0.5, 0.5) * (b * b))) / (y_45_scale * y_45_scale)))) * sqrt(8.0)) * (y_45_scale * y_45_scale)) / b);
	} else {
		tmp = 0.25 * ((pow(y_45_scale, 2.0) * sqrt(0.0)) / pow(b, 2.0));
	}
	return tmp;
}

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b <= 1.55e-162)
		tmp = Float64(Float64(0.25 / b) * Float64(Float64(Float64(Float64(Float64(b * b) * sqrt(Float64(Float64(0.0 * Float64(fma(cos(Float64(Float64(pi * angle) * 0.011111111111111112)), 0.5, 0.5) * Float64(b * b))) / Float64(y_45_scale * y_45_scale)))) * sqrt(8.0)) * Float64(y_45_scale * y_45_scale)) / b));
	else
		tmp = Float64(0.25 * Float64(Float64((y_45_scale ^ 2.0) * sqrt(0.0)) / (b ^ 2.0)));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b, 1.55e-162], N[(N[(0.25 / b), $MachinePrecision] * N[(N[(N[(N[(N[(b * b), $MachinePrecision] * N[Sqrt[N[(N[(0.0 * N[(N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[Power[y$45$scale, 2.0], $MachinePrecision] * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5499999999999999e-162

   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 a around 0

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

   4. Applied rewrites 0.7%

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

   5. Taylor expanded in x-scale 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

   7. Applied rewrites 3.7%

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

   9. Applied rewrites 2.5%

      \[\leadsto 0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(e^{\log \left({\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{4}\right) \cdot 0.5} - \sqrt{{b}^{4} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{y-scale}^{2}}}}{{b}^{2}} \]

   11. Applied rewrites 11.2%

       \[\leadsto \frac{0.25}{b} \cdot \color{blue}{\frac{\left(\left(\left(b \cdot b\right) \cdot \sqrt{\frac{0 \cdot \left(\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\right)}{y-scale \cdot y-scale}}\right) \cdot \sqrt{8}\right) \cdot \left(y-scale \cdot y-scale\right)}{b}} \]

   if 1.5499999999999999e-162 < 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 a around 0

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

   4. Applied rewrites 0.7%

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

   5. Taylor expanded in x-scale 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

   7. Applied rewrites 3.7%

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

   9. Applied rewrites 2.5%

      \[\leadsto 0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(e^{\log \left({\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{4}\right) \cdot 0.5} - \sqrt{{b}^{4} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{y-scale}^{2}}}}{{b}^{2}} \]

   10. Taylor expanded in b around 0

       \[\leadsto \frac{1}{4} \cdot \frac{{y-scale}^{2} \cdot \sqrt{0}}{{b}^{2}} \]
   11. Step-by-step derivation

   12. Applied rewrites 15.4%

       \[\leadsto 0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{0}}{{b}^{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 15.4% accurate, 10.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \frac{1}{x-scale \cdot x-scale}\\ \mathbf{if}\;b \leq 8.5 \cdot 10^{-175}:\\ \;\;\;\;\frac{0.25}{a\_m} \cdot \frac{\left(b \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\sqrt{\left(\frac{\left(\left(a\_m \cdot a\_m\right) \cdot a\_m\right) \cdot a\_m}{y-scale \cdot x-scale} \cdot \frac{t\_0 - t\_0}{y-scale \cdot x-scale}\right) \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right)}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{0}}{{b}^{2}}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* x-scale x-scale))))
   (if (<= b 8.5e-175)
     (*
      (/ 0.25 a_m)
      (/
       (*
        (* b (* x-scale x-scale))
        (*
         (sqrt
          (*
           (*
            (/ (* (* (* a_m a_m) a_m) a_m) (* y-scale x-scale))
            (/ (- t_0 t_0) (* y-scale x-scale)))
           8.0))
         (* y-scale y-scale)))
       a_m))
     (* 0.25 (/ (* (pow y-scale 2.0) (sqrt 0.0)) (pow b 2.0))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 1.0 / (x_45_scale * x_45_scale);
	double tmp;
	if (b <= 8.5e-175) {
		tmp = (0.25 / a_m) * (((b * (x_45_scale * x_45_scale)) * (sqrt(((((((a_m * a_m) * a_m) * a_m) / (y_45_scale * x_45_scale)) * ((t_0 - t_0) / (y_45_scale * x_45_scale))) * 8.0)) * (y_45_scale * y_45_scale))) / a_m);
	} else {
		tmp = 0.25 * ((pow(y_45_scale, 2.0) * sqrt(0.0)) / pow(b, 2.0));
	}
	return tmp;
}

Fortran

a_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_m, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (x_45scale * x_45scale)
    if (b <= 8.5d-175) then
        tmp = (0.25d0 / a_m) * (((b * (x_45scale * x_45scale)) * (sqrt(((((((a_m * a_m) * a_m) * a_m) / (y_45scale * x_45scale)) * ((t_0 - t_0) / (y_45scale * x_45scale))) * 8.0d0)) * (y_45scale * y_45scale))) / a_m)
    else
        tmp = 0.25d0 * (((y_45scale ** 2.0d0) * sqrt(0.0d0)) / (b ** 2.0d0))
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 1.0 / (x_45_scale * x_45_scale);
	double tmp;
	if (b <= 8.5e-175) {
		tmp = (0.25 / a_m) * (((b * (x_45_scale * x_45_scale)) * (Math.sqrt(((((((a_m * a_m) * a_m) * a_m) / (y_45_scale * x_45_scale)) * ((t_0 - t_0) / (y_45_scale * x_45_scale))) * 8.0)) * (y_45_scale * y_45_scale))) / a_m);
	} else {
		tmp = 0.25 * ((Math.pow(y_45_scale, 2.0) * Math.sqrt(0.0)) / Math.pow(b, 2.0));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = 1.0 / (x_45_scale * x_45_scale)
	tmp = 0
	if b <= 8.5e-175:
		tmp = (0.25 / a_m) * (((b * (x_45_scale * x_45_scale)) * (math.sqrt(((((((a_m * a_m) * a_m) * a_m) / (y_45_scale * x_45_scale)) * ((t_0 - t_0) / (y_45_scale * x_45_scale))) * 8.0)) * (y_45_scale * y_45_scale))) / a_m)
	else:
		tmp = 0.25 * ((math.pow(y_45_scale, 2.0) * math.sqrt(0.0)) / math.pow(b, 2.0))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(1.0 / Float64(x_45_scale * x_45_scale))
	tmp = 0.0
	if (b <= 8.5e-175)
		tmp = Float64(Float64(0.25 / a_m) * Float64(Float64(Float64(b * Float64(x_45_scale * x_45_scale)) * Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(a_m * a_m) * a_m) * a_m) / Float64(y_45_scale * x_45_scale)) * Float64(Float64(t_0 - t_0) / Float64(y_45_scale * x_45_scale))) * 8.0)) * Float64(y_45_scale * y_45_scale))) / a_m));
	else
		tmp = Float64(0.25 * Float64(Float64((y_45_scale ^ 2.0) * sqrt(0.0)) / (b ^ 2.0)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = 1.0 / (x_45_scale * x_45_scale);
	tmp = 0.0;
	if (b <= 8.5e-175)
		tmp = (0.25 / a_m) * (((b * (x_45_scale * x_45_scale)) * (sqrt(((((((a_m * a_m) * a_m) * a_m) / (y_45_scale * x_45_scale)) * ((t_0 - t_0) / (y_45_scale * x_45_scale))) * 8.0)) * (y_45_scale * y_45_scale))) / a_m);
	else
		tmp = 0.25 * (((y_45_scale ^ 2.0) * sqrt(0.0)) / (b ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(1.0 / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 8.5e-175], N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(N[(b * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[(N[(N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] * a$95$m), $MachinePrecision] * a$95$m), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 - t$95$0), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[Power[y$45$scale, 2.0], $MachinePrecision] * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 8.5000000000000005e-175

   1. Initial program 0.1%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 0.5%

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

   6. Applied rewrites 1.0%

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

   7. Taylor expanded in angle around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 1.7

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

   9. Applied rewrites 1.7%

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

   11. Applied rewrites 6.9%

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

   if 8.5000000000000005e-175 < 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 a around 0

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

   4. Applied rewrites 0.7%

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

   5. Taylor expanded in x-scale 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

   7. Applied rewrites 3.7%

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

   9. Applied rewrites 2.5%

      \[\leadsto 0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(e^{\log \left({\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{4}\right) \cdot 0.5} - \sqrt{{b}^{4} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{y-scale}^{2}}}}{{b}^{2}} \]

   10. Taylor expanded in b around 0

       \[\leadsto \frac{1}{4} \cdot \frac{{y-scale}^{2} \cdot \sqrt{0}}{{b}^{2}} \]
   11. Step-by-step derivation

   12. Applied rewrites 15.4%

       \[\leadsto 0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{0}}{{b}^{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 12.7% accurate, 11.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ \begin{array}{l} t_0 := \frac{1}{x-scale \cdot x-scale}\\ \mathbf{if}\;b \leq 2.3 \cdot 10^{-165}:\\ \;\;\;\;\frac{0.25}{\frac{a\_m}{\left(\left(\left(\left(a\_m \cdot a\_m\right) \cdot \sqrt{\frac{t\_0 - t\_0}{\left(\left(x-scale \cdot x-scale\right) \cdot y-scale\right) \cdot y-scale} \cdot 8}\right) \cdot y-scale\right) \cdot y-scale\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot b\right)} \cdot a\_m}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{0}}{{b}^{2}}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* x-scale x-scale))))
   (if (<= b 2.3e-165)
     (/
      0.25
      (*
       (/
        a_m
        (*
         (*
          (*
           (*
            (* a_m a_m)
            (sqrt
             (*
              (/ (- t_0 t_0) (* (* (* x-scale x-scale) y-scale) y-scale))
              8.0)))
           y-scale)
          y-scale)
         (* (* x-scale x-scale) b)))
       a_m))
     (* 0.25 (/ (* (pow y-scale 2.0) (sqrt 0.0)) (pow b 2.0))))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 1.0 / (x_45_scale * x_45_scale);
	double tmp;
	if (b <= 2.3e-165) {
		tmp = 0.25 / ((a_m / (((((a_m * a_m) * sqrt((((t_0 - t_0) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * 8.0))) * y_45_scale) * y_45_scale) * ((x_45_scale * x_45_scale) * b))) * a_m);
	} else {
		tmp = 0.25 * ((pow(y_45_scale, 2.0) * sqrt(0.0)) / pow(b, 2.0));
	}
	return tmp;
}

Fortran

a_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_m, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (x_45scale * x_45scale)
    if (b <= 2.3d-165) then
        tmp = 0.25d0 / ((a_m / (((((a_m * a_m) * sqrt((((t_0 - t_0) / (((x_45scale * x_45scale) * y_45scale) * y_45scale)) * 8.0d0))) * y_45scale) * y_45scale) * ((x_45scale * x_45scale) * b))) * a_m)
    else
        tmp = 0.25d0 * (((y_45scale ** 2.0d0) * sqrt(0.0d0)) / (b ** 2.0d0))
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 1.0 / (x_45_scale * x_45_scale);
	double tmp;
	if (b <= 2.3e-165) {
		tmp = 0.25 / ((a_m / (((((a_m * a_m) * Math.sqrt((((t_0 - t_0) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * 8.0))) * y_45_scale) * y_45_scale) * ((x_45_scale * x_45_scale) * b))) * a_m);
	} else {
		tmp = 0.25 * ((Math.pow(y_45_scale, 2.0) * Math.sqrt(0.0)) / Math.pow(b, 2.0));
	}
	return tmp;
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	t_0 = 1.0 / (x_45_scale * x_45_scale)
	tmp = 0
	if b <= 2.3e-165:
		tmp = 0.25 / ((a_m / (((((a_m * a_m) * math.sqrt((((t_0 - t_0) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * 8.0))) * y_45_scale) * y_45_scale) * ((x_45_scale * x_45_scale) * b))) * a_m)
	else:
		tmp = 0.25 * ((math.pow(y_45_scale, 2.0) * math.sqrt(0.0)) / math.pow(b, 2.0))
	return tmp

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(1.0 / Float64(x_45_scale * x_45_scale))
	tmp = 0.0
	if (b <= 2.3e-165)
		tmp = Float64(0.25 / Float64(Float64(a_m / Float64(Float64(Float64(Float64(Float64(a_m * a_m) * sqrt(Float64(Float64(Float64(t_0 - t_0) / Float64(Float64(Float64(x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * 8.0))) * y_45_scale) * y_45_scale) * Float64(Float64(x_45_scale * x_45_scale) * b))) * a_m));
	else
		tmp = Float64(0.25 * Float64(Float64((y_45_scale ^ 2.0) * sqrt(0.0)) / (b ^ 2.0)));
	end
	return tmp
end

MATLAB

a_m = abs(a);
function tmp_2 = code(a_m, b, angle, x_45_scale, y_45_scale)
	t_0 = 1.0 / (x_45_scale * x_45_scale);
	tmp = 0.0;
	if (b <= 2.3e-165)
		tmp = 0.25 / ((a_m / (((((a_m * a_m) * sqrt((((t_0 - t_0) / (((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)) * 8.0))) * y_45_scale) * y_45_scale) * ((x_45_scale * x_45_scale) * b))) * a_m);
	else
		tmp = 0.25 * (((y_45_scale ^ 2.0) * sqrt(0.0)) / (b ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(1.0 / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.3e-165], N[(0.25 / N[(N[(a$95$m / N[(N[(N[(N[(N[(a$95$m * a$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(t$95$0 - t$95$0), $MachinePrecision] / N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[Power[y$45$scale, 2.0], $MachinePrecision] * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.3e-165

   1. Initial program 0.1%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 0.5%

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

   6. Applied rewrites 1.0%

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

   7. Taylor expanded in angle around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 1.7

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

   9. Applied rewrites 1.7%

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

   11. Applied rewrites 5.4%

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

   if 2.3e-165 < 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 a around 0

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

   4. Applied rewrites 0.7%

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

   5. Taylor expanded in x-scale 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

   7. Applied rewrites 3.7%

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

   9. Applied rewrites 2.5%

      \[\leadsto 0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(e^{\log \left({\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{4}\right) \cdot 0.5} - \sqrt{{b}^{4} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{y-scale}^{2}}}}{{b}^{2}} \]

   10. Taylor expanded in b around 0

       \[\leadsto \frac{1}{4} \cdot \frac{{y-scale}^{2} \cdot \sqrt{0}}{{b}^{2}} \]
   11. Step-by-step derivation

   12. Applied rewrites 15.4%

       \[\leadsto 0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{0}}{{b}^{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 11.9% accurate, 17.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ 0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{0}}{{b}^{2}} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (* 0.25 (/ (* (pow y-scale 2.0) (sqrt 0.0)) (pow b 2.0))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * ((pow(y_45_scale, 2.0) * sqrt(0.0)) / pow(b, 2.0));
}

Fortran

a_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_m, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = 0.25d0 * (((y_45scale ** 2.0d0) * sqrt(0.0d0)) / (b ** 2.0d0))
end function

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * ((Math.pow(y_45_scale, 2.0) * Math.sqrt(0.0)) / Math.pow(b, 2.0));
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	return 0.25 * ((math.pow(y_45_scale, 2.0) * math.sqrt(0.0)) / math.pow(b, 2.0))

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	return Float64(0.25 * Float64(Float64((y_45_scale ^ 2.0) * sqrt(0.0)) / (b ^ 2.0)))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.25 * (((y_45_scale ^ 2.0) * sqrt(0.0)) / (b ^ 2.0));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := N[(0.25 * N[(N[(N[Power[y$45$scale, 2.0], $MachinePrecision] * N[Sqrt[0.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $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 a around 0

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

4. Applied rewrites 0.7%

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

5. Taylor expanded in x-scale 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

7. Applied rewrites 3.7%

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

9. Applied rewrites 2.5%

   \[\leadsto 0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(e^{\log \left({\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{4}\right) \cdot 0.5} - \sqrt{{b}^{4} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{y-scale}^{2}}}}{{b}^{2}} \]

10. Taylor expanded in b around 0

    \[\leadsto \frac{1}{4} \cdot \frac{{y-scale}^{2} \cdot \sqrt{0}}{{b}^{2}} \]
11. Step-by-step derivation

12. Applied rewrites 15.4%

    \[\leadsto 0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{0}}{{b}^{2}} \]
13. Add Preprocessing

Alternative 7: 0.0% accurate, 18.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ 0.25 \cdot \frac{{y-scale}^{22} \cdot \mathsf{NaN}}{{b}^{2}} \end{array} \]

FPCore

a_m = (fabs.f64 a)
(FPCore (a_m b angle x-scale y-scale)
 :precision binary64
 (* 0.25 (/ (* (pow y-scale 22.0) NAN) (pow b 2.0))))

C

a_m = fabs(a);
double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * ((pow(y_45_scale, 22.0) * ((double) NAN)) / pow(b, 2.0));
}

Java

a_m = Math.abs(a);
public static double code(double a_m, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * ((Math.pow(y_45_scale, 22.0) * Double.NaN) / Math.pow(b, 2.0));
}

Python

a_m = math.fabs(a)
def code(a_m, b, angle, x_45_scale, y_45_scale):
	return 0.25 * ((math.pow(y_45_scale, 22.0) * math.nan) / math.pow(b, 2.0))

Julia

a_m = abs(a)
function code(a_m, b, angle, x_45_scale, y_45_scale)
	return Float64(0.25 * Float64(Float64((y_45_scale ^ 22.0) * NaN) / (b ^ 2.0)))
end

MATLAB

a_m = abs(a);
function tmp = code(a_m, b, angle, x_45_scale, y_45_scale)
	tmp = 0.25 * (((y_45_scale ^ 22.0) * NaN) / (b ^ 2.0));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale_, y$45$scale_] := N[(0.25 * N[(N[(N[Power[y$45$scale, 22.0], $MachinePrecision] * Indeterminate), $MachinePrecision] / N[Power[b, 2.0], $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 a around 0

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

4. Applied rewrites 0.7%

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

5. Taylor expanded in x-scale 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

7. Applied rewrites 3.7%

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

9. Applied rewrites 2.5%

   \[\leadsto 0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(e^{\log \left({\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{4}\right) \cdot 0.5} - \sqrt{{b}^{4} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{y-scale}^{2}}}}{{b}^{2}} \]

10. Taylor expanded in y-scale around 0

    \[\leadsto \frac{1}{4} \cdot \frac{{y-scale}^{22} \cdot \mathsf{NAN}\left(\right)}{{b}^{2}} \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \frac{{y-scale}^{22} \cdot \mathsf{NAN}\left(\right)}{{b}^{2}} \]

    2. lower-pow.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \frac{{y-scale}^{22} \cdot \mathsf{NAN}\left(\right)}{{b}^{2}} \]

    3. lower-NAN.f64 0.0

       \[\leadsto 0.25 \cdot \frac{{y-scale}^{22} \cdot \mathsf{NaN}}{{b}^{2}} \]

12. Applied rewrites 0.0%

    \[\leadsto 0.25 \cdot \frac{{y-scale}^{22} \cdot \mathsf{NaN}}{{b}^{2}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(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))))