Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.3% → 90.6%

Time: 15.1s

Alternatives: 7

Speedup: 20.4×

• 34.4% 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: 90.6% accurate, 20.1× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (* a b) (* (* a (/ b (* y-scale x-scale))) (/ -4.0 (* y-scale x-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (a * b) * ((a * (b / (y_45_scale * x_45_scale))) * (-4.0 / (y_45_scale * x_45_scale)));
}

Fortran

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)
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
    code = (a * b) * ((a * (b / (y_45scale * x_45scale))) * ((-4.0d0) / (y_45scale * x_45scale)))
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (a * b) * ((a * (b / (y_45_scale * x_45_scale))) * (-4.0 / (y_45_scale * x_45_scale)));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (a * b) * ((a * (b / (y_45_scale * x_45_scale))) * (-4.0 / (y_45_scale * x_45_scale)))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(a * b) * Float64(Float64(a * Float64(b / Float64(y_45_scale * x_45_scale))) * Float64(-4.0 / Float64(y_45_scale * x_45_scale))))
end

MATLAB

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

Wolfram

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

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 47.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-prod-down N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 64.4

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-pow.f64 N/A

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

   18. pow2 N/A

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

   19. lift-pow.f64 N/A

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

   20. pow2 N/A

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

   21. unswap-sqr N/A

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

   22. lower-*.f64 N/A

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

   23. lower-*.f64 N/A

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

   24. lower-*.f64 82.9

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

6. Applied rewrites 82.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 90.6

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

8. Applied rewrites 90.6%

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

Alternative 2: 89.9% accurate, 20.1× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* a (* b (* (/ -4.0 (* y-scale x-scale)) (* (/ b (* y-scale x-scale)) a)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return a * (b * ((-4.0 / (y_45_scale * x_45_scale)) * ((b / (y_45_scale * x_45_scale)) * a)));
}

Fortran

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)
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
    code = a * (b * (((-4.0d0) / (y_45scale * x_45scale)) * ((b / (y_45scale * x_45scale)) * a)))
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return a * (b * ((-4.0 / (y_45_scale * x_45_scale)) * ((b / (y_45_scale * x_45_scale)) * a)));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return a * (b * ((-4.0 / (y_45_scale * x_45_scale)) * ((b / (y_45_scale * x_45_scale)) * a)))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(a * Float64(b * Float64(Float64(-4.0 / Float64(y_45_scale * x_45_scale)) * Float64(Float64(b / Float64(y_45_scale * x_45_scale)) * a))))
end

MATLAB

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

Wolfram

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

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 47.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-prod-down N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 64.4

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-pow.f64 N/A

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

   18. pow2 N/A

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

   19. lift-pow.f64 N/A

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

   20. pow2 N/A

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

   21. unswap-sqr N/A

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

   22. lower-*.f64 N/A

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

   23. lower-*.f64 N/A

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

   24. lower-*.f64 82.9

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

6. Applied rewrites 82.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 90.6

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

8. Applied rewrites 90.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 89.9

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 89.9

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 89.9

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

10. Applied rewrites 89.9%

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

Alternative 3: 77.8% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale\\ \mathbf{if}\;x-scale \leq 9 \cdot 10^{-100}:\\ \;\;\;\;-4 \cdot \frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{t\_0}\\ \mathbf{elif}\;x-scale \leq 6 \cdot 10^{+110}:\\ \;\;\;\;-4 \cdot \frac{a \cdot \left(b \cdot \frac{a \cdot b}{\left(y-scale \cdot x-scale\right) \cdot x-scale}\right)}{y-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{b}{t\_0} \cdot \left(a \cdot b\right)\right) \cdot a\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* (* y-scale x-scale) y-scale) x-scale)))
   (if (<= x-scale 9e-100)
     (* -4.0 (/ (* (* b a) (* b a)) t_0))
     (if (<= x-scale 6e+110)
       (*
        -4.0
        (/ (* a (* b (/ (* a b) (* (* y-scale x-scale) x-scale)))) y-scale))
       (* (* (* (/ b t_0) (* a b)) a) -4.0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale;
	double tmp;
	if (x_45_scale <= 9e-100) {
		tmp = -4.0 * (((b * a) * (b * a)) / t_0);
	} else if (x_45_scale <= 6e+110) {
		tmp = -4.0 * ((a * (b * ((a * b) / ((y_45_scale * x_45_scale) * x_45_scale)))) / y_45_scale);
	} else {
		tmp = (((b / t_0) * (a * b)) * a) * -4.0;
	}
	return tmp;
}

Fortran

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)
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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y_45scale * x_45scale) * y_45scale) * x_45scale
    if (x_45scale <= 9d-100) then
        tmp = (-4.0d0) * (((b * a) * (b * a)) / t_0)
    else if (x_45scale <= 6d+110) then
        tmp = (-4.0d0) * ((a * (b * ((a * b) / ((y_45scale * x_45scale) * x_45scale)))) / y_45scale)
    else
        tmp = (((b / t_0) * (a * b)) * a) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale;
	double tmp;
	if (x_45_scale <= 9e-100) {
		tmp = -4.0 * (((b * a) * (b * a)) / t_0);
	} else if (x_45_scale <= 6e+110) {
		tmp = -4.0 * ((a * (b * ((a * b) / ((y_45_scale * x_45_scale) * x_45_scale)))) / y_45_scale);
	} else {
		tmp = (((b / t_0) * (a * b)) * a) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = ((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale
	tmp = 0
	if x_45_scale <= 9e-100:
		tmp = -4.0 * (((b * a) * (b * a)) / t_0)
	elif x_45_scale <= 6e+110:
		tmp = -4.0 * ((a * (b * ((a * b) / ((y_45_scale * x_45_scale) * x_45_scale)))) / y_45_scale)
	else:
		tmp = (((b / t_0) * (a * b)) * a) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)
	tmp = 0.0
	if (x_45_scale <= 9e-100)
		tmp = Float64(-4.0 * Float64(Float64(Float64(b * a) * Float64(b * a)) / t_0));
	elseif (x_45_scale <= 6e+110)
		tmp = Float64(-4.0 * Float64(Float64(a * Float64(b * Float64(Float64(a * b) / Float64(Float64(y_45_scale * x_45_scale) * x_45_scale)))) / y_45_scale));
	else
		tmp = Float64(Float64(Float64(Float64(b / t_0) * Float64(a * b)) * a) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = ((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale;
	tmp = 0.0;
	if (x_45_scale <= 9e-100)
		tmp = -4.0 * (((b * a) * (b * a)) / t_0);
	elseif (x_45_scale <= 6e+110)
		tmp = -4.0 * ((a * (b * ((a * b) / ((y_45_scale * x_45_scale) * x_45_scale)))) / y_45_scale);
	else
		tmp = (((b / t_0) * (a * b)) * a) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]}, If[LessEqual[x$45$scale, 9e-100], N[(-4.0 * N[(N[(N[(b * a), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 6e+110], N[(-4.0 * N[(N[(a * N[(b * N[(N[(a * b), $MachinePrecision] / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b / t$95$0), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * -4.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < 9.0000000000000002e-100

   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. 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

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 60.4

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-prod-down N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 74.7

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

   6. Applied rewrites 74.7%

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

   if 9.0000000000000002e-100 < x-scale < 6.00000000000000014e110

   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. 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

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-prod-down N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-pow.f64 N/A

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

      18. pow2 N/A

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

      19. lift-pow.f64 N/A

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

      20. pow2 N/A

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

      21. unswap-sqr N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 82.9

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

   6. Applied rewrites 82.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 75.8

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

   8. Applied rewrites 79.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 79.6

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

   10. Applied rewrites 79.6%

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

   if 6.00000000000000014e110 < x-scale

   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. 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

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-prod-down N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-pow.f64 N/A

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

      18. pow2 N/A

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

      19. lift-pow.f64 N/A

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

      20. pow2 N/A

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

      21. unswap-sqr N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 82.9

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

   6. Applied rewrites 82.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

   8. Applied rewrites 74.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 77.8

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

   10. Applied rewrites 77.8%

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

Alternative 4: 77.7% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{-124}:\\ \;\;\;\;-4 \cdot \frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot \left(\left(\left(\frac{b}{\left(y-scale \cdot x-scale\right) \cdot x-scale} \cdot a\right) \cdot b\right) \cdot a\right)}{y-scale}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= a 1.5e-124)
   (* -4.0 (/ (* (* b a) (* b a)) (* (* (* y-scale x-scale) y-scale) x-scale)))
   (/
    (* -4.0 (* (* (* (/ b (* (* y-scale x-scale) x-scale)) a) b) a))
    y-scale)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 1.5e-124) {
		tmp = -4.0 * (((b * a) * (b * a)) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale));
	} else {
		tmp = (-4.0 * ((((b / ((y_45_scale * x_45_scale) * x_45_scale)) * a) * b) * a)) / y_45_scale;
	}
	return tmp;
}

Fortran

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)
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
    real(8) :: tmp
    if (a <= 1.5d-124) then
        tmp = (-4.0d0) * (((b * a) * (b * a)) / (((y_45scale * x_45scale) * y_45scale) * x_45scale))
    else
        tmp = ((-4.0d0) * ((((b / ((y_45scale * x_45scale) * x_45scale)) * a) * b) * a)) / y_45scale
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 1.5e-124) {
		tmp = -4.0 * (((b * a) * (b * a)) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale));
	} else {
		tmp = (-4.0 * ((((b / ((y_45_scale * x_45_scale) * x_45_scale)) * a) * b) * a)) / y_45_scale;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a <= 1.5e-124:
		tmp = -4.0 * (((b * a) * (b * a)) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))
	else:
		tmp = (-4.0 * ((((b / ((y_45_scale * x_45_scale) * x_45_scale)) * a) * b) * a)) / y_45_scale
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a <= 1.5e-124)
		tmp = Float64(-4.0 * Float64(Float64(Float64(b * a) * Float64(b * a)) / Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)));
	else
		tmp = Float64(Float64(-4.0 * Float64(Float64(Float64(Float64(b / Float64(Float64(y_45_scale * x_45_scale) * x_45_scale)) * a) * b) * a)) / y_45_scale);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a <= 1.5e-124)
		tmp = -4.0 * (((b * a) * (b * a)) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale));
	else
		tmp = (-4.0 * ((((b / ((y_45_scale * x_45_scale) * x_45_scale)) * a) * b) * a)) / y_45_scale;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a, 1.5e-124], N[(-4.0 * N[(N[(N[(b * a), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * N[(N[(N[(N[(b / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.5e-124

   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. 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

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 60.4

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-prod-down N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 74.7

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

   6. Applied rewrites 74.7%

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

   if 1.5e-124 < a

   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. 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

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-prod-down N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-pow.f64 N/A

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

      18. pow2 N/A

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

      19. lift-pow.f64 N/A

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

      20. pow2 N/A

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

      21. unswap-sqr N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 82.9

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

   6. Applied rewrites 82.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 75.8

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

   8. Applied rewrites 79.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 79.1

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 79.1

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 79.1

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 79.1

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

   10. Applied rewrites 79.1%

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

Alternative 5: 77.3% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.1 \cdot 10^{-108}:\\ \;\;\;\;-4 \cdot \frac{\left(b \cdot a\right) \cdot \left(b \cdot a\right)}{\left(\left(y-scale \cdot x-scale\right) \cdot y-scale\right) \cdot x-scale}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\frac{b}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)} \cdot b\right) \cdot a\right) \cdot a\right) \cdot -4\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (if (<= a 3.1e-108)
   (* -4.0 (/ (* (* b a) (* b a)) (* (* (* y-scale x-scale) y-scale) x-scale)))
   (*
    (* (* (* (/ b (* (* y-scale x-scale) (* y-scale x-scale))) b) a) a)
    -4.0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 3.1e-108) {
		tmp = -4.0 * (((b * a) * (b * a)) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale));
	} else {
		tmp = ((((b / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b) * a) * a) * -4.0;
	}
	return tmp;
}

Fortran

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)
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
    real(8) :: tmp
    if (a <= 3.1d-108) then
        tmp = (-4.0d0) * (((b * a) * (b * a)) / (((y_45scale * x_45scale) * y_45scale) * x_45scale))
    else
        tmp = ((((b / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * b) * a) * a) * (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 3.1e-108) {
		tmp = -4.0 * (((b * a) * (b * a)) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale));
	} else {
		tmp = ((((b / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b) * a) * a) * -4.0;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a <= 3.1e-108:
		tmp = -4.0 * (((b * a) * (b * a)) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale))
	else:
		tmp = ((((b / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b) * a) * a) * -4.0
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a <= 3.1e-108)
		tmp = Float64(-4.0 * Float64(Float64(Float64(b * a) * Float64(b * a)) / Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(b / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * b) * a) * a) * -4.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a <= 3.1e-108)
		tmp = -4.0 * (((b * a) * (b * a)) / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale));
	else
		tmp = ((((b / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b) * a) * a) * -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a, 3.1e-108], N[(-4.0 * N[(N[(N[(b * a), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(b / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * -4.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.10000000000000014e-108

   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. 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

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 60.4

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

      11. lift-*.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow-prod-down N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 74.7

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

   6. Applied rewrites 74.7%

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

   if 3.10000000000000014e-108 < a

   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. 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

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

   4. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow-prod-down N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 64.4

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-pow.f64 N/A

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

      18. pow2 N/A

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

      19. lift-pow.f64 N/A

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

      20. pow2 N/A

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

      21. unswap-sqr N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 82.9

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

   6. Applied rewrites 82.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

   8. Applied rewrites 74.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-*.f64 77.3

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

   10. Applied rewrites 77.3%

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

Alternative 6: 76.6% accurate, 20.4× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (* (* (/ b (* (* (* y-scale x-scale) y-scale) x-scale)) (* a b)) a) -4.0))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (((b / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * (a * b)) * a) * -4.0;
}

Fortran

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)
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
    code = (((b / (((y_45scale * x_45scale) * y_45scale) * x_45scale)) * (a * b)) * a) * (-4.0d0)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (((b / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * (a * b)) * a) * -4.0;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (((b / (((y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * (a * b)) * a) * -4.0

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(Float64(Float64(b / Float64(Float64(Float64(y_45_scale * x_45_scale) * y_45_scale) * x_45_scale)) * Float64(a * b)) * a) * -4.0)
end

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(N[(b / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] * a), $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. 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

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 47.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-prod-down N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 64.4

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-pow.f64 N/A

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

   18. pow2 N/A

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

   19. lift-pow.f64 N/A

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

   20. pow2 N/A

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

   21. unswap-sqr N/A

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

   22. lower-*.f64 N/A

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

   23. lower-*.f64 N/A

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

   24. lower-*.f64 82.9

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

6. Applied rewrites 82.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

8. Applied rewrites 74.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 77.8

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

10. Applied rewrites 77.8%

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

Alternative 7: 76.2% accurate, 20.4× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (* (* (* (/ b (* (* y-scale x-scale) (* y-scale x-scale))) b) a) a) -4.0))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((((b / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b) * a) * a) * -4.0;
}

Fortran

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)
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
    code = ((((b / ((y_45scale * x_45scale) * (y_45scale * x_45scale))) * b) * a) * a) * (-4.0d0)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((((b / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b) * a) * a) * -4.0;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return ((((b / ((y_45_scale * x_45_scale) * (y_45_scale * x_45_scale))) * b) * a) * a) * -4.0

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(Float64(Float64(Float64(b / Float64(Float64(y_45_scale * x_45_scale) * Float64(y_45_scale * x_45_scale))) * b) * a) * a) * -4.0)
end

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(N[(N[(b / N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision] * a), $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. 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

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

4. Applied rewrites 47.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-prod-down N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 64.4

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-pow.f64 N/A

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

   18. pow2 N/A

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

   19. lift-pow.f64 N/A

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

   20. pow2 N/A

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

   21. unswap-sqr N/A

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

   22. lower-*.f64 N/A

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

   23. lower-*.f64 N/A

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

   24. lower-*.f64 82.9

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

6. Applied rewrites 82.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

8. Applied rewrites 74.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lower-*.f64 77.3

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

10. Applied rewrites 77.3%

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

Reproduce

herbie shell --seed 2025150 
(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))))