Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 25.0% → 79.1%

Time: 12.2s

Alternatives: 6

Speedup: 20.4×

• 33.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

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 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Java

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

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 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

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(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
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 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
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[(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]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * 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]), $MachinePrecision] * 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]), $MachinePrecision]), $MachinePrecision]]]]]

Initial Program: 25.0% accurate, 1.0× speedup

Math

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

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
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

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 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Java

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

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 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

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(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
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 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
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[(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]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * 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]), $MachinePrecision] * 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]), $MachinePrecision]), $MachinePrecision]]]]]

Alternative 1: 79.1% accurate, 14.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \left(x-scale\_m \cdot y-scale\right) \cdot \left(x-scale\_m \cdot y-scale\right)\\ t_1 := \left(a \cdot b\right) \cdot \left(a \cdot b\right)\\ \mathbf{if}\;x-scale\_m \leq 8.5 \cdot 10^{-166}:\\ \;\;\;\;-4 \cdot \frac{t\_1}{t\_0}\\ \mathbf{elif}\;x-scale\_m \leq 3.05 \cdot 10^{+124}:\\ \;\;\;\;-4 \cdot \left(\frac{a \cdot b}{x-scale\_m \cdot x-scale\_m} \cdot \frac{a \cdot b}{y-scale \cdot y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{1}{\frac{t\_0}{t\_1}}\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0 (* (* x-scale_m y-scale) (* x-scale_m y-scale)))
        (t_1 (* (* a b) (* a b))))
   (if (<= x-scale_m 8.5e-166)
     (* -4.0 (/ t_1 t_0))
     (if (<= x-scale_m 3.05e+124)
       (*
        -4.0
        (*
         (/ (* a b) (* x-scale_m x-scale_m))
         (/ (* a b) (* y-scale y-scale))))
       (* -4.0 (/ 1.0 (/ t_0 t_1)))))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = (x_45_scale_m * y_45_scale) * (x_45_scale_m * y_45_scale);
	double t_1 = (a * b) * (a * b);
	double tmp;
	if (x_45_scale_m <= 8.5e-166) {
		tmp = -4.0 * (t_1 / t_0);
	} else if (x_45_scale_m <= 3.05e+124) {
		tmp = -4.0 * (((a * b) / (x_45_scale_m * x_45_scale_m)) * ((a * b) / (y_45_scale * y_45_scale)));
	} else {
		tmp = -4.0 * (1.0 / (t_0 / t_1));
	}
	return tmp;
}

Fortran

x-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_45scale_m * y_45scale) * (x_45scale_m * y_45scale)
    t_1 = (a * b) * (a * b)
    if (x_45scale_m <= 8.5d-166) then
        tmp = (-4.0d0) * (t_1 / t_0)
    else if (x_45scale_m <= 3.05d+124) then
        tmp = (-4.0d0) * (((a * b) / (x_45scale_m * x_45scale_m)) * ((a * b) / (y_45scale * y_45scale)))
    else
        tmp = (-4.0d0) * (1.0d0 / (t_0 / t_1))
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = (x_45_scale_m * y_45_scale) * (x_45_scale_m * y_45_scale);
	double t_1 = (a * b) * (a * b);
	double tmp;
	if (x_45_scale_m <= 8.5e-166) {
		tmp = -4.0 * (t_1 / t_0);
	} else if (x_45_scale_m <= 3.05e+124) {
		tmp = -4.0 * (((a * b) / (x_45_scale_m * x_45_scale_m)) * ((a * b) / (y_45_scale * y_45_scale)));
	} else {
		tmp = -4.0 * (1.0 / (t_0 / t_1));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	t_0 = (x_45_scale_m * y_45_scale) * (x_45_scale_m * y_45_scale)
	t_1 = (a * b) * (a * b)
	tmp = 0
	if x_45_scale_m <= 8.5e-166:
		tmp = -4.0 * (t_1 / t_0)
	elif x_45_scale_m <= 3.05e+124:
		tmp = -4.0 * (((a * b) / (x_45_scale_m * x_45_scale_m)) * ((a * b) / (y_45_scale * y_45_scale)))
	else:
		tmp = -4.0 * (1.0 / (t_0 / t_1))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(Float64(x_45_scale_m * y_45_scale) * Float64(x_45_scale_m * y_45_scale))
	t_1 = Float64(Float64(a * b) * Float64(a * b))
	tmp = 0.0
	if (x_45_scale_m <= 8.5e-166)
		tmp = Float64(-4.0 * Float64(t_1 / t_0));
	elseif (x_45_scale_m <= 3.05e+124)
		tmp = Float64(-4.0 * Float64(Float64(Float64(a * b) / Float64(x_45_scale_m * x_45_scale_m)) * Float64(Float64(a * b) / Float64(y_45_scale * y_45_scale))));
	else
		tmp = Float64(-4.0 * Float64(1.0 / Float64(t_0 / t_1)));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = (x_45_scale_m * y_45_scale) * (x_45_scale_m * y_45_scale);
	t_1 = (a * b) * (a * b);
	tmp = 0.0;
	if (x_45_scale_m <= 8.5e-166)
		tmp = -4.0 * (t_1 / t_0);
	elseif (x_45_scale_m <= 3.05e+124)
		tmp = -4.0 * (((a * b) / (x_45_scale_m * x_45_scale_m)) * ((a * b) / (y_45_scale * y_45_scale)));
	else
		tmp = -4.0 * (1.0 / (t_0 / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(N[(x$45$scale$95$m * y$45$scale), $MachinePrecision] * N[(x$45$scale$95$m * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 8.5e-166], N[(-4.0 * N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale$95$m, 3.05e+124], N[(-4.0 * N[(N[(N[(a * b), $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(1.0 / N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < 8.5e-166

   1. Initial program 25.0%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 48.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 60.9

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

   6. Applied rewrites 60.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 78.6

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

   8. Applied rewrites 78.6%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 78.6

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 78.6

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

   10. Applied rewrites 78.6%

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

   if 8.5e-166 < x-scale < 3.0500000000000001e124

   1. Initial program 25.0%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 48.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 60.9

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

   6. Applied rewrites 60.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 78.6

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

   8. Applied rewrites 78.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 66.5

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

   10. Applied rewrites 66.5%

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

   if 3.0500000000000001e124 < x-scale

   1. Initial program 25.0%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 48.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 60.9

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

   6. Applied rewrites 60.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 78.6

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

   8. Applied rewrites 78.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-prod-down N/A

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

      4. lower-/.f64 N/A

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

      5. div-flip N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow-prod-down N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 78.4

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 78.4

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

   10. Applied rewrites 78.4%

       \[\leadsto -4 \cdot \frac{1}{\color{blue}{\frac{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 79.0% accurate, 14.4× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \left(x-scale\_m \cdot y-scale\right) \cdot \left(x-scale\_m \cdot y-scale\right)\\ t_1 := \left(a \cdot b\right) \cdot \left(a \cdot b\right)\\ \mathbf{if}\;x-scale\_m \leq 8.5 \cdot 10^{-166}:\\ \;\;\;\;-4 \cdot \frac{t\_1}{t\_0}\\ \mathbf{elif}\;x-scale\_m \leq 3.05 \cdot 10^{+124}:\\ \;\;\;\;-4 \cdot \left(\frac{a \cdot b}{x-scale\_m \cdot x-scale\_m} \cdot \frac{a \cdot b}{y-scale \cdot y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(t\_1 \cdot \frac{1}{t\_0}\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0 (* (* x-scale_m y-scale) (* x-scale_m y-scale)))
        (t_1 (* (* a b) (* a b))))
   (if (<= x-scale_m 8.5e-166)
     (* -4.0 (/ t_1 t_0))
     (if (<= x-scale_m 3.05e+124)
       (*
        -4.0
        (*
         (/ (* a b) (* x-scale_m x-scale_m))
         (/ (* a b) (* y-scale y-scale))))
       (* -4.0 (* t_1 (/ 1.0 t_0)))))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = (x_45_scale_m * y_45_scale) * (x_45_scale_m * y_45_scale);
	double t_1 = (a * b) * (a * b);
	double tmp;
	if (x_45_scale_m <= 8.5e-166) {
		tmp = -4.0 * (t_1 / t_0);
	} else if (x_45_scale_m <= 3.05e+124) {
		tmp = -4.0 * (((a * b) / (x_45_scale_m * x_45_scale_m)) * ((a * b) / (y_45_scale * y_45_scale)));
	} else {
		tmp = -4.0 * (t_1 * (1.0 / t_0));
	}
	return tmp;
}

Fortran

x-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_45scale_m * y_45scale) * (x_45scale_m * y_45scale)
    t_1 = (a * b) * (a * b)
    if (x_45scale_m <= 8.5d-166) then
        tmp = (-4.0d0) * (t_1 / t_0)
    else if (x_45scale_m <= 3.05d+124) then
        tmp = (-4.0d0) * (((a * b) / (x_45scale_m * x_45scale_m)) * ((a * b) / (y_45scale * y_45scale)))
    else
        tmp = (-4.0d0) * (t_1 * (1.0d0 / t_0))
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = (x_45_scale_m * y_45_scale) * (x_45_scale_m * y_45_scale);
	double t_1 = (a * b) * (a * b);
	double tmp;
	if (x_45_scale_m <= 8.5e-166) {
		tmp = -4.0 * (t_1 / t_0);
	} else if (x_45_scale_m <= 3.05e+124) {
		tmp = -4.0 * (((a * b) / (x_45_scale_m * x_45_scale_m)) * ((a * b) / (y_45_scale * y_45_scale)));
	} else {
		tmp = -4.0 * (t_1 * (1.0 / t_0));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	t_0 = (x_45_scale_m * y_45_scale) * (x_45_scale_m * y_45_scale)
	t_1 = (a * b) * (a * b)
	tmp = 0
	if x_45_scale_m <= 8.5e-166:
		tmp = -4.0 * (t_1 / t_0)
	elif x_45_scale_m <= 3.05e+124:
		tmp = -4.0 * (((a * b) / (x_45_scale_m * x_45_scale_m)) * ((a * b) / (y_45_scale * y_45_scale)))
	else:
		tmp = -4.0 * (t_1 * (1.0 / t_0))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(Float64(x_45_scale_m * y_45_scale) * Float64(x_45_scale_m * y_45_scale))
	t_1 = Float64(Float64(a * b) * Float64(a * b))
	tmp = 0.0
	if (x_45_scale_m <= 8.5e-166)
		tmp = Float64(-4.0 * Float64(t_1 / t_0));
	elseif (x_45_scale_m <= 3.05e+124)
		tmp = Float64(-4.0 * Float64(Float64(Float64(a * b) / Float64(x_45_scale_m * x_45_scale_m)) * Float64(Float64(a * b) / Float64(y_45_scale * y_45_scale))));
	else
		tmp = Float64(-4.0 * Float64(t_1 * Float64(1.0 / t_0)));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = (x_45_scale_m * y_45_scale) * (x_45_scale_m * y_45_scale);
	t_1 = (a * b) * (a * b);
	tmp = 0.0;
	if (x_45_scale_m <= 8.5e-166)
		tmp = -4.0 * (t_1 / t_0);
	elseif (x_45_scale_m <= 3.05e+124)
		tmp = -4.0 * (((a * b) / (x_45_scale_m * x_45_scale_m)) * ((a * b) / (y_45_scale * y_45_scale)));
	else
		tmp = -4.0 * (t_1 * (1.0 / t_0));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(N[(x$45$scale$95$m * y$45$scale), $MachinePrecision] * N[(x$45$scale$95$m * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 8.5e-166], N[(-4.0 * N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale$95$m, 3.05e+124], N[(-4.0 * N[(N[(N[(a * b), $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(t$95$1 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < 8.5e-166

   1. Initial program 25.0%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 48.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 60.9

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

   6. Applied rewrites 60.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 78.6

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

   8. Applied rewrites 78.6%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 78.6

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 78.6

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

   10. Applied rewrites 78.6%

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

   if 8.5e-166 < x-scale < 3.0500000000000001e124

   1. Initial program 25.0%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 48.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 60.9

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

   6. Applied rewrites 60.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 78.6

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

   8. Applied rewrites 78.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 66.5

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

   10. Applied rewrites 66.5%

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

   if 3.0500000000000001e124 < x-scale

   1. Initial program 25.0%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 48.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 60.9

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

   6. Applied rewrites 60.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 78.6

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

   8. Applied rewrites 78.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-prod-down N/A

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

      4. lower-/.f64 N/A

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

      5. mult-flip N/A

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

      6. lower-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 78.3

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

   10. Applied rewrites 78.3%

       \[\leadsto -4 \cdot \left(\left(\left(a \cdot b\right) \cdot \left(a \cdot b\right)\right) \cdot \color{blue}{\frac{1}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 79.0% accurate, 15.5× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := -4 \cdot \frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{\left(x-scale\_m \cdot y-scale\right) \cdot \left(x-scale\_m \cdot y-scale\right)}\\ \mathbf{if}\;x-scale\_m \leq 8.5 \cdot 10^{-166}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x-scale\_m \leq 3.05 \cdot 10^{+124}:\\ \;\;\;\;-4 \cdot \left(\frac{a \cdot b}{x-scale\_m \cdot x-scale\_m} \cdot \frac{a \cdot b}{y-scale \cdot y-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0
         (*
          -4.0
          (/
           (* (* a b) (* a b))
           (* (* x-scale_m y-scale) (* x-scale_m y-scale))))))
   (if (<= x-scale_m 8.5e-166)
     t_0
     (if (<= x-scale_m 3.05e+124)
       (*
        -4.0
        (*
         (/ (* a b) (* x-scale_m x-scale_m))
         (/ (* a b) (* y-scale y-scale))))
       t_0))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = -4.0 * (((a * b) * (a * b)) / ((x_45_scale_m * y_45_scale) * (x_45_scale_m * y_45_scale)));
	double tmp;
	if (x_45_scale_m <= 8.5e-166) {
		tmp = t_0;
	} else if (x_45_scale_m <= 3.05e+124) {
		tmp = -4.0 * (((a * b) / (x_45_scale_m * x_45_scale_m)) * ((a * b) / (y_45_scale * y_45_scale)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

x-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-4.0d0) * (((a * b) * (a * b)) / ((x_45scale_m * y_45scale) * (x_45scale_m * y_45scale)))
    if (x_45scale_m <= 8.5d-166) then
        tmp = t_0
    else if (x_45scale_m <= 3.05d+124) then
        tmp = (-4.0d0) * (((a * b) / (x_45scale_m * x_45scale_m)) * ((a * b) / (y_45scale * y_45scale)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = -4.0 * (((a * b) * (a * b)) / ((x_45_scale_m * y_45_scale) * (x_45_scale_m * y_45_scale)));
	double tmp;
	if (x_45_scale_m <= 8.5e-166) {
		tmp = t_0;
	} else if (x_45_scale_m <= 3.05e+124) {
		tmp = -4.0 * (((a * b) / (x_45_scale_m * x_45_scale_m)) * ((a * b) / (y_45_scale * y_45_scale)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	t_0 = -4.0 * (((a * b) * (a * b)) / ((x_45_scale_m * y_45_scale) * (x_45_scale_m * y_45_scale)))
	tmp = 0
	if x_45_scale_m <= 8.5e-166:
		tmp = t_0
	elif x_45_scale_m <= 3.05e+124:
		tmp = -4.0 * (((a * b) / (x_45_scale_m * x_45_scale_m)) * ((a * b) / (y_45_scale * y_45_scale)))
	else:
		tmp = t_0
	return tmp

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(-4.0 * Float64(Float64(Float64(a * b) * Float64(a * b)) / Float64(Float64(x_45_scale_m * y_45_scale) * Float64(x_45_scale_m * y_45_scale))))
	tmp = 0.0
	if (x_45_scale_m <= 8.5e-166)
		tmp = t_0;
	elseif (x_45_scale_m <= 3.05e+124)
		tmp = Float64(-4.0 * Float64(Float64(Float64(a * b) / Float64(x_45_scale_m * x_45_scale_m)) * Float64(Float64(a * b) / Float64(y_45_scale * y_45_scale))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale)
	t_0 = -4.0 * (((a * b) * (a * b)) / ((x_45_scale_m * y_45_scale) * (x_45_scale_m * y_45_scale)));
	tmp = 0.0;
	if (x_45_scale_m <= 8.5e-166)
		tmp = t_0;
	elseif (x_45_scale_m <= 3.05e+124)
		tmp = -4.0 * (((a * b) / (x_45_scale_m * x_45_scale_m)) * ((a * b) / (y_45_scale * y_45_scale)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(-4.0 * N[(N[(N[(a * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale$95$m * y$45$scale), $MachinePrecision] * N[(x$45$scale$95$m * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 8.5e-166], t$95$0, If[LessEqual[x$45$scale$95$m, 3.05e+124], N[(-4.0 * N[(N[(N[(a * b), $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 8.5e-166 or 3.0500000000000001e124 < x-scale

   1. Initial program 25.0%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 48.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 60.9

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

   6. Applied rewrites 60.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 78.6

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

   8. Applied rewrites 78.6%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 78.6

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 78.6

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

   10. Applied rewrites 78.6%

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

   if 8.5e-166 < x-scale < 3.0500000000000001e124

   1. Initial program 25.0%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   4. Applied rewrites 48.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 60.9

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

   6. Applied rewrites 60.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 78.6

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

   8. Applied rewrites 78.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 66.5

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

   10. Applied rewrites 66.5%

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

Alternative 4: 78.6% accurate, 20.4× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ -4 \cdot \frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{\left(x-scale\_m \cdot y-scale\right) \cdot \left(x-scale\_m \cdot y-scale\right)} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (*
  -4.0
  (/ (* (* a b) (* a b)) (* (* x-scale_m y-scale) (* x-scale_m y-scale)))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return -4.0 * (((a * b) * (a * b)) / ((x_45_scale_m * y_45_scale) * (x_45_scale_m * y_45_scale)));
}

Fortran

x-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    code = (-4.0d0) * (((a * b) * (a * b)) / ((x_45scale_m * y_45scale) * (x_45scale_m * y_45scale)))
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return -4.0 * (((a * b) * (a * b)) / ((x_45_scale_m * y_45_scale) * (x_45_scale_m * y_45_scale)));
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	return -4.0 * (((a * b) * (a * b)) / ((x_45_scale_m * y_45_scale) * (x_45_scale_m * y_45_scale)))

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	return Float64(-4.0 * Float64(Float64(Float64(a * b) * Float64(a * b)) / Float64(Float64(x_45_scale_m * y_45_scale) * Float64(x_45_scale_m * y_45_scale))))
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = -4.0 * (((a * b) * (a * b)) / ((x_45_scale_m * y_45_scale) * (x_45_scale_m * y_45_scale)));
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := N[(-4.0 * N[(N[(N[(a * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale$95$m * y$45$scale), $MachinePrecision] * N[(x$45$scale$95$m * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.0%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 48.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. pow2 N/A

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

   6. pow-prod-down N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-*.f64 60.9

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

6. Applied rewrites 60.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. pow2 N/A

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

   6. pow-prod-down N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-*.f64 78.6

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

8. Applied rewrites 78.6%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 78.6

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 78.6

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

10. Applied rewrites 78.6%

    \[\leadsto -4 \cdot \color{blue}{\frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}} \]
11. Add Preprocessing

Alternative 5: 60.9% accurate, 20.4× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ -4 \cdot \frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{\left(x-scale\_m \cdot x-scale\_m\right) \cdot \left(y-scale \cdot y-scale\right)} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (*
  -4.0
  (/ (* (* a b) (* a b)) (* (* x-scale_m x-scale_m) (* y-scale y-scale)))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return -4.0 * (((a * b) * (a * b)) / ((x_45_scale_m * x_45_scale_m) * (y_45_scale * y_45_scale)));
}

Fortran

x-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    code = (-4.0d0) * (((a * b) * (a * b)) / ((x_45scale_m * x_45scale_m) * (y_45scale * y_45scale)))
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return -4.0 * (((a * b) * (a * b)) / ((x_45_scale_m * x_45_scale_m) * (y_45_scale * y_45_scale)));
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	return -4.0 * (((a * b) * (a * b)) / ((x_45_scale_m * x_45_scale_m) * (y_45_scale * y_45_scale)))

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	return Float64(-4.0 * Float64(Float64(Float64(a * b) * Float64(a * b)) / Float64(Float64(x_45_scale_m * x_45_scale_m) * Float64(y_45_scale * y_45_scale))))
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = -4.0 * (((a * b) * (a * b)) / ((x_45_scale_m * x_45_scale_m) * (y_45_scale * y_45_scale)));
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := N[(-4.0 * N[(N[(N[(a * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.0%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 48.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. pow2 N/A

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

   6. pow-prod-down N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-*.f64 60.9

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

6. Applied rewrites 60.9%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 60.9

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

8. Applied rewrites 60.9%

   \[\leadsto -4 \cdot \frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{\color{blue}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}} \]
9. Add Preprocessing

Alternative 6: 48.4% accurate, 20.4× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ -4 \cdot \frac{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}{\left(x-scale\_m \cdot x-scale\_m\right) \cdot \left(y-scale \cdot y-scale\right)} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
(FPCore (a b angle x-scale_m y-scale)
 :precision binary64
 (*
  -4.0
  (/ (* (* a a) (* b b)) (* (* x-scale_m x-scale_m) (* y-scale y-scale)))))

C

x-scale_m = fabs(x_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return -4.0 * (((a * a) * (b * b)) / ((x_45_scale_m * x_45_scale_m) * (y_45_scale * y_45_scale)));
}

Fortran

x-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale_m, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale
    code = (-4.0d0) * (((a * a) * (b * b)) / ((x_45scale_m * x_45scale_m) * (y_45scale * y_45scale)))
end function

Java

x-scale_m = Math.abs(x_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale) {
	return -4.0 * (((a * a) * (b * b)) / ((x_45_scale_m * x_45_scale_m) * (y_45_scale * y_45_scale)));
}

Python

x-scale_m = math.fabs(x_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale):
	return -4.0 * (((a * a) * (b * b)) / ((x_45_scale_m * x_45_scale_m) * (y_45_scale * y_45_scale)))

Julia

x-scale_m = abs(x_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale)
	return Float64(-4.0 * Float64(Float64(Float64(a * a) * Float64(b * b)) / Float64(Float64(x_45_scale_m * x_45_scale_m) * Float64(y_45_scale * y_45_scale))))
end

MATLAB

x-scale_m = abs(x_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale)
	tmp = -4.0 * (((a * a) * (b * b)) / ((x_45_scale_m * x_45_scale_m) * (y_45_scale * y_45_scale)));
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale_] := N[(-4.0 * N[(N[(N[(a * a), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.0%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

4. Applied rewrites 48.4%

   \[\leadsto \color{blue}{-4 \cdot \frac{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}{\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot y-scale\right)}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(FPCore (a b angle x-scale y-scale)
  :name "Simplification of discriminant from scale-rotated-ellipse"
  :precision binary64
  (- (* (/ (/ (* (* (* 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 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale)) (* (* 4.0 (/ (/ (+ (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))))