b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 47.4%

Time: 42.8s

Alternatives: 3

Speedup: 484.7×

• could not determine a ground truth

  1. a = 7.371126152930094e-52
  2. b = 6.924353718730208e+230
  3. angle = 3.015342613138877e+91
  4. x-scale = -8.893812630507476e-5
  5. y-scale = 4.0505916213840876e+63

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: 47.4% accurate, 55.9× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2.6 \cdot 10^{-26}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left(b\_m \cdot b\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= b_m 2.6e-26)
   (* (* 0.25 (* y-scale_m (sqrt 8.0))) (sqrt (* 2.0 (* b_m b_m))))
   (* a_m x-scale_m)))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b_m <= 2.6e-26) {
		tmp = (0.25 * (y_45_scale_m * sqrt(8.0))) * sqrt((2.0 * (b_m * b_m)));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
a_m = abs(a)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (b_m <= 2.6d-26) then
        tmp = (0.25d0 * (y_45scale_m * sqrt(8.0d0))) * sqrt((2.0d0 * (b_m * b_m)))
    else
        tmp = a_m * x_45scale_m
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b_m <= 2.6e-26) {
		tmp = (0.25 * (y_45_scale_m * Math.sqrt(8.0))) * Math.sqrt((2.0 * (b_m * b_m)));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if b_m <= 2.6e-26:
		tmp = (0.25 * (y_45_scale_m * math.sqrt(8.0))) * math.sqrt((2.0 * (b_m * b_m)))
	else:
		tmp = a_m * x_45_scale_m
	return tmp

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b_m <= 2.6e-26)
		tmp = Float64(Float64(0.25 * Float64(y_45_scale_m * sqrt(8.0))) * sqrt(Float64(2.0 * Float64(b_m * b_m))));
	else
		tmp = Float64(a_m * x_45_scale_m);
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
b_m = abs(b);
a_m = abs(a);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (b_m <= 2.6e-26)
		tmp = (0.25 * (y_45_scale_m * sqrt(8.0))) * sqrt((2.0 * (b_m * b_m)));
	else
		tmp = a_m * x_45_scale_m;
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b$95$m, 2.6e-26], N[(N[(0.25 * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6000000000000001e-26

   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. Add Preprocessing

   3. Taylor expanded in x-scale around inf

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

   5. Simplified 1.5%

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

   6. Taylor expanded in angle around 0

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

   8. Simplified 1.5%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, -0.5 \cdot \frac{\mathsf{fma}\left(-2, \mathsf{fma}\left(a \cdot a, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}, \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)}{y-scale \cdot y-scale}\right) \cdot \mathsf{fma}\left(a \cdot a, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\color{blue}{1}}^{2} \cdot \left(b \cdot b\right)\right), \frac{4 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\mathsf{fma}\left(a \cdot a, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}, \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)}{y-scale \cdot y-scale}\right)}\right)\right)} \]

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 32.8

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \color{blue}{\left(b \cdot b\right)}} \]

   11. Simplified 32.8%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \left(b \cdot b\right)}} \]

   if 2.6000000000000001e-26 < 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. Add Preprocessing

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 3.7

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

   5. Simplified 3.7%

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

   6. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{-4 \cdot \left({a}^{3} \cdot {b}^{2}\right)}{x-scale \cdot {y-scale}^{2}}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(\mathsf{neg}\left(a\right)\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{-4 \cdot \left({a}^{3} \cdot {b}^{2}\right)}{x-scale \cdot {y-scale}^{2}}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(\mathsf{neg}\left(a\right)\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   8. Simplified 3.5%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 3.5

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

   11. Simplified 3.5%

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

   12. Taylor expanded in a around 0

       \[\leadsto \color{blue}{a \cdot x-scale} \]
   13. Step-by-step derivation

       1. lower-*.f64 24.0

          \[\leadsto \color{blue}{a \cdot x-scale} \]

   14. Simplified 24.0%

       \[\leadsto \color{blue}{a \cdot x-scale} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 48.9% accurate, 61.9× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 2.5 \cdot 10^{-37}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right) \cdot \left(b\_m \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot x-scale\_m\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 2.5e-37)
   (* (* 0.25 (* y-scale_m (sqrt 8.0))) (* b_m (sqrt 2.0)))
   (* a_m x-scale_m)))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 2.5e-37) {
		tmp = (0.25 * (y_45_scale_m * sqrt(8.0))) * (b_m * sqrt(2.0));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
a_m = abs(a)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (y_45scale_m <= 2.5d-37) then
        tmp = (0.25d0 * (y_45scale_m * sqrt(8.0d0))) * (b_m * sqrt(2.0d0))
    else
        tmp = a_m * x_45scale_m
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 2.5e-37) {
		tmp = (0.25 * (y_45_scale_m * Math.sqrt(8.0))) * (b_m * Math.sqrt(2.0));
	} else {
		tmp = a_m * x_45_scale_m;
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 2.5e-37:
		tmp = (0.25 * (y_45_scale_m * math.sqrt(8.0))) * (b_m * math.sqrt(2.0))
	else:
		tmp = a_m * x_45_scale_m
	return tmp

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 2.5e-37)
		tmp = Float64(Float64(0.25 * Float64(y_45_scale_m * sqrt(8.0))) * Float64(b_m * sqrt(2.0)));
	else
		tmp = Float64(a_m * x_45_scale_m);
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
b_m = abs(b);
a_m = abs(a);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (y_45_scale_m <= 2.5e-37)
		tmp = (0.25 * (y_45_scale_m * sqrt(8.0))) * (b_m * sqrt(2.0));
	else
		tmp = a_m * x_45_scale_m;
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 2.5e-37], N[(N[(0.25 * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m * x$45$scale$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.4999999999999999e-37

   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. Add Preprocessing

   3. Taylor expanded in x-scale around inf

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

   5. Simplified 2.1%

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

   6. Taylor expanded in angle around 0

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

   8. Simplified 2.2%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\mathsf{fma}\left(a \cdot a, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, -0.5 \cdot \frac{\mathsf{fma}\left(-2, \mathsf{fma}\left(a \cdot a, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}, \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)}{y-scale \cdot y-scale}\right) \cdot \mathsf{fma}\left(a \cdot a, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\color{blue}{1}}^{2} \cdot \left(b \cdot b\right)\right), \frac{4 \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left({\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\mathsf{fma}\left(a \cdot a, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{y-scale \cdot y-scale}, \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(b \cdot b\right)}{y-scale \cdot y-scale}\right)}\right)\right)} \]

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 26.5

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(b \cdot \color{blue}{\sqrt{2}}\right) \]

   11. Simplified 26.5%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(b \cdot \sqrt{2}\right)} \]

   if 2.4999999999999999e-37 < y-scale

   1. Initial program 0.1%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 3.9

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

   5. Simplified 3.9%

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

   6. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{-4 \cdot \left({a}^{3} \cdot {b}^{2}\right)}{x-scale \cdot {y-scale}^{2}}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(\mathsf{neg}\left(a\right)\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{-4 \cdot \left({a}^{3} \cdot {b}^{2}\right)}{x-scale \cdot {y-scale}^{2}}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(\mathsf{neg}\left(a\right)\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   8. Simplified 4.8%

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

   9. Taylor expanded in b around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 1.9

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

   11. Simplified 1.9%

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

   12. Taylor expanded in a around 0

       \[\leadsto \color{blue}{a \cdot x-scale} \]
   13. Step-by-step derivation

       1. lower-*.f64 22.5

          \[\leadsto \color{blue}{a \cdot x-scale} \]

   14. Simplified 22.5%

       \[\leadsto \color{blue}{a \cdot x-scale} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 36.5% accurate, 484.7× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ x-scale_m = \left|x-scale\right| \\ b_m = \left|b\right| \\ a_m = \left|a\right| \\ a\_m \cdot x-scale\_m \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
x-scale_m = (fabs.f64 x-scale)
b_m = (fabs.f64 b)
a_m = (fabs.f64 a)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (* a_m x-scale_m))

C

y-scale_m = fabs(y_45_scale);
x-scale_m = fabs(x_45_scale);
b_m = fabs(b);
a_m = fabs(a);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return a_m * x_45_scale_m;
}

Fortran

y-scale_m = abs(y_45scale)
x-scale_m = abs(x_45scale)
b_m = abs(b)
a_m = abs(a)
real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = a_m * x_45scale_m
end function

Java

y-scale_m = Math.abs(y_45_scale);
x-scale_m = Math.abs(x_45_scale);
b_m = Math.abs(b);
a_m = Math.abs(a);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return a_m * x_45_scale_m;
}

Python

y-scale_m = math.fabs(y_45_scale)
x-scale_m = math.fabs(x_45_scale)
b_m = math.fabs(b)
a_m = math.fabs(a)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	return a_m * x_45_scale_m

Julia

y-scale_m = abs(y_45_scale)
x-scale_m = abs(x_45_scale)
b_m = abs(b)
a_m = abs(a)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return Float64(a_m * x_45_scale_m)
end

MATLAB

y-scale_m = abs(y_45_scale);
x-scale_m = abs(x_45_scale);
b_m = abs(b);
a_m = abs(a);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = a_m * x_45_scale_m;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
a_m = N[Abs[a], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(a$95$m * x$45$scale$95$m), $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. Add Preprocessing

3. Taylor expanded in angle around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 3.0

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

5. Simplified 3.0%

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

6. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{-4 \cdot \left({a}^{3} \cdot {b}^{2}\right)}{x-scale \cdot {y-scale}^{2}}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(\mathsf{neg}\left(a\right)\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{-4 \cdot \left({a}^{3} \cdot {b}^{2}\right)}{x-scale \cdot {y-scale}^{2}}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(\mathsf{neg}\left(a\right)\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

8. Simplified 3.7%

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

9. Taylor expanded in b around 0

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

    1. *-commutative N/A

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

    2. associate-*l* N/A

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

    3. lower-*.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f64 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f64 1.9

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

11. Simplified 1.9%

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

12. Taylor expanded in a around 0

    \[\leadsto \color{blue}{a \cdot x-scale} \]
13. Step-by-step derivation

    1. lower-*.f64 23.6

       \[\leadsto \color{blue}{a \cdot x-scale} \]

14. Simplified 23.6%

    \[\leadsto \color{blue}{a \cdot x-scale} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(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))))