Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5

Percentage Accurate: 44.1% → 56.8%

Time: 7.7s

Alternatives: 7

Speedup: 244.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \frac{\tan t\_0}{\sin t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0)))) (/ (tan t_0) (sin t_0))))

C

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	return tan(t_0) / sin(t_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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = x / (y * 2.0d0)
    code = tan(t_0) / sin(t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	return Math.tan(t_0) / Math.sin(t_0);
}

Python

def code(x, y):
	t_0 = x / (y * 2.0)
	return math.tan(t_0) / math.sin(t_0)

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	return Float64(tan(t_0) / sin(t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = x / (y * 2.0);
	tmp = tan(t_0) / sin(t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]]

Initial Program: 44.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ \frac{\tan t\_0}{\sin t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0)))) (/ (tan t_0) (sin t_0))))

C

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	return tan(t_0) / sin(t_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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = x / (y * 2.0d0)
    code = tan(t_0) / sin(t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	return Math.tan(t_0) / Math.sin(t_0);
}

Python

def code(x, y):
	t_0 = x / (y * 2.0)
	return math.tan(t_0) / math.sin(t_0)

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	return Float64(tan(t_0) / sin(t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = x / (y * 2.0);
	tmp = tan(t_0) / sin(t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 56.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{x\_m}{y\_m \cdot 2}\\ t_1 := \sin t\_0\\ \mathbf{if}\;\frac{\tan t\_0}{t\_1} \leq 2.75:\\ \;\;\;\;\frac{\frac{\sin \left(\frac{x\_m}{2 \cdot y\_m}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) + \frac{\frac{x\_m}{y\_m}}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (/ x_m (* y_m 2.0))) (t_1 (sin t_0)))
   (if (<= (/ (tan t_0) t_1) 2.75)
     (/
      (/
       (sin (/ x_m (* 2.0 y_m)))
       (sin (+ (+ (PI) (/ (/ x_m y_m) 2.0)) (/ (PI) 2.0))))
      (- t_1))
     1.0)))

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64))))) < 2.75

   1. Initial program 61.9%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-tan.f64 N/A

         \[\leadsto \frac{\color{blue}{\tan \left(\frac{x}{y \cdot 2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      2. tan-+PI-rev N/A

         \[\leadsto \frac{\color{blue}{\tan \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      3. tan-quot N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}{\cos \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      4. sin-+PI-rev N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(\sin \left(\frac{x}{y \cdot 2}\right)\right)}}{\cos \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(\color{blue}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)}{\cos \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(\sin \left(\frac{x}{y \cdot 2}\right)\right)}{\cos \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{-\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{\color{blue}{y \cdot 2}}\right)}{\cos \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{\color{blue}{2 \cdot y}}\right)}{\cos \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{\color{blue}{2 \cdot y}}\right)}{\cos \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\color{blue}{\cos \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\cos \color{blue}{\left(\mathsf{PI}\left(\right) + \frac{x}{y \cdot 2}\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\cos \color{blue}{\left(\mathsf{PI}\left(\right) + \frac{x}{y \cdot 2}\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      14. lower-PI.f64 61.7

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\cos \left(\color{blue}{\mathsf{PI}\left(\right)} + \frac{x}{y \cdot 2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\cos \left(\mathsf{PI}\left(\right) + \frac{x}{\color{blue}{y \cdot 2}}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\cos \left(\mathsf{PI}\left(\right) + \frac{x}{\color{blue}{2 \cdot y}}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      17. lower-*.f64 61.7

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\cos \left(\mathsf{PI}\left(\right) + \frac{x}{\color{blue}{2 \cdot y}}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   4. Applied rewrites 61.7%

      \[\leadsto \frac{\color{blue}{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\cos \left(\mathsf{PI}\left(\right) + \frac{x}{2 \cdot y}\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\color{blue}{\cos \left(\mathsf{PI}\left(\right) + \frac{x}{2 \cdot y}\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\color{blue}{\sin \left(\left(\mathsf{PI}\left(\right) + \frac{x}{2 \cdot y}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\color{blue}{\sin \left(\left(\mathsf{PI}\left(\right) + \frac{x}{2 \cdot y}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\sin \color{blue}{\left(\left(\mathsf{PI}\left(\right) + \frac{x}{2 \cdot y}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) + \color{blue}{\frac{x}{2 \cdot y}}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) + \frac{x}{\color{blue}{2 \cdot y}}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) + \frac{x}{\color{blue}{y \cdot 2}}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) + \color{blue}{\frac{\frac{x}{y}}{2}}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) + \frac{\color{blue}{\frac{x}{y}}}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) + \color{blue}{\frac{\frac{x}{y}}{2}}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      11. lift-PI.f64 N/A

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) + \frac{\frac{x}{y}}{2}\right) + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      12. lower-/.f64 61.0

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) + \frac{\frac{x}{y}}{2}\right) + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   6. Applied rewrites 61.0%

      \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\color{blue}{\sin \left(\left(\mathsf{PI}\left(\right) + \frac{\frac{x}{y}}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   if 2.75 < (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))))

   1. Initial program 1.8%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 40.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 2.75:\\ \;\;\;\;\frac{\frac{\sin \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) + \frac{\frac{x}{y}}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{-\sin \left(\frac{x}{y \cdot 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{x\_m}{y\_m \cdot 2}\\ \mathbf{if}\;\frac{\tan t\_0}{\sin t\_0} \leq 2.75:\\ \;\;\;\;\frac{-1}{\sin \left(\mathsf{fma}\left(1.5, \mathsf{PI}\left(\right), \frac{x\_m}{y\_m} \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (/ x_m (* y_m 2.0))))
   (if (<= (/ (tan t_0) (sin t_0)) 2.75)
     (/ -1.0 (sin (fma 1.5 (PI) (* (/ x_m y_m) 0.5))))
     1.0)))

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64))))) < 2.75

   1. Initial program 61.9%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-tan.f64 N/A

         \[\leadsto \frac{\color{blue}{\tan \left(\frac{x}{y \cdot 2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      2. tan-+PI-rev N/A

         \[\leadsto \frac{\color{blue}{\tan \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      3. tan-quot N/A

         \[\leadsto \frac{\color{blue}{\frac{\sin \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}{\cos \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      4. sin-+PI-rev N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(\sin \left(\frac{x}{y \cdot 2}\right)\right)}}{\cos \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{neg}\left(\color{blue}{\sin \left(\frac{x}{y \cdot 2}\right)}\right)}{\cos \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(\sin \left(\frac{x}{y \cdot 2}\right)\right)}{\cos \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{-\sin \left(\frac{x}{y \cdot 2}\right)}}{\cos \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{\color{blue}{y \cdot 2}}\right)}{\cos \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{\color{blue}{2 \cdot y}}\right)}{\cos \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{\color{blue}{2 \cdot y}}\right)}{\cos \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      11. lower-cos.f64 N/A

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\color{blue}{\cos \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\cos \color{blue}{\left(\mathsf{PI}\left(\right) + \frac{x}{y \cdot 2}\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      13. lower-+.f64 N/A

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\cos \color{blue}{\left(\mathsf{PI}\left(\right) + \frac{x}{y \cdot 2}\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      14. lower-PI.f64 61.7

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\cos \left(\color{blue}{\mathsf{PI}\left(\right)} + \frac{x}{y \cdot 2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\cos \left(\mathsf{PI}\left(\right) + \frac{x}{\color{blue}{y \cdot 2}}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      16. *-commutative N/A

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\cos \left(\mathsf{PI}\left(\right) + \frac{x}{\color{blue}{2 \cdot y}}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      17. lower-*.f64 61.7

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\cos \left(\mathsf{PI}\left(\right) + \frac{x}{\color{blue}{2 \cdot y}}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   4. Applied rewrites 61.7%

      \[\leadsto \frac{\color{blue}{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\cos \left(\mathsf{PI}\left(\right) + \frac{x}{2 \cdot y}\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\color{blue}{\cos \left(\mathsf{PI}\left(\right) + \frac{x}{2 \cdot y}\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\color{blue}{\sin \left(\left(\mathsf{PI}\left(\right) + \frac{x}{2 \cdot y}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\color{blue}{\sin \left(\left(\mathsf{PI}\left(\right) + \frac{x}{2 \cdot y}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\sin \color{blue}{\left(\left(\mathsf{PI}\left(\right) + \frac{x}{2 \cdot y}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) + \color{blue}{\frac{x}{2 \cdot y}}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) + \frac{x}{\color{blue}{2 \cdot y}}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) + \frac{x}{\color{blue}{y \cdot 2}}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      8. associate-/r* N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) + \color{blue}{\frac{\frac{x}{y}}{2}}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) + \frac{\color{blue}{\frac{x}{y}}}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) + \color{blue}{\frac{\frac{x}{y}}{2}}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      11. lift-PI.f64 N/A

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) + \frac{\frac{x}{y}}{2}\right) + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

      12. lower-/.f64 61.0

          \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\sin \left(\left(\mathsf{PI}\left(\right) + \frac{\frac{x}{y}}{2}\right) + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   6. Applied rewrites 61.0%

      \[\leadsto \frac{\frac{-\sin \left(\frac{x}{2 \cdot y}\right)}{\color{blue}{\sin \left(\left(\mathsf{PI}\left(\right) + \frac{\frac{x}{y}}{2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{\sin \left(\mathsf{PI}\left(\right) + \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{x}{y}\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\sin \left(\mathsf{PI}\left(\right) + \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{x}{y}\right)\right)}} \]

      2. lower-sin.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin \left(\mathsf{PI}\left(\right) + \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \frac{1}{2} \cdot \frac{x}{y}\right)\right)}} \]

      3. associate-+r+ N/A

         \[\leadsto \frac{-1}{\sin \color{blue}{\left(\left(\mathsf{PI}\left(\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \frac{x}{y}\right)}} \]

      4. distribute-rgt1-in N/A

         \[\leadsto \frac{-1}{\sin \left(\color{blue}{\left(\frac{1}{2} + 1\right) \cdot \mathsf{PI}\left(\right)} + \frac{1}{2} \cdot \frac{x}{y}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{-1}{\sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2} + 1, \mathsf{PI}\left(\right), \frac{1}{2} \cdot \frac{x}{y}\right)\right)}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{-1}{\sin \left(\mathsf{fma}\left(\color{blue}{\frac{3}{2}}, \mathsf{PI}\left(\right), \frac{1}{2} \cdot \frac{x}{y}\right)\right)} \]

      7. lower-PI.f64 N/A

         \[\leadsto \frac{-1}{\sin \left(\mathsf{fma}\left(\frac{3}{2}, \color{blue}{\mathsf{PI}\left(\right)}, \frac{1}{2} \cdot \frac{x}{y}\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{-1}{\sin \left(\mathsf{fma}\left(\frac{3}{2}, \mathsf{PI}\left(\right), \color{blue}{\frac{x}{y} \cdot \frac{1}{2}}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{-1}{\sin \left(\mathsf{fma}\left(\frac{3}{2}, \mathsf{PI}\left(\right), \color{blue}{\frac{x}{y} \cdot \frac{1}{2}}\right)\right)} \]

      10. lower-/.f64 61.1

          \[\leadsto \frac{-1}{\sin \left(\mathsf{fma}\left(1.5, \mathsf{PI}\left(\right), \color{blue}{\frac{x}{y}} \cdot 0.5\right)\right)} \]

   9. Applied rewrites 61.1%

      \[\leadsto \color{blue}{\frac{-1}{\sin \left(\mathsf{fma}\left(1.5, \mathsf{PI}\left(\right), \frac{x}{y} \cdot 0.5\right)\right)}} \]

   if 2.75 < (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))))

   1. Initial program 1.8%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 40.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 2.75:\\ \;\;\;\;\frac{-1}{\sin \left(\mathsf{fma}\left(1.5, \mathsf{PI}\left(\right), \frac{x}{y} \cdot 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{x\_m}{y\_m \cdot 2}\\ \mathbf{if}\;\frac{\tan t\_0}{\sin t\_0} \leq 2.75:\\ \;\;\;\;\frac{1}{\sin \left(0.5 \cdot \left(\mathsf{PI}\left(\right) - \frac{x\_m}{y\_m}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (/ x_m (* y_m 2.0))))
   (if (<= (/ (tan t_0) (sin t_0)) 2.75)
     (/ 1.0 (sin (* 0.5 (- (PI) (/ x_m y_m)))))
     1.0)))

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64))))) < 2.75

   1. Initial program 61.9%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 58.7%

      \[\leadsto \color{blue}{1} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{\cos \left(\frac{1}{2} \cdot \frac{x}{y}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\cos \left(\frac{1}{2} \cdot \frac{x}{y}\right)}} \]

      2. cos-neg-rev N/A

         \[\leadsto \frac{1}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{x}{y}\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\cos \left(\color{blue}{\frac{-1}{2}} \cdot \frac{x}{y}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{\frac{-1}{2} \cdot x}{y}\right)}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\cos \left(\frac{\frac{-1}{2} \cdot x}{y}\right)}} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{-1}{2} \cdot \frac{x}{y}\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{-1}{2} \cdot \frac{x}{y}\right)}} \]

      9. lower-/.f64 61.9

         \[\leadsto \frac{1}{\cos \left(-0.5 \cdot \color{blue}{\frac{x}{y}}\right)} \]

   8. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\frac{1}{\cos \left(-0.5 \cdot \frac{x}{y}\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.9%

       \[\leadsto \frac{1}{\sin \left(\mathsf{fma}\left(\frac{x}{y}, -0.5, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{1}{\sin \left(\frac{-1}{2} \cdot \frac{x}{y} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 60.9%

       \[\leadsto \frac{1}{\sin \left(0.5 \cdot \left(\mathsf{PI}\left(\right) - \frac{x}{y}\right)\right)} \]

   if 2.75 < (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))))

   1. Initial program 1.8%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 40.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 2.75:\\ \;\;\;\;\frac{1}{\sin \left(0.5 \cdot \left(\mathsf{PI}\left(\right) - \frac{x}{y}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{x\_m}{y\_m \cdot 2}\\ \mathbf{if}\;\frac{\tan t\_0}{\sin t\_0} \leq 2.5:\\ \;\;\;\;\frac{-1}{\cos \left(\mathsf{fma}\left(\frac{0.5}{y\_m}, x\_m, \mathsf{PI}\left(\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (/ x_m (* y_m 2.0))))
   (if (<= (/ (tan t_0) (sin t_0)) 2.5)
     (/ -1.0 (cos (fma (/ 0.5 y_m) x_m (PI))))
     1.0)))

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64))))) < 2.5

   1. Initial program 62.4%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 59.2%

      \[\leadsto \color{blue}{1} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{\cos \left(\frac{1}{2} \cdot \frac{x}{y}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\cos \left(\frac{1}{2} \cdot \frac{x}{y}\right)}} \]

      2. cos-neg-rev N/A

         \[\leadsto \frac{1}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{x}{y}\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\cos \left(\color{blue}{\frac{-1}{2}} \cdot \frac{x}{y}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{\frac{-1}{2} \cdot x}{y}\right)}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\cos \left(\frac{\frac{-1}{2} \cdot x}{y}\right)}} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{-1}{2} \cdot \frac{x}{y}\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{-1}{2} \cdot \frac{x}{y}\right)}} \]

      9. lower-/.f64 62.4

         \[\leadsto \frac{1}{\cos \left(-0.5 \cdot \color{blue}{\frac{x}{y}}\right)} \]

   8. Applied rewrites 62.4%

      \[\leadsto \color{blue}{\frac{1}{\cos \left(-0.5 \cdot \frac{x}{y}\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 61.4%

       \[\leadsto \frac{1}{\sin \left(\mathsf{fma}\left(\frac{x}{y}, -0.5, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 62.5%

       \[\leadsto \frac{-1}{\color{blue}{\cos \left(\mathsf{fma}\left(\frac{0.5}{y}, x, \mathsf{PI}\left(\right)\right)\right)}} \]

   if 2.5 < (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))))

   1. Initial program 2.2%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 40.3%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 2.5:\\ \;\;\;\;\frac{-1}{\cos \left(\mathsf{fma}\left(\frac{0.5}{y}, x, \mathsf{PI}\left(\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{x\_m}{y\_m \cdot 2}\\ \mathbf{if}\;\frac{\tan t\_0}{\sin t\_0} \leq 2:\\ \;\;\;\;\frac{1}{\cos \left(-0.5 \cdot \frac{x\_m}{y\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (/ x_m (* y_m 2.0))))
   (if (<= (/ (tan t_0) (sin t_0)) 2.0)
     (/ 1.0 (cos (* -0.5 (/ x_m y_m))))
     1.0)))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double t_0 = x_m / (y_m * 2.0);
	double tmp;
	if ((tan(t_0) / sin(t_0)) <= 2.0) {
		tmp = 1.0 / cos((-0.5 * (x_m / y_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m =     private
x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_m, y_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_m / (y_m * 2.0d0)
    if ((tan(t_0) / sin(t_0)) <= 2.0d0) then
        tmp = 1.0d0 / cos(((-0.5d0) * (x_m / y_m)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double t_0 = x_m / (y_m * 2.0);
	double tmp;
	if ((Math.tan(t_0) / Math.sin(t_0)) <= 2.0) {
		tmp = 1.0 / Math.cos((-0.5 * (x_m / y_m)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	t_0 = x_m / (y_m * 2.0)
	tmp = 0
	if (math.tan(t_0) / math.sin(t_0)) <= 2.0:
		tmp = 1.0 / math.cos((-0.5 * (x_m / y_m)))
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	t_0 = Float64(x_m / Float64(y_m * 2.0))
	tmp = 0.0
	if (Float64(tan(t_0) / sin(t_0)) <= 2.0)
		tmp = Float64(1.0 / cos(Float64(-0.5 * Float64(x_m / y_m))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	t_0 = x_m / (y_m * 2.0);
	tmp = 0.0;
	if ((tan(t_0) / sin(t_0)) <= 2.0)
		tmp = 1.0 / cos((-0.5 * (x_m / y_m)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := Block[{t$95$0 = N[(x$95$m / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], N[(1.0 / N[Cos[N[(-0.5 * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64))))) < 2

   1. Initial program 64.3%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 61.0%

      \[\leadsto \color{blue}{1} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{\cos \left(\frac{1}{2} \cdot \frac{x}{y}\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\cos \left(\frac{1}{2} \cdot \frac{x}{y}\right)}} \]

      2. cos-neg-rev N/A

         \[\leadsto \frac{1}{\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{x}{y}\right)\right)}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{x}{y}\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\cos \left(\color{blue}{\frac{-1}{2}} \cdot \frac{x}{y}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{\frac{-1}{2} \cdot x}{y}\right)}} \]

      6. lower-cos.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\cos \left(\frac{\frac{-1}{2} \cdot x}{y}\right)}} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{-1}{2} \cdot \frac{x}{y}\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{1}{\cos \color{blue}{\left(\frac{-1}{2} \cdot \frac{x}{y}\right)}} \]

      9. lower-/.f64 64.3

         \[\leadsto \frac{1}{\cos \left(-0.5 \cdot \color{blue}{\frac{x}{y}}\right)} \]

   8. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\frac{1}{\cos \left(-0.5 \cdot \frac{x}{y}\right)}} \]

   if 2 < (/.f64 (tan.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))) (sin.f64 (/.f64 x (*.f64 y #s(literal 2 binary64)))))

   1. Initial program 2.6%

      \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 38.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \leq 2:\\ \;\;\;\;\frac{1}{\cos \left(-0.5 \cdot \frac{x}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.4% accurate, 244.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ 1 \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m) :precision binary64 1.0)

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	return 1.0;
}

Fortran

y_m =     private
x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_m, y_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    code = 1.0d0
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	return 1.0;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	return 1.0

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	return 1.0
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp = code(x_m, y_m)
	tmp = 1.0;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := 1.0

Derivation

1. Initial program 43.3%

   \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 53.2%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 53.2%

   \[\leadsto 1 \]
7. Add Preprocessing

Alternative 7: 3.1% accurate, 244.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m) :precision binary64 0.0)

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	return 0.0;
}

Fortran

y_m =     private
x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_m, y_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    code = 0.0d0
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	return 0.0;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	return 0.0

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	return 0.0
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp = code(x_m, y_m)
	tmp = 0.0;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := 0.0

Derivation

1. Initial program 43.3%

   \[\frac{\tan \left(\frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-tan.f64 N/A

      \[\leadsto \frac{\color{blue}{\tan \left(\frac{x}{y \cdot 2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   2. tan-+PI-rev N/A

      \[\leadsto \frac{\color{blue}{\tan \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   3. lower-tan.f64 N/A

      \[\leadsto \frac{\color{blue}{\tan \left(\frac{x}{y \cdot 2} + \mathsf{PI}\left(\right)\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   4. +-commutative N/A

      \[\leadsto \frac{\tan \color{blue}{\left(\mathsf{PI}\left(\right) + \frac{x}{y \cdot 2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{\tan \color{blue}{\left(\mathsf{PI}\left(\right) + \frac{x}{y \cdot 2}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   6. lower-PI.f64 8.6

      \[\leadsto \frac{\tan \left(\color{blue}{\mathsf{PI}\left(\right)} + \frac{x}{y \cdot 2}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\tan \left(\mathsf{PI}\left(\right) + \frac{x}{\color{blue}{y \cdot 2}}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\tan \left(\mathsf{PI}\left(\right) + \frac{x}{\color{blue}{2 \cdot y}}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

   9. lower-*.f64 8.6

      \[\leadsto \frac{\tan \left(\mathsf{PI}\left(\right) + \frac{x}{\color{blue}{2 \cdot y}}\right)}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

4. Applied rewrites 8.6%

   \[\leadsto \frac{\color{blue}{\tan \left(\mathsf{PI}\left(\right) + \frac{x}{2 \cdot y}\right)}}{\sin \left(\frac{x}{y \cdot 2}\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{2 \cdot \frac{y \cdot \sin \mathsf{PI}\left(\right)}{x \cdot \cos \mathsf{PI}\left(\right)}} \]
6. Step-by-step derivation

   1. cos-PI N/A

      \[\leadsto 2 \cdot \frac{y \cdot \sin \mathsf{PI}\left(\right)}{x \cdot \color{blue}{-1}} \]

   2. times-frac N/A

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{\sin \mathsf{PI}\left(\right)}{-1}\right)} \]

   3. sin-PI N/A

      \[\leadsto 2 \cdot \left(\frac{y}{x} \cdot \frac{\color{blue}{0}}{-1}\right) \]

   4. metadata-eval N/A

      \[\leadsto 2 \cdot \left(\frac{y}{x} \cdot \color{blue}{0}\right) \]

   5. mul0-rgt N/A

      \[\leadsto 2 \cdot \color{blue}{0} \]

   6. metadata-eval 3.1

      \[\leadsto \color{blue}{0} \]

7. Applied rewrites 3.1%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 55.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot 2}\\ t_1 := \sin t\_0\\ \mathbf{if}\;y < -1.2303690911306994 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y < -9.102852406811914 \cdot 10^{-222}:\\ \;\;\;\;\frac{t\_1}{t\_1 \cdot \log \left(e^{\cos t\_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y 2.0))) (t_1 (sin t_0)))
   (if (< y -1.2303690911306994e+114)
     1.0
     (if (< y -9.102852406811914e-222)
       (/ t_1 (* t_1 (log (exp (cos t_0)))))
       1.0))))

C

double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double t_1 = sin(t_0);
	double tmp;
	if (y < -1.2303690911306994e+114) {
		tmp = 1.0;
	} else if (y < -9.102852406811914e-222) {
		tmp = t_1 / (t_1 * log(exp(cos(t_0))));
	} else {
		tmp = 1.0;
	}
	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(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / (y * 2.0d0)
    t_1 = sin(t_0)
    if (y < (-1.2303690911306994d+114)) then
        tmp = 1.0d0
    else if (y < (-9.102852406811914d-222)) then
        tmp = t_1 / (t_1 * log(exp(cos(t_0))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * 2.0);
	double t_1 = Math.sin(t_0);
	double tmp;
	if (y < -1.2303690911306994e+114) {
		tmp = 1.0;
	} else if (y < -9.102852406811914e-222) {
		tmp = t_1 / (t_1 * Math.log(Math.exp(Math.cos(t_0))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * 2.0)
	t_1 = math.sin(t_0)
	tmp = 0
	if y < -1.2303690911306994e+114:
		tmp = 1.0
	elif y < -9.102852406811914e-222:
		tmp = t_1 / (t_1 * math.log(math.exp(math.cos(t_0))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * 2.0))
	t_1 = sin(t_0)
	tmp = 0.0
	if (y < -1.2303690911306994e+114)
		tmp = 1.0;
	elseif (y < -9.102852406811914e-222)
		tmp = Float64(t_1 / Float64(t_1 * log(exp(cos(t_0)))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * 2.0);
	t_1 = sin(t_0);
	tmp = 0.0;
	if (y < -1.2303690911306994e+114)
		tmp = 1.0;
	elseif (y < -9.102852406811914e-222)
		tmp = t_1 / (t_1 * log(exp(cos(t_0))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[Less[y, -1.2303690911306994e+114], 1.0, If[Less[y, -9.102852406811914e-222], N[(t$95$1 / N[(t$95$1 * N[Log[N[Exp[N[Cos[t$95$0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]]]

Reproduce

herbie shell --seed 2025015 
(FPCore (x y)
  :name "Diagrams.TwoD.Layout.CirclePacking:approxRadius from diagrams-contrib-1.3.0.5"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -1230369091130699400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) 1 (if (< y -4551426203405957/500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (sin (/ x (* y 2))) (* (sin (/ x (* y 2))) (log (exp (cos (/ x (* y 2))))))) 1)))

  (/ (tan (/ x (* y 2.0))) (sin (/ x (* y 2.0)))))