Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 3.8s

Alternatives: 14

Speedup: 1.0×

• 50.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y) {
	return cosh(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 = cosh(x) * (sin(y) / y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return cosh(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 = cosh(x) * (sin(y) / y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return cosh(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 = cosh(x) * (sin(y) / y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999997430856:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cosh x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* y (* y -0.16666666666666666)))
     (if (<= t_1 0.9999999997430856)
       (* (fma (* x x) 0.5 1.0) t_0)
       (* (* 2.0 (cosh x)) 0.5)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * (y * (y * -0.16666666666666666));
	} else if (t_1 <= 0.9999999997430856) {
		tmp = fma((x * x), 0.5, 1.0) * t_0;
	} else {
		tmp = (2.0 * cosh(x)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * Float64(y * Float64(y * -0.16666666666666666)));
	elseif (t_1 <= 0.9999999997430856)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * t_0);
	else
		tmp = Float64(Float64(2.0 * cosh(x)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999997430856], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -inf.0

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 64.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 63.1

         \[\leadsto \cosh x \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \]

   7. Applied rewrites 63.1%

      \[\leadsto \cosh x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)} \]

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

         \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right) \]

      4. lift-*.f64 13.6

         \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

   10. Applied rewrites 13.6%

       \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{-0.16666666666666666}\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right) \]

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 13.6

          \[\leadsto \cosh x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right) \]

   12. Applied rewrites 13.6%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.999999999743085621

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \frac{1}{2} + 1\right) \cdot \frac{\sin y}{y} \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 75.8

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

   4. Applied rewrites 75.8%

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

   if 0.999999999743085621 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      9. cosh-undef N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      11. lift-cosh.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      4. cosh-undef-rev N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      5. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + \frac{1}{e^{x}}}{y} \]

      6. +-commutative N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{\frac{1}{e^{x}} + e^{x}}{y} \]

      7. div-add N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \left(\frac{e^{-x}}{y} + \frac{e^{x}}{y}\right) \]

      13. lower-/.f64 N/A

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

      14. lower-exp.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \left(\frac{e^{-x}}{y} + \frac{e^{x}}{y}\right) \]

   6. Applied rewrites 99.8%

      \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \left(\frac{e^{-x}}{y} + \color{blue}{\frac{e^{x}}{y}}\right) \]

   7. Taylor expanded in y around 0

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

      1. cosh-undef-rev N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \cosh x\right) \]

      2. *-commutative N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      5. lift-cosh.f64 64.0

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot 0.5 \]

   9. Applied rewrites 64.0%

      \[\leadsto \left(2 \cdot \cosh x\right) \cdot \color{blue}{0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999997430856:\\ \;\;\;\;\frac{\left(\sin y \cdot 0.5\right) \cdot \mathsf{fma}\left(x, x, 2\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cosh x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ (sin y) y))))
   (if (<= t_0 (- INFINITY))
     (* (cosh x) (* y (* y -0.16666666666666666)))
     (if (<= t_0 0.9999999997430856)
       (/ (* (* (sin y) 0.5) (fma x x 2.0)) y)
       (* (* 2.0 (cosh x)) 0.5)))))

C

double code(double x, double y) {
	double t_0 = cosh(x) * (sin(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = cosh(x) * (y * (y * -0.16666666666666666));
	} else if (t_0 <= 0.9999999997430856) {
		tmp = ((sin(y) * 0.5) * fma(x, x, 2.0)) / y;
	} else {
		tmp = (2.0 * cosh(x)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cosh(x) * Float64(sin(y) / y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(cosh(x) * Float64(y * Float64(y * -0.16666666666666666)));
	elseif (t_0 <= 0.9999999997430856)
		tmp = Float64(Float64(Float64(sin(y) * 0.5) * fma(x, x, 2.0)) / y);
	else
		tmp = Float64(Float64(2.0 * cosh(x)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999997430856], N[(N[(N[(N[Sin[y], $MachinePrecision] * 0.5), $MachinePrecision] * N[(x * x + 2.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -inf.0

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 64.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 63.1

         \[\leadsto \cosh x \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \]

   7. Applied rewrites 63.1%

      \[\leadsto \cosh x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)} \]

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

         \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right) \]

      4. lift-*.f64 13.6

         \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

   10. Applied rewrites 13.6%

       \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{-0.16666666666666666}\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right) \]

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 13.6

          \[\leadsto \cosh x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right) \]

   12. Applied rewrites 13.6%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.999999999743085621

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      9. cosh-undef N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      11. lift-cosh.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{{x}^{2} + 2}{y} \]

      2. pow2 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{x \cdot x + 2}{y} \]

      3. lower-fma.f64 81.4

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

   7. Applied rewrites 81.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-*.f64 75.8

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

   9. Applied rewrites 75.8%

      \[\leadsto \frac{\left(\sin y \cdot 0.5\right) \cdot \mathsf{fma}\left(x, x, 2\right)}{\color{blue}{y}} \]

   if 0.999999999743085621 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      9. cosh-undef N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      11. lift-cosh.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      4. cosh-undef-rev N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      5. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + \frac{1}{e^{x}}}{y} \]

      6. +-commutative N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{\frac{1}{e^{x}} + e^{x}}{y} \]

      7. div-add N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \left(\frac{e^{-x}}{y} + \frac{e^{x}}{y}\right) \]

      13. lower-/.f64 N/A

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

      14. lower-exp.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \left(\frac{e^{-x}}{y} + \frac{e^{x}}{y}\right) \]

   6. Applied rewrites 99.8%

      \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \left(\frac{e^{-x}}{y} + \color{blue}{\frac{e^{x}}{y}}\right) \]

   7. Taylor expanded in y around 0

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

      1. cosh-undef-rev N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \cosh x\right) \]

      2. *-commutative N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      5. lift-cosh.f64 64.0

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot 0.5 \]

   9. Applied rewrites 64.0%

      \[\leadsto \left(2 \cdot \cosh x\right) \cdot \color{blue}{0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999999997430856:\\ \;\;\;\;\left(\sin y \cdot 0.5\right) \cdot \frac{\mathsf{fma}\left(x, x, 2\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cosh x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ (sin y) y))))
   (if (<= t_0 (- INFINITY))
     (* (cosh x) (* y (* y -0.16666666666666666)))
     (if (<= t_0 0.9999999997430856)
       (* (* (sin y) 0.5) (/ (fma x x 2.0) y))
       (* (* 2.0 (cosh x)) 0.5)))))

C

double code(double x, double y) {
	double t_0 = cosh(x) * (sin(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = cosh(x) * (y * (y * -0.16666666666666666));
	} else if (t_0 <= 0.9999999997430856) {
		tmp = (sin(y) * 0.5) * (fma(x, x, 2.0) / y);
	} else {
		tmp = (2.0 * cosh(x)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cosh(x) * Float64(sin(y) / y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(cosh(x) * Float64(y * Float64(y * -0.16666666666666666)));
	elseif (t_0 <= 0.9999999997430856)
		tmp = Float64(Float64(sin(y) * 0.5) * Float64(fma(x, x, 2.0) / y));
	else
		tmp = Float64(Float64(2.0 * cosh(x)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999999997430856], N[(N[(N[Sin[y], $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(x * x + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -inf.0

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 64.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 63.1

         \[\leadsto \cosh x \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \]

   7. Applied rewrites 63.1%

      \[\leadsto \cosh x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)} \]

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

         \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right) \]

      4. lift-*.f64 13.6

         \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

   10. Applied rewrites 13.6%

       \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{-0.16666666666666666}\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right) \]

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 13.6

          \[\leadsto \cosh x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right) \]

   12. Applied rewrites 13.6%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.999999999743085621

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      9. cosh-undef N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      11. lift-cosh.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{{x}^{2} + 2}{y} \]

      2. pow2 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{x \cdot x + 2}{y} \]

      3. lower-fma.f64 81.4

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

   7. Applied rewrites 81.4%

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

   if 0.999999999743085621 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      9. cosh-undef N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      11. lift-cosh.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      4. cosh-undef-rev N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      5. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + \frac{1}{e^{x}}}{y} \]

      6. +-commutative N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{\frac{1}{e^{x}} + e^{x}}{y} \]

      7. div-add N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \left(\frac{e^{-x}}{y} + \frac{e^{x}}{y}\right) \]

      13. lower-/.f64 N/A

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

      14. lower-exp.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \left(\frac{e^{-x}}{y} + \frac{e^{x}}{y}\right) \]

   6. Applied rewrites 99.8%

      \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \left(\frac{e^{-x}}{y} + \color{blue}{\frac{e^{x}}{y}}\right) \]

   7. Taylor expanded in y around 0

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

      1. cosh-undef-rev N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \cosh x\right) \]

      2. *-commutative N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      5. lift-cosh.f64 64.0

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot 0.5 \]

   9. Applied rewrites 64.0%

      \[\leadsto \left(2 \cdot \cosh x\right) \cdot \color{blue}{0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999997430856:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cosh x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* y (* y -0.16666666666666666)))
     (if (<= t_1 0.9999999997430856) t_0 (* (* 2.0 (cosh x)) 0.5)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * (y * (y * -0.16666666666666666));
	} else if (t_1 <= 0.9999999997430856) {
		tmp = t_0;
	} else {
		tmp = (2.0 * cosh(x)) * 0.5;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) / y;
	double t_1 = Math.cosh(x) * t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.cosh(x) * (y * (y * -0.16666666666666666));
	} else if (t_1 <= 0.9999999997430856) {
		tmp = t_0;
	} else {
		tmp = (2.0 * Math.cosh(x)) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) / y
	t_1 = math.cosh(x) * t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = math.cosh(x) * (y * (y * -0.16666666666666666))
	elif t_1 <= 0.9999999997430856:
		tmp = t_0
	else:
		tmp = (2.0 * math.cosh(x)) * 0.5
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * Float64(y * Float64(y * -0.16666666666666666)));
	elseif (t_1 <= 0.9999999997430856)
		tmp = t_0;
	else
		tmp = Float64(Float64(2.0 * cosh(x)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) / y;
	t_1 = cosh(x) * t_0;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = cosh(x) * (y * (y * -0.16666666666666666));
	elseif (t_1 <= 0.9999999997430856)
		tmp = t_0;
	else
		tmp = (2.0 * cosh(x)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999997430856], t$95$0, N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -inf.0

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 64.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 63.1

         \[\leadsto \cosh x \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \]

   7. Applied rewrites 63.1%

      \[\leadsto \cosh x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)} \]

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

         \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right) \]

      4. lift-*.f64 13.6

         \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

   10. Applied rewrites 13.6%

       \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{-0.16666666666666666}\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right) \]

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 13.6

          \[\leadsto \cosh x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right) \]

   12. Applied rewrites 13.6%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.999999999743085621

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\sin y}{y} \]

      2. lift-/.f64 50.3

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

   4. Applied rewrites 50.3%

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

   if 0.999999999743085621 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      9. cosh-undef N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      11. lift-cosh.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      4. cosh-undef-rev N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      5. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + \frac{1}{e^{x}}}{y} \]

      6. +-commutative N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{\frac{1}{e^{x}} + e^{x}}{y} \]

      7. div-add N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \left(\frac{e^{-x}}{y} + \frac{e^{x}}{y}\right) \]

      13. lower-/.f64 N/A

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

      14. lower-exp.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \left(\frac{e^{-x}}{y} + \frac{e^{x}}{y}\right) \]

   6. Applied rewrites 99.8%

      \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \left(\frac{e^{-x}}{y} + \color{blue}{\frac{e^{x}}{y}}\right) \]

   7. Taylor expanded in y around 0

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

      1. cosh-undef-rev N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \cosh x\right) \]

      2. *-commutative N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      5. lift-cosh.f64 64.0

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot 0.5 \]

   9. Applied rewrites 64.0%

      \[\leadsto \left(2 \cdot \cosh x\right) \cdot \color{blue}{0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 76.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\cosh x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cosh x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-153)
   (* (cosh x) (* y (* y -0.16666666666666666)))
   (* (* 2.0 (cosh x)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-153) {
		tmp = cosh(x) * (y * (y * -0.16666666666666666));
	} else {
		tmp = (2.0 * cosh(x)) * 0.5;
	}
	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 ((cosh(x) * (sin(y) / y)) <= (-1d-153)) then
        tmp = cosh(x) * (y * (y * (-0.16666666666666666d0)))
    else
        tmp = (2.0d0 * cosh(x)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.cosh(x) * (Math.sin(y) / y)) <= -1e-153) {
		tmp = Math.cosh(x) * (y * (y * -0.16666666666666666));
	} else {
		tmp = (2.0 * Math.cosh(x)) * 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.cosh(x) * (math.sin(y) / y)) <= -1e-153:
		tmp = math.cosh(x) * (y * (y * -0.16666666666666666))
	else:
		tmp = (2.0 * math.cosh(x)) * 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-153)
		tmp = Float64(cosh(x) * Float64(y * Float64(y * -0.16666666666666666)));
	else
		tmp = Float64(Float64(2.0 * cosh(x)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((cosh(x) * (sin(y) / y)) <= -1e-153)
		tmp = cosh(x) * (y * (y * -0.16666666666666666));
	else
		tmp = (2.0 * cosh(x)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-153], N[(N[Cosh[x], $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000004e-153

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 64.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 63.1

         \[\leadsto \cosh x \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \]

   7. Applied rewrites 63.1%

      \[\leadsto \cosh x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)} \]

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

         \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right) \]

      4. lift-*.f64 13.6

         \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \]

   10. Applied rewrites 13.6%

       \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot \color{blue}{-0.16666666666666666}\right) \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \cosh x \cdot \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right) \]

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 13.6

          \[\leadsto \cosh x \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right)\right) \]

   12. Applied rewrites 13.6%

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

   if -1.00000000000000004e-153 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      9. cosh-undef N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      11. lift-cosh.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      4. cosh-undef-rev N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      5. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + \frac{1}{e^{x}}}{y} \]

      6. +-commutative N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{\frac{1}{e^{x}} + e^{x}}{y} \]

      7. div-add N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \left(\frac{e^{-x}}{y} + \frac{e^{x}}{y}\right) \]

      13. lower-/.f64 N/A

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

      14. lower-exp.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \left(\frac{e^{-x}}{y} + \frac{e^{x}}{y}\right) \]

   6. Applied rewrites 99.8%

      \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \left(\frac{e^{-x}}{y} + \color{blue}{\frac{e^{x}}{y}}\right) \]

   7. Taylor expanded in y around 0

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

      1. cosh-undef-rev N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \cosh x\right) \]

      2. *-commutative N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      5. lift-cosh.f64 64.0

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot 0.5 \]

   9. Applied rewrites 64.0%

      \[\leadsto \left(2 \cdot \cosh x\right) \cdot \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 72.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \cosh x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-153)
   (/ (* (fma (* y y) -0.16666666666666666 1.0) y) y)
   (* (* 2.0 (cosh x)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-153) {
		tmp = (fma((y * y), -0.16666666666666666, 1.0) * y) / y;
	} else {
		tmp = (2.0 * cosh(x)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-153)
		tmp = Float64(Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * y) / y);
	else
		tmp = Float64(Float64(2.0 * cosh(x)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-153], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000004e-153

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\sin y}{y} \]

      2. lift-/.f64 50.3

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

   4. Applied rewrites 50.3%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

         \[\leadsto \frac{\left(1 + \left(y \cdot y\right) \cdot \frac{-1}{6}\right) \cdot y}{y} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left(\left(y \cdot y\right) \cdot \frac{-1}{6} + 1\right) \cdot y}{y} \]

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 34.7

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}{y} \]

   7. Applied rewrites 34.7%

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}{y} \]

   if -1.00000000000000004e-153 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      9. cosh-undef N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      11. lift-cosh.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      4. cosh-undef-rev N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      5. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + \frac{1}{e^{x}}}{y} \]

      6. +-commutative N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{\frac{1}{e^{x}} + e^{x}}{y} \]

      7. div-add N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \left(\frac{e^{-x}}{y} + \frac{e^{x}}{y}\right) \]

      13. lower-/.f64 N/A

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

      14. lower-exp.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \left(\frac{e^{-x}}{y} + \frac{e^{x}}{y}\right) \]

   6. Applied rewrites 99.8%

      \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \left(\frac{e^{-x}}{y} + \color{blue}{\frac{e^{x}}{y}}\right) \]

   7. Taylor expanded in y around 0

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

      1. cosh-undef-rev N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \cosh x\right) \]

      2. *-commutative N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      5. lift-cosh.f64 64.0

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot 0.5 \]

   9. Applied rewrites 64.0%

      \[\leadsto \left(2 \cdot \cosh x\right) \cdot \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 64.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-301}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}{y}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-194}:\\ \;\;\;\;\frac{\left(0.5 \cdot y\right) \cdot \mathsf{fma}\left(x, x, 2\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -1e-301)
     (/ (* (fma (* y y) -0.16666666666666666 1.0) y) y)
     (if (<= t_0 4e-194)
       (/ (* (* 0.5 y) (fma x x 2.0)) y)
       (fma (fma 0.041666666666666664 (* x x) 0.5) (* x x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -1e-301) {
		tmp = (fma((y * y), -0.16666666666666666, 1.0) * y) / y;
	} else if (t_0 <= 4e-194) {
		tmp = ((0.5 * y) * fma(x, x, 2.0)) / y;
	} else {
		tmp = fma(fma(0.041666666666666664, (x * x), 0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -1e-301)
		tmp = Float64(Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * y) / y);
	elseif (t_0 <= 4e-194)
		tmp = Float64(Float64(Float64(0.5 * y) * fma(x, x, 2.0)) / y);
	else
		tmp = fma(fma(0.041666666666666664, Float64(x * x), 0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-301], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 4e-194], N[(N[(N[(0.5 * y), $MachinePrecision] * N[(x * x + 2.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.00000000000000007e-301

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\sin y}{y} \]

      2. lift-/.f64 50.3

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

   4. Applied rewrites 50.3%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

         \[\leadsto \frac{\left(1 + \left(y \cdot y\right) \cdot \frac{-1}{6}\right) \cdot y}{y} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left(\left(y \cdot y\right) \cdot \frac{-1}{6} + 1\right) \cdot y}{y} \]

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 34.7

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}{y} \]

   7. Applied rewrites 34.7%

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}{y} \]

   if -1.00000000000000007e-301 < (/.f64 (sin.f64 y) y) < 4.00000000000000007e-194

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      9. cosh-undef N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      11. lift-cosh.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{{x}^{2} + 2}{y} \]

      2. pow2 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{x \cdot x + 2}{y} \]

      3. lower-fma.f64 81.4

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

   7. Applied rewrites 81.4%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 51.6

         \[\leadsto \left(0.5 \cdot y\right) \cdot \frac{\mathsf{fma}\left(x, \color{blue}{x}, 2\right)}{y} \]

   10. Applied rewrites 51.6%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 48.9

          \[\leadsto \frac{\left(0.5 \cdot y\right) \cdot \mathsf{fma}\left(x, x, 2\right)}{y} \]

   12. Applied rewrites 48.9%

       \[\leadsto \frac{\left(0.5 \cdot y\right) \cdot \mathsf{fma}\left(x, x, 2\right)}{\color{blue}{y}} \]

   if 4.00000000000000007e-194 < (/.f64 (sin.f64 y) y)

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      9. cosh-undef N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      11. lift-cosh.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      4. cosh-undef-rev N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      5. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + \frac{1}{e^{x}}}{y} \]

      6. +-commutative N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{\frac{1}{e^{x}} + e^{x}}{y} \]

      7. div-add N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \left(\frac{e^{-x}}{y} + \frac{e^{x}}{y}\right) \]

      13. lower-/.f64 N/A

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

      14. lower-exp.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \left(\frac{e^{-x}}{y} + \frac{e^{x}}{y}\right) \]

   6. Applied rewrites 99.8%

      \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \left(\frac{e^{-x}}{y} + \color{blue}{\frac{e^{x}}{y}}\right) \]

   7. Taylor expanded in y around 0

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

      1. cosh-undef-rev N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \cosh x\right) \]

      2. *-commutative N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      5. lift-cosh.f64 64.0

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot 0.5 \]

   9. Applied rewrites 64.0%

      \[\leadsto \left(2 \cdot \cosh x\right) \cdot \color{blue}{0.5} \]

   10. Taylor expanded in x around 0

       \[\leadsto 1 + {x}^{2} \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1 \]

       2. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2} + 1 \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}, {x}^{2}, 1\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right) \]

       6. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), {x}^{2}, 1\right) \]

       8. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right) \]

       9. lower-*.f64 55.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right) \]

   12. Applied rewrites 55.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot \color{blue}{x}, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 62.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-301}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\left(0.5 \cdot y\right) \cdot \mathsf{fma}\left(x, x, 2\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{\mathsf{fma}\left(x, x, 2\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (if (<= t_0 -1e-301)
     (/ (* (fma (* y y) -0.16666666666666666 1.0) y) y)
     (if (<= t_0 5e-25)
       (/ (* (* 0.5 y) (fma x x 2.0)) y)
       (* (* 0.5 y) (/ (fma x x 2.0) y))))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double tmp;
	if (t_0 <= -1e-301) {
		tmp = (fma((y * y), -0.16666666666666666, 1.0) * y) / y;
	} else if (t_0 <= 5e-25) {
		tmp = ((0.5 * y) * fma(x, x, 2.0)) / y;
	} else {
		tmp = (0.5 * y) * (fma(x, x, 2.0) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (t_0 <= -1e-301)
		tmp = Float64(Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * y) / y);
	elseif (t_0 <= 5e-25)
		tmp = Float64(Float64(Float64(0.5 * y) * fma(x, x, 2.0)) / y);
	else
		tmp = Float64(Float64(0.5 * y) * Float64(fma(x, x, 2.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-301], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 5e-25], N[(N[(N[(0.5 * y), $MachinePrecision] * N[(x * x + 2.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] * N[(N[(x * x + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 y) y) < -1.00000000000000007e-301

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\sin y}{y} \]

      2. lift-/.f64 50.3

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

   4. Applied rewrites 50.3%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

         \[\leadsto \frac{\left(1 + \left(y \cdot y\right) \cdot \frac{-1}{6}\right) \cdot y}{y} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left(\left(y \cdot y\right) \cdot \frac{-1}{6} + 1\right) \cdot y}{y} \]

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 34.7

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}{y} \]

   7. Applied rewrites 34.7%

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}{y} \]

   if -1.00000000000000007e-301 < (/.f64 (sin.f64 y) y) < 4.99999999999999962e-25

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      9. cosh-undef N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      11. lift-cosh.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{{x}^{2} + 2}{y} \]

      2. pow2 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{x \cdot x + 2}{y} \]

      3. lower-fma.f64 81.4

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

   7. Applied rewrites 81.4%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 51.6

         \[\leadsto \left(0.5 \cdot y\right) \cdot \frac{\mathsf{fma}\left(x, \color{blue}{x}, 2\right)}{y} \]

   10. Applied rewrites 51.6%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 48.9

          \[\leadsto \frac{\left(0.5 \cdot y\right) \cdot \mathsf{fma}\left(x, x, 2\right)}{y} \]

   12. Applied rewrites 48.9%

       \[\leadsto \frac{\left(0.5 \cdot y\right) \cdot \mathsf{fma}\left(x, x, 2\right)}{\color{blue}{y}} \]

   if 4.99999999999999962e-25 < (/.f64 (sin.f64 y) y)

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      9. cosh-undef N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      11. lift-cosh.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{{x}^{2} + 2}{y} \]

      2. pow2 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{x \cdot x + 2}{y} \]

      3. lower-fma.f64 81.4

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

   7. Applied rewrites 81.4%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 51.6

         \[\leadsto \left(0.5 \cdot y\right) \cdot \frac{\mathsf{fma}\left(x, \color{blue}{x}, 2\right)}{y} \]

   10. Applied rewrites 51.6%

       \[\leadsto \left(0.5 \cdot y\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 60.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -1 \cdot 10^{-153}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{\mathsf{fma}\left(x, x, 2\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -1e-153)
   (/ (* (fma (* y y) -0.16666666666666666 1.0) y) y)
   (* (* 0.5 y) (/ (fma x x 2.0) y))))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -1e-153) {
		tmp = (fma((y * y), -0.16666666666666666, 1.0) * y) / y;
	} else {
		tmp = (0.5 * y) * (fma(x, x, 2.0) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -1e-153)
		tmp = Float64(Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * y) / y);
	else
		tmp = Float64(Float64(0.5 * y) * Float64(fma(x, x, 2.0) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -1e-153], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] * N[(N[(x * x + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -1.00000000000000004e-153

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

         \[\leadsto \frac{\sin y}{y} \]

      2. lift-/.f64 50.3

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

   4. Applied rewrites 50.3%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

         \[\leadsto \frac{\left(1 + \left(y \cdot y\right) \cdot \frac{-1}{6}\right) \cdot y}{y} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left(\left(y \cdot y\right) \cdot \frac{-1}{6} + 1\right) \cdot y}{y} \]

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 34.7

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}{y} \]

   7. Applied rewrites 34.7%

      \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot y}{y} \]

   if -1.00000000000000004e-153 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      9. cosh-undef N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      11. lift-cosh.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{{x}^{2} + 2}{y} \]

      2. pow2 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{x \cdot x + 2}{y} \]

      3. lower-fma.f64 81.4

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

   7. Applied rewrites 81.4%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 51.6

         \[\leadsto \left(0.5 \cdot y\right) \cdot \frac{\mathsf{fma}\left(x, \color{blue}{x}, 2\right)}{y} \]

   10. Applied rewrites 51.6%

       \[\leadsto \left(0.5 \cdot y\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 51.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \frac{x \cdot x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 2.0) (fma (* x x) 0.5 1.0) (* (* 0.5 y) (/ (* x x) y))))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 2.0) {
		tmp = fma((x * x), 0.5, 1.0);
	} else {
		tmp = (0.5 * y) * ((x * x) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 2.0)
		tmp = fma(Float64(x * x), 0.5, 1.0);
	else
		tmp = Float64(Float64(0.5 * y) * Float64(Float64(x * x) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 2.0], N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision], N[(N[(0.5 * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 2

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      9. cosh-undef N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      11. lift-cosh.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      3. lift-cosh.f64 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      4. cosh-undef-rev N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      5. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + \frac{1}{e^{x}}}{y} \]

      6. +-commutative N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{\frac{1}{e^{x}} + e^{x}}{y} \]

      7. div-add N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. rec-exp N/A

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

      11. lower-exp.f64 N/A

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

      12. lower-neg.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \left(\frac{e^{-x}}{y} + \frac{e^{x}}{y}\right) \]

      13. lower-/.f64 N/A

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

      14. lower-exp.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \left(\frac{e^{-x}}{y} + \frac{e^{x}}{y}\right) \]

   6. Applied rewrites 99.8%

      \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \left(\frac{e^{-x}}{y} + \color{blue}{\frac{e^{x}}{y}}\right) \]

   7. Taylor expanded in y around 0

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

      1. cosh-undef-rev N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \cosh x\right) \]

      2. *-commutative N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

      5. lift-cosh.f64 64.0

         \[\leadsto \left(2 \cdot \cosh x\right) \cdot 0.5 \]

   9. Applied rewrites 64.0%

      \[\leadsto \left(2 \cdot \cosh x\right) \cdot \color{blue}{0.5} \]

   10. Taylor expanded in x around 0

       \[\leadsto 1 + \frac{1}{2} \cdot \color{blue}{{x}^{2}} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{2} \cdot {x}^{2} + 1 \]

       2. *-commutative N/A

          \[\leadsto {x}^{2} \cdot \frac{1}{2} + 1 \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right) \]

       4. pow2 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \]

       5. lower-*.f64 45.9

          \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \]

   12. Applied rewrites 45.9%

       \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \]

   if 2 < (cosh.f64 x)

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. rec-exp N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

      9. cosh-undef N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

      11. lift-cosh.f64 99.8

          \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{{x}^{2} + 2}{y} \]

      2. pow2 N/A

         \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{x \cdot x + 2}{y} \]

      3. lower-fma.f64 81.4

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

   7. Applied rewrites 81.4%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 51.6

         \[\leadsto \left(0.5 \cdot y\right) \cdot \frac{\mathsf{fma}\left(x, \color{blue}{x}, 2\right)}{y} \]

   10. Applied rewrites 51.6%

       \[\leadsto \left(0.5 \cdot y\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}}{y} \]

   11. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. lower-*.f64 28.1

          \[\leadsto \left(0.5 \cdot y\right) \cdot \frac{x \cdot x}{y} \]

   13. Applied rewrites 28.1%

       \[\leadsto \left(0.5 \cdot y\right) \cdot \frac{x \cdot x}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 51.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot y\right) \cdot \frac{\mathsf{fma}\left(x, x, 2\right)}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (* 0.5 y) (/ (fma x x 2.0) y)))

C

double code(double x, double y) {
	return (0.5 * y) * (fma(x, x, 2.0) / y);
}

Julia

function code(x, y)
	return Float64(Float64(0.5 * y) * Float64(fma(x, x, 2.0) / y))
end

Wolfram

code[x_, y_] := N[(N[(0.5 * y), $MachinePrecision] * N[(N[(x * x + 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]

2. Taylor expanded in x around inf

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. rec-exp N/A

      \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

   9. cosh-undef N/A

      \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   11. lift-cosh.f64 99.8

       \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y} \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y}} \]

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{{x}^{2} + 2}{y} \]

   2. pow2 N/A

      \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{x \cdot x + 2}{y} \]

   3. lower-fma.f64 81.4

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

7. Applied rewrites 81.4%

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

8. Taylor expanded in y around 0

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

   1. lower-*.f64 51.6

      \[\leadsto \left(0.5 \cdot y\right) \cdot \frac{\mathsf{fma}\left(x, \color{blue}{x}, 2\right)}{y} \]

10. Applied rewrites 51.6%

    \[\leadsto \left(0.5 \cdot y\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}}{y} \]
11. Add Preprocessing

Alternative 13: 45.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (* x x) 0.5 1.0))

C

double code(double x, double y) {
	return fma((x * x), 0.5, 1.0);
}

Julia

function code(x, y)
	return fma(Float64(x * x), 0.5, 1.0)
end

Wolfram

code[x_, y_] := N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]

2. Taylor expanded in x around inf

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. rec-exp N/A

      \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

   9. cosh-undef N/A

      \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   11. lift-cosh.f64 99.8

       \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y} \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\left(\sin y \cdot 0.5\right) \cdot \frac{2 \cdot \cosh x}{y}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

      \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   3. lift-cosh.f64 N/A

      \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{2 \cdot \cosh x}{y} \]

   4. cosh-undef-rev N/A

      \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{y} \]

   5. rec-exp N/A

      \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{e^{x} + \frac{1}{e^{x}}}{y} \]

   6. +-commutative N/A

      \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \frac{\frac{1}{e^{x}} + e^{x}}{y} \]

   7. div-add N/A

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

   8. lower-+.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. rec-exp N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-neg.f64 N/A

       \[\leadsto \left(\sin y \cdot \frac{1}{2}\right) \cdot \left(\frac{e^{-x}}{y} + \frac{e^{x}}{y}\right) \]

   13. lower-/.f64 N/A

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

   14. lower-exp.f64 99.8

       \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \left(\frac{e^{-x}}{y} + \frac{e^{x}}{y}\right) \]

6. Applied rewrites 99.8%

   \[\leadsto \left(\sin y \cdot 0.5\right) \cdot \left(\frac{e^{-x}}{y} + \color{blue}{\frac{e^{x}}{y}}\right) \]

7. Taylor expanded in y around 0

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

   1. cosh-undef-rev N/A

      \[\leadsto \frac{1}{2} \cdot \left(2 \cdot \cosh x\right) \]

   2. *-commutative N/A

      \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(2 \cdot \cosh x\right) \cdot \frac{1}{2} \]

   5. lift-cosh.f64 64.0

      \[\leadsto \left(2 \cdot \cosh x\right) \cdot 0.5 \]

9. Applied rewrites 64.0%

   \[\leadsto \left(2 \cdot \cosh x\right) \cdot \color{blue}{0.5} \]

10. Taylor expanded in x around 0

    \[\leadsto 1 + \frac{1}{2} \cdot \color{blue}{{x}^{2}} \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{1}{2} \cdot {x}^{2} + 1 \]

    2. *-commutative N/A

       \[\leadsto {x}^{2} \cdot \frac{1}{2} + 1 \]

    3. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{2}, \frac{1}{2}, 1\right) \]

    4. pow2 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{2}, 1\right) \]

    5. lower-*.f64 45.9

       \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \]

12. Applied rewrites 45.9%

    \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \]
13. Add Preprocessing

Alternative 14: 27.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \frac{y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 1.0 (/ y y)))

C

double code(double x, double y) {
	return 1.0 * (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 = 1.0d0 * (y / y)
end function

Java

public static double code(double x, double y) {
	return 1.0 * (y / y);
}

Python

def code(x, y):
	return 1.0 * (y / y)

Julia

function code(x, y)
	return Float64(1.0 * Float64(y / y))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 * (y / y);
end

Wolfram

code[x_, y_] := N[(1.0 * N[(y / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]

2. Taylor expanded in y around 0

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

4. Applied rewrites 64.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \cdot \frac{y}{y} \]
6. Step-by-step derivation

7. Applied rewrites 27.1%

   \[\leadsto \color{blue}{1} \cdot \frac{y}{y} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025139 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64
  (* (cosh x) (/ (sin y) y)))