Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 6.3s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x \cdot \frac{\sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (/ (sin y) y)))

C

double code(double x, double y) {
	return x * (sin(y) / y);
}

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
    code = x * (sin(y) / y)
end function

Java

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

Python

def code(x, y):
	return x * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(x * Float64(sin(y) / y))
end

MATLAB

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

Wolfram

code[x_, y_] := N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{\sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (/ (sin y) y)))

C

double code(double x, double y) {
	return x * (sin(y) / y);
}

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
    code = x * (sin(y) / y)
end function

Java

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

Python

def code(x, y):
	return x * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(x * Float64(sin(y) / y))
end

MATLAB

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

Wolfram

code[x_, y_] := N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{\sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (/ (sin y) y)))

C

double code(double x, double y) {
	return x * (sin(y) / y);
}

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
    code = x * (sin(y) / y)
end function

Java

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

Python

def code(x, y):
	return x * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(x * Float64(sin(y) / y))
end

MATLAB

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

Wolfram

code[x_, y_] := N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x \cdot \frac{\sin y}{y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 63.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-35}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.008333333333333333 \cdot y, y, -0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (sin y) y) 1e-35)
   0.0
   (*
    x
    (fma
     (* y y)
     (fma (* 0.008333333333333333 y) y -0.16666666666666666)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(y) / y) <= 1e-35) {
		tmp = 0.0;
	} else {
		tmp = x * fma((y * y), fma((0.008333333333333333 * y), y, -0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 1e-35)
		tmp = 0.0;
	else
		tmp = Float64(x * fma(Float64(y * y), fma(Float64(0.008333333333333333 * y), y, -0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 1e-35], 0.0, N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(0.008333333333333333 * y), $MachinePrecision] * y + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 1.00000000000000001e-35

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{\sin y}{y}} \]

      2. lift-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{\sin y}{y}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \sin y}{y}} \]

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{\sin y} \cdot x}{y} \]

      12. lower-*.f64 99.6

          \[\leadsto \frac{\color{blue}{\sin y \cdot x}}{y} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin y \cdot x}{y}} \]
   5. Step-by-step derivation

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-+PI-rev N/A

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

      8. sin-neg-rev N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-PI.f64 31.3

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

   6. Applied rewrites 31.3%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. sin-neg N/A

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

      4. sin-PI N/A

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

      5. metadata-eval N/A

         \[\leadsto \color{blue}{0} \cdot \frac{x}{y} \]

      6. mul0-lft 29.4

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

   9. Applied rewrites 29.4%

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

   if 1.00000000000000001e-35 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right) + x} \]

      2. +-commutative N/A

         \[\leadsto {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left(x \cdot {y}^{2}\right) + \frac{-1}{6} \cdot x\right)} + x \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right) \cdot {y}^{2} + \left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2}\right)} + x \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right)} + \left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2}\right) + x \]

      5. associate-+l+ N/A

         \[\leadsto \color{blue}{{y}^{2} \cdot \left(\frac{1}{120} \cdot \left(x \cdot {y}^{2}\right)\right) + \left(\left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2} + x\right)} \]

      6. associate-*r* N/A

         \[\leadsto {y}^{2} \cdot \color{blue}{\left(\left(\frac{1}{120} \cdot x\right) \cdot {y}^{2}\right)} + \left(\left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2} + x\right) \]

      7. *-commutative N/A

         \[\leadsto {y}^{2} \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot x\right)\right)} + \left(\left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2} + x\right) \]

      8. associate-*r* N/A

         \[\leadsto {y}^{2} \cdot \color{blue}{\left(\left({y}^{2} \cdot \frac{1}{120}\right) \cdot x\right)} + \left(\left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2} + x\right) \]

      9. *-commutative N/A

         \[\leadsto {y}^{2} \cdot \left(\color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)} \cdot x\right) + \left(\left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2} + x\right) \]

      10. associate-*r* N/A

          \[\leadsto \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x} + \left(\left(\frac{-1}{6} \cdot x\right) \cdot {y}^{2} + x\right) \]

      11. associate-*r* N/A

          \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x + \left(\color{blue}{\frac{-1}{6} \cdot \left(x \cdot {y}^{2}\right)} + x\right) \]

      12. *-commutative N/A

          \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x + \left(\frac{-1}{6} \cdot \color{blue}{\left({y}^{2} \cdot x\right)} + x\right) \]

      13. associate-*r* N/A

          \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x + \left(\color{blue}{\left(\frac{-1}{6} \cdot {y}^{2}\right) \cdot x} + x\right) \]

      14. distribute-lft1-in N/A

          \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x + \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2} + 1\right) \cdot x} \]

      15. +-commutative N/A

          \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2}\right)\right) \cdot x + \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \cdot x \]

   5. Applied rewrites 95.6%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.008333333333333333 \cdot y, y, -0.16666666666666666\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-35}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.008333333333333333 \cdot y, y, -0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-64}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (/ (sin y) y) 1e-64) 0.0 x))

C

double code(double x, double y) {
	double tmp;
	if ((sin(y) / y) <= 1e-64) {
		tmp = 0.0;
	} else {
		tmp = x;
	}
	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) :: tmp
    if ((sin(y) / y) <= 1d-64) then
        tmp = 0.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sin(y) / y) <= 1e-64) {
		tmp = 0.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sin(y) / y) <= 1e-64:
		tmp = 0.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 1e-64)
		tmp = 0.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sin(y) / y) <= 1e-64)
		tmp = 0.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 1e-64], 0.0, x]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 9.99999999999999965e-65

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{\sin y}{y}} \]

      2. lift-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{\sin y}{y}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \sin y}{y}} \]

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{\sin y} \cdot x}{y} \]

      12. lower-*.f64 99.6

          \[\leadsto \frac{\color{blue}{\sin y \cdot x}}{y} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin y \cdot x}{y}} \]
   5. Step-by-step derivation

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-+PI-rev N/A

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

      8. sin-neg-rev N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-PI.f64 32.9

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

   6. Applied rewrites 32.9%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. sin-neg N/A

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

      4. sin-PI N/A

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

      5. metadata-eval N/A

         \[\leadsto \color{blue}{0} \cdot \frac{x}{y} \]

      6. mul0-lft 30.8

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

   9. Applied rewrites 30.8%

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

   if 9.99999999999999965e-65 < (/.f64 (sin.f64 y) y)

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{\sin y}{y}} \]

      2. lift-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{\sin y}{y}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \sin y}{y}} \]

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{\sin y} \cdot x}{y} \]

      12. lower-*.f64 79.0

          \[\leadsto \frac{\color{blue}{\sin y \cdot x}}{y} \]

   4. Applied rewrites 79.0%

      \[\leadsto \color{blue}{\frac{\sin y \cdot x}{y}} \]
   5. Step-by-step derivation

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-+PI-rev N/A

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

      8. sin-neg-rev N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-PI.f64 6.2

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

   6. Applied rewrites 6.2%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 91.2%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-64}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10500000000:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y, y, -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 10500000000.0)
   (fma
    (* (* y y) x)
    (fma
     (* (fma -0.0001984126984126984 (* y y) 0.008333333333333333) y)
     y
     -0.16666666666666666)
    x)
   0.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 10500000000.0) {
		tmp = fma(((y * y) * x), fma((fma(-0.0001984126984126984, (y * y), 0.008333333333333333) * y), y, -0.16666666666666666), x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 10500000000.0)
		tmp = fma(Float64(Float64(y * y) * x), fma(Float64(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333) * y), y, -0.16666666666666666), x);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 10500000000.0], N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * y), $MachinePrecision] * y + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.05e10

   1. Initial program 99.8%

      \[x \cdot \frac{\sin y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(\frac{-1}{6} \cdot x + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{120} \cdot x\right)\right)} \]

   5. Applied rewrites 68.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y, y, -0.16666666666666666\right), x\right)} \]

   if 1.05e10 < y

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{\sin y}{y}} \]

      2. lift-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{\sin y}{y}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \sin y}{y}} \]

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{\sin y} \cdot x}{y} \]

      12. lower-*.f64 99.6

          \[\leadsto \frac{\color{blue}{\sin y \cdot x}}{y} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin y \cdot x}{y}} \]
   5. Step-by-step derivation

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-+PI-rev N/A

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

      8. sin-neg-rev N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-PI.f64 21.4

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

   6. Applied rewrites 21.4%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. sin-neg N/A

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

      4. sin-PI N/A

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

      5. metadata-eval N/A

         \[\leadsto \color{blue}{0} \cdot \frac{x}{y} \]

      6. mul0-lft 22.7

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

   9. Applied rewrites 22.7%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10500000000:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y, y, -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.3% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10500000000:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, -0.16666666666666666 \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 10500000000.0) (fma (* x y) (* -0.16666666666666666 y) x) 0.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 10500000000.0) {
		tmp = fma((x * y), (-0.16666666666666666 * y), x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 10500000000.0)
		tmp = fma(Float64(x * y), Float64(-0.16666666666666666 * y), x);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 10500000000.0], N[(N[(x * y), $MachinePrecision] * N[(-0.16666666666666666 * y), $MachinePrecision] + x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.05e10

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{\sin y}{y}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin y}{y}} \cdot x \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin y \cdot x}{y}} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\sin y \cdot \frac{x}{y}} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \sin y} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y} \cdot \sin y} \]

      8. lower-/.f64 85.2

         \[\leadsto \color{blue}{\frac{x}{y}} \cdot \sin y \]

   4. Applied rewrites 85.2%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \sin y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{-1}{6} \cdot \left(x \cdot {y}^{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{6} \cdot \left(x \cdot {y}^{2}\right) + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot {y}^{2}\right) \cdot \frac{-1}{6}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {y}^{2}, \frac{-1}{6}, x\right)} \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{y}^{2} \cdot x}, \frac{-1}{6}, x\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{y}^{2} \cdot x}, \frac{-1}{6}, x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y \cdot y\right)} \cdot x, \frac{-1}{6}, x\right) \]

      7. lower-*.f64 68.7

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y \cdot y\right)} \cdot x, -0.16666666666666666, x\right) \]

   7. Applied rewrites 68.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, -0.16666666666666666, x\right)} \]

   9. Applied rewrites 68.7%

      \[\leadsto \mathsf{fma}\left(x \cdot y, \color{blue}{-0.16666666666666666 \cdot y}, x\right) \]

   if 1.05e10 < y

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{\sin y}{y}} \]

      2. lift-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{\sin y}{y}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \sin y}{y}} \]

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. remove-double-neg N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. remove-double-neg N/A

          \[\leadsto \frac{\color{blue}{\sin y} \cdot x}{y} \]

      12. lower-*.f64 99.6

          \[\leadsto \frac{\color{blue}{\sin y \cdot x}}{y} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin y \cdot x}{y}} \]
   5. Step-by-step derivation

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-+PI-rev N/A

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

      8. sin-neg-rev N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-PI.f64 21.4

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

   6. Applied rewrites 21.4%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. sin-neg N/A

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

      4. sin-PI N/A

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

      5. metadata-eval N/A

         \[\leadsto \color{blue}{0} \cdot \frac{x}{y} \]

      6. mul0-lft 22.7

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

   9. Applied rewrites 22.7%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10500000000:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, -0.16666666666666666 \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 15.8% accurate, 117.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.0)

C

double code(double x, double y) {
	return 0.0;
}

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
    code = 0.0d0
end function

Java

public static double code(double x, double y) {
	return 0.0;
}

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, y_] := 0.0

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \frac{\sin y}{y}} \]

   2. lift-/.f64 N/A

      \[\leadsto x \cdot \color{blue}{\frac{\sin y}{y}} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{x \cdot \sin y}{y}} \]

   4. frac-2neg N/A

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

   5. frac-2neg N/A

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

   6. remove-double-neg N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

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

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

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

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

   11. remove-double-neg N/A

       \[\leadsto \frac{\color{blue}{\sin y} \cdot x}{y} \]

   12. lower-*.f64 87.7

       \[\leadsto \frac{\color{blue}{\sin y \cdot x}}{y} \]

4. Applied rewrites 87.7%

   \[\leadsto \color{blue}{\frac{\sin y \cdot x}{y}} \]
5. Step-by-step derivation

   1. remove-double-neg N/A

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

   2. mul-1-neg N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lift-sin.f64 N/A

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

   7. sin-+PI-rev N/A

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

   8. sin-neg-rev N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. lower-PI.f64 17.4

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

6. Applied rewrites 17.4%

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

7. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. sin-neg N/A

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

   4. sin-PI N/A

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

   5. metadata-eval N/A

      \[\leadsto \color{blue}{0} \cdot \frac{x}{y} \]

   6. mul0-lft 15.2

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

9. Applied rewrites 15.2%

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

10. Final simplification 15.2%

    \[\leadsto 0 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024350 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))