Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.3% → 77.9%

Time: 9.4s

Alternatives: 4

Speedup: 20.4×

• 34.9% 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: 24.3% 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: 77.9% accurate, 17.2× speedup

Math

\[\begin{array}{l} y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 6.8 \cdot 10^{-232}:\\ \;\;\;\;\left(\frac{a \cdot a}{x-scale \cdot x-scale} \cdot \left(\frac{b}{y-scale\_m} \cdot \frac{b}{y-scale\_m}\right)\right) \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{\left(x-scale \cdot y-scale\_m\right) \cdot \left(x-scale \cdot y-scale\_m\right)} \cdot -4\\ \end{array} \end{array} \]

FPCore

y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale y-scale_m)
 :precision binary64
 (if (<= y-scale_m 6.8e-232)
   (*
    (* (/ (* a a) (* x-scale x-scale)) (* (/ b y-scale_m) (/ b y-scale_m)))
    -4.0)
   (*
    (/ (* (* a b) (* a b)) (* (* x-scale y-scale_m) (* x-scale y-scale_m)))
    -4.0)))

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 6.8e-232) {
		tmp = (((a * a) / (x_45_scale * x_45_scale)) * ((b / y_45_scale_m) * (b / y_45_scale_m))) * -4.0;
	} else {
		tmp = (((a * b) * (a * b)) / ((x_45_scale * y_45_scale_m) * (x_45_scale * y_45_scale_m))) * -4.0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (y_45scale_m <= 6.8d-232) then
        tmp = (((a * a) / (x_45scale * x_45scale)) * ((b / y_45scale_m) * (b / y_45scale_m))) * (-4.0d0)
    else
        tmp = (((a * b) * (a * b)) / ((x_45scale * y_45scale_m) * (x_45scale * y_45scale_m))) * (-4.0d0)
    end if
    code = tmp
end function

Java

y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 6.8e-232) {
		tmp = (((a * a) / (x_45_scale * x_45_scale)) * ((b / y_45_scale_m) * (b / y_45_scale_m))) * -4.0;
	} else {
		tmp = (((a * b) * (a * b)) / ((x_45_scale * y_45_scale_m) * (x_45_scale * y_45_scale_m))) * -4.0;
	}
	return tmp;
}

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 6.8e-232:
		tmp = (((a * a) / (x_45_scale * x_45_scale)) * ((b / y_45_scale_m) * (b / y_45_scale_m))) * -4.0
	else:
		tmp = (((a * b) * (a * b)) / ((x_45_scale * y_45_scale_m) * (x_45_scale * y_45_scale_m))) * -4.0
	return tmp

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 6.8e-232)
		tmp = Float64(Float64(Float64(Float64(a * a) / Float64(x_45_scale * x_45_scale)) * Float64(Float64(b / y_45_scale_m) * Float64(b / y_45_scale_m))) * -4.0);
	else
		tmp = Float64(Float64(Float64(Float64(a * b) * Float64(a * b)) / Float64(Float64(x_45_scale * y_45_scale_m) * Float64(x_45_scale * y_45_scale_m))) * -4.0);
	end
	return tmp
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = 0.0;
	if (y_45_scale_m <= 6.8e-232)
		tmp = (((a * a) / (x_45_scale * x_45_scale)) * ((b / y_45_scale_m) * (b / y_45_scale_m))) * -4.0;
	else
		tmp = (((a * b) * (a * b)) / ((x_45_scale * y_45_scale_m) * (x_45_scale * y_45_scale_m))) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 6.8e-232], N[(N[(N[(N[(a * a), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(b / y$45$scale$95$m), $MachinePrecision] * N[(b / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(N[(N[(a * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale$95$m), $MachinePrecision] * N[(x$45$scale * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 6.8000000000000004e-232

   1. Initial program 6.6%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 41.6%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 39.9

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

   7. Applied rewrites 39.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. pow2 N/A

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

      21. lift-*.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 40.1

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

   9. Applied rewrites 40.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 59.3

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

   11. Applied rewrites 59.3%

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

   if 6.8000000000000004e-232 < y-scale

   1. Initial program 27.0%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 49.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 49.1

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

   7. Applied rewrites 49.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 60.9

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

   9. Applied rewrites 60.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. unswap-sqr N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 77.8

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

       8. lift-*.f64 N/A

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

       9. lift-pow.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 77.8

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

   11. Applied rewrites 77.8%

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

Alternative 2: 75.4% accurate, 20.4× speedup

Math

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

FPCore

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

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return (((a * b) * (a * b)) / ((x_45_scale * y_45_scale_m) * (x_45_scale * y_45_scale_m))) * -4.0;
}

Fortran

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

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    code = (((a * b) * (a * b)) / ((x_45scale * y_45scale_m) * (x_45scale * y_45scale_m))) * (-4.0d0)
end function

Java

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

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	return (((a * b) * (a * b)) / ((x_45_scale * y_45_scale_m) * (x_45_scale * y_45_scale_m))) * -4.0

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	return Float64(Float64(Float64(Float64(a * b) * Float64(a * b)) / Float64(Float64(x_45_scale * y_45_scale_m) * Float64(x_45_scale * y_45_scale_m))) * -4.0)
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = (((a * b) * (a * b)) / ((x_45_scale * y_45_scale_m) * (x_45_scale * y_45_scale_m))) * -4.0;
end

Wolfram

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

Derivation

1. Initial program 24.3%

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 48.5%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 47.8

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

7. Applied rewrites 47.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. pow-prod-down N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-*.f64 60.8

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

9. Applied rewrites 60.8%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. unswap-sqr N/A

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

    5. lower-*.f64 N/A

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

    6. lower-*.f64 N/A

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

    7. lower-*.f64 77.9

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

    8. lift-*.f64 N/A

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

    9. lift-pow.f64 N/A

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

    10. unpow2 N/A

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

    11. lower-*.f64 N/A

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

    12. lift-*.f64 N/A

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

    13. lift-*.f64 77.9

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

11. Applied rewrites 77.9%

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

Alternative 3: 60.8% accurate, 20.4× speedup

Math

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

FPCore

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

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return (((a * a) * (b * b)) / ((x_45_scale * y_45_scale_m) * (x_45_scale * y_45_scale_m))) * -4.0;
}

Fortran

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

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    code = (((a * a) * (b * b)) / ((x_45scale * y_45scale_m) * (x_45scale * y_45scale_m))) * (-4.0d0)
end function

Java

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

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	return (((a * a) * (b * b)) / ((x_45_scale * y_45_scale_m) * (x_45_scale * y_45_scale_m))) * -4.0

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	return Float64(Float64(Float64(Float64(a * a) * Float64(b * b)) / Float64(Float64(x_45_scale * y_45_scale_m) * Float64(x_45_scale * y_45_scale_m))) * -4.0)
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = (((a * a) * (b * b)) / ((x_45_scale * y_45_scale_m) * (x_45_scale * y_45_scale_m))) * -4.0;
end

Wolfram

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

Derivation

1. Initial program 24.3%

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 48.5%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 47.8

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

7. Applied rewrites 47.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. pow-prod-down N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-*.f64 60.8

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

9. Applied rewrites 60.8%

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

    1. lift-*.f64 N/A

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

    2. lift-pow.f64 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 60.8

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

11. Applied rewrites 60.8%

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

Alternative 4: 47.8% accurate, 20.4× speedup

Math

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

FPCore

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

C

y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale, double y_45_scale_m) {
	return (((a * a) * (b * b)) / ((x_45_scale * x_45_scale) * (y_45_scale_m * y_45_scale_m))) * -4.0;
}

Fortran

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

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

real(8) function code(a, b, angle, x_45scale, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale_m
    code = (((a * a) * (b * b)) / ((x_45scale * x_45scale) * (y_45scale_m * y_45scale_m))) * (-4.0d0)
end function

Java

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

Python

y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale, y_45_scale_m):
	return (((a * a) * (b * b)) / ((x_45_scale * x_45_scale) * (y_45_scale_m * y_45_scale_m))) * -4.0

Julia

y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale, y_45_scale_m)
	return Float64(Float64(Float64(Float64(a * a) * Float64(b * b)) / Float64(Float64(x_45_scale * x_45_scale) * Float64(y_45_scale_m * y_45_scale_m))) * -4.0)
end

MATLAB

y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale, y_45_scale_m)
	tmp = (((a * a) * (b * b)) / ((x_45_scale * x_45_scale) * (y_45_scale_m * y_45_scale_m))) * -4.0;
end

Wolfram

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

Derivation

1. Initial program 24.3%

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 48.5%

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

5. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 47.8

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

7. Applied rewrites 47.8%

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

Reproduce

herbie shell --seed 2025117 
(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))))