Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.9% → 99.8%

Time: 4.4s

Alternatives: 17

Speedup: 1.4×

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sinh.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-sinh.f64 N/A

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

   10. lift-sin.f64 99.5

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\sinh y}{x} \cdot \sin x} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ t_1 := 2 \cdot \sinh y\\ \mathbf{if}\;t\_0 \leq -500000:\\ \;\;\;\;t\_1 \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)) (t_1 (* 2.0 (sinh y))))
   (if (<= t_0 -500000.0)
     (* t_1 0.5)
     (if (<= t_0 4e-8)
       (* (/ (sin x) x) y)
       (* t_1 (fma (* x x) -0.08333333333333333 0.5))))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double t_1 = 2.0 * sinh(y);
	double tmp;
	if (t_0 <= -500000.0) {
		tmp = t_1 * 0.5;
	} else if (t_0 <= 4e-8) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = t_1 * fma((x * x), -0.08333333333333333, 0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	t_1 = Float64(2.0 * sinh(y))
	tmp = 0.0
	if (t_0 <= -500000.0)
		tmp = Float64(t_1 * 0.5);
	elseif (t_0 <= 4e-8)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = Float64(t_1 * fma(Float64(x * x), -0.08333333333333333, 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -500000.0], N[(t$95$1 * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 4e-8], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[(t$95$1 * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -5e5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 77.6

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 77.6%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   if -5e5 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.0000000000000001e-8

   1. Initial program 78.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 4.0000000000000001e-8 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {x}^{2}\right) \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + \color{blue}{\frac{1}{2}} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      4. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      5. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      7. lift-sinh.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 77.9

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   5. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 86.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -500000 \lor \neg \left(t\_0 \leq 4 \cdot 10^{-8}\right):\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (or (<= t_0 -500000.0) (not (<= t_0 4e-8)))
     (* (* 2.0 (sinh y)) 0.5)
     (* (/ (sin x) x) y))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if ((t_0 <= -500000.0) || !(t_0 <= 4e-8)) {
		tmp = (2.0 * sinh(y)) * 0.5;
	} else {
		tmp = (sin(x) / x) * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(x) * sinh(y)) / x
    if ((t_0 <= (-500000.0d0)) .or. (.not. (t_0 <= 4d-8))) then
        tmp = (2.0d0 * sinh(y)) * 0.5d0
    else
        tmp = (sin(x) / x) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (Math.sin(x) * Math.sinh(y)) / x;
	double tmp;
	if ((t_0 <= -500000.0) || !(t_0 <= 4e-8)) {
		tmp = (2.0 * Math.sinh(y)) * 0.5;
	} else {
		tmp = (Math.sin(x) / x) * y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sin(x) * math.sinh(y)) / x
	tmp = 0
	if (t_0 <= -500000.0) or not (t_0 <= 4e-8):
		tmp = (2.0 * math.sinh(y)) * 0.5
	else:
		tmp = (math.sin(x) / x) * y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if ((t_0 <= -500000.0) || !(t_0 <= 4e-8))
		tmp = Float64(Float64(2.0 * sinh(y)) * 0.5);
	else
		tmp = Float64(Float64(sin(x) / x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sin(x) * sinh(y)) / x;
	tmp = 0.0;
	if ((t_0 <= -500000.0) || ~((t_0 <= 4e-8)))
		tmp = (2.0 * sinh(y)) * 0.5;
	else
		tmp = (sin(x) / x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -500000.0], N[Not[LessEqual[t$95$0, 4e-8]], $MachinePrecision]], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -5e5 or 4.0000000000000001e-8 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 77.0

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   if -5e5 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.0000000000000001e-8

   1. Initial program 78.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -500000 \lor \neg \left(\frac{\sin x \cdot \sinh y}{x} \leq 4 \cdot 10^{-8}\right):\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0064:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.0064)
   (* (* 2.0 (sinh y)) 0.5)
   (/
    (*
     (sin x)
     (*
      (fma
       (fma
        (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
        (* y y)
        0.16666666666666666)
       (* y y)
       1.0)
      y))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.0064) {
		tmp = (2.0 * sinh(y)) * 0.5;
	} else {
		tmp = (sin(x) * (fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0) * y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.0064)
		tmp = Float64(Float64(2.0 * sinh(y)) * 0.5);
	else
		tmp = Float64(Float64(sin(x) * Float64(fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.0064], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00640000000000000031

   1. Initial program 87.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 72.0

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   if 0.00640000000000000031 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{2}, 1\right) \cdot y\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}, {y}^{2}, 1\right) \cdot y\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left(\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2} + \frac{1}{6}, {y}^{2}, 1\right) \cdot y\right)}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, {y}^{2}, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot y\right)}{x} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}, {y}^{2}, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot y\right)}{x} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, {y}^{2}, \frac{1}{120}\right), {y}^{2}, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot y\right)}{x} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right), {y}^{2}, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot y\right)}{x} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right), {y}^{2}, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot y\right)}{x} \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot y\right)}{x} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot y\right)}{x} \]

      15. unpow2 N/A

          \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right), y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right) \cdot y\right)}{x} \]

      16. lower-*.f64 88.4

          \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right)}{x} \]

   5. Applied rewrites 88.4%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right)}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 92.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \sinh y\\ t_1 := \left(\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\right) \cdot \sin x\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -0.095:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{elif}\;y \leq 0.195 \lor \neg \left(y \leq 1.6 \cdot 10^{+147}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 2.0 (sinh y)))
        (t_1 (* (* (/ (fma (* y y) 0.16666666666666666 1.0) x) y) (sin x))))
   (if (<= y -3.2e+132)
     t_1
     (if (<= y -0.095)
       (* t_0 (fma (* x x) -0.08333333333333333 0.5))
       (if (or (<= y 0.195) (not (<= y 1.6e+147))) t_1 (* t_0 0.5))))))

C

double code(double x, double y) {
	double t_0 = 2.0 * sinh(y);
	double t_1 = ((fma((y * y), 0.16666666666666666, 1.0) / x) * y) * sin(x);
	double tmp;
	if (y <= -3.2e+132) {
		tmp = t_1;
	} else if (y <= -0.095) {
		tmp = t_0 * fma((x * x), -0.08333333333333333, 0.5);
	} else if ((y <= 0.195) || !(y <= 1.6e+147)) {
		tmp = t_1;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 * sinh(y))
	t_1 = Float64(Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) / x) * y) * sin(x))
	tmp = 0.0
	if (y <= -3.2e+132)
		tmp = t_1;
	elseif (y <= -0.095)
		tmp = Float64(t_0 * fma(Float64(x * x), -0.08333333333333333, 0.5));
	elseif ((y <= 0.195) || !(y <= 1.6e+147))
		tmp = t_1;
	else
		tmp = Float64(t_0 * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.2e+132], t$95$1, If[LessEqual[y, -0.095], N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 0.195], N[Not[LessEqual[y, 1.6e+147]], $MachinePrecision]], t$95$1, N[(t$95$0 * 0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.1999999999999997e132 or -0.095000000000000001 < y < 0.19500000000000001 or 1.59999999999999989e147 < y

   1. Initial program 86.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 99.0

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

   7. Applied rewrites 99.0%

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

   if -3.1999999999999997e132 < y < -0.095000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {x}^{2}\right) \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + \color{blue}{\frac{1}{2}} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      4. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      5. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      7. lift-sinh.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 81.5

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   5. Applied rewrites 81.5%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)} \]

   if 0.19500000000000001 < y < 1.59999999999999989e147

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 80.0

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+132}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\right) \cdot \sin x\\ \mathbf{elif}\;y \leq -0.095:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{elif}\;y \leq 0.195 \lor \neg \left(y \leq 1.6 \cdot 10^{+147}\right):\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\right) \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ t_1 := \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ t_2 := \frac{\left(t\_1 \cdot \sin x\right) \cdot y}{x}\\ \mathbf{if}\;y \leq -4 \cdot 10^{+106}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -0.095:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.195:\\ \;\;\;\;\left(\frac{t\_1}{x} \cdot y\right) \cdot \sin x\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+103}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 2.0 (sinh y)) (fma (* x x) -0.08333333333333333 0.5)))
        (t_1 (fma (* y y) 0.16666666666666666 1.0))
        (t_2 (/ (* (* t_1 (sin x)) y) x)))
   (if (<= y -4e+106)
     t_2
     (if (<= y -0.095)
       t_0
       (if (<= y 0.195)
         (* (* (/ t_1 x) y) (sin x))
         (if (<= y 1.5e+103) t_0 t_2))))))

C

double code(double x, double y) {
	double t_0 = (2.0 * sinh(y)) * fma((x * x), -0.08333333333333333, 0.5);
	double t_1 = fma((y * y), 0.16666666666666666, 1.0);
	double t_2 = ((t_1 * sin(x)) * y) / x;
	double tmp;
	if (y <= -4e+106) {
		tmp = t_2;
	} else if (y <= -0.095) {
		tmp = t_0;
	} else if (y <= 0.195) {
		tmp = ((t_1 / x) * y) * sin(x);
	} else if (y <= 1.5e+103) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 * sinh(y)) * fma(Float64(x * x), -0.08333333333333333, 0.5))
	t_1 = fma(Float64(y * y), 0.16666666666666666, 1.0)
	t_2 = Float64(Float64(Float64(t_1 * sin(x)) * y) / x)
	tmp = 0.0
	if (y <= -4e+106)
		tmp = t_2;
	elseif (y <= -0.095)
		tmp = t_0;
	elseif (y <= 0.195)
		tmp = Float64(Float64(Float64(t_1 / x) * y) * sin(x));
	elseif (y <= 1.5e+103)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 * N[Sin[x], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, -4e+106], t$95$2, If[LessEqual[y, -0.095], t$95$0, If[LessEqual[y, 0.195], N[(N[(N[(t$95$1 / x), $MachinePrecision] * y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+103], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.00000000000000036e106 or 1.5e103 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sin.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -4.00000000000000036e106 < y < -0.095000000000000001 or 0.19500000000000001 < y < 1.5e103

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {x}^{2}\right) \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + \color{blue}{\frac{1}{2}} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      4. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      5. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      7. lift-sinh.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 79.6

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   5. Applied rewrites 79.6%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)} \]

   if -0.095000000000000001 < y < 0.19500000000000001

   1. Initial program 78.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 99.2

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

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\right)} \cdot \sin x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0064:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.0064)
   (* (* 2.0 (sinh y)) 0.5)
   (/
    (*
     (sin x)
     (*
      (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0)
      y))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.0064) {
		tmp = (2.0 * sinh(y)) * 0.5;
	} else {
		tmp = (sin(x) * (fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0) * y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.0064)
		tmp = Float64(Float64(2.0 * sinh(y)) * 0.5);
	else
		tmp = Float64(Float64(sin(x) * Float64(fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0) * y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.0064], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00640000000000000031

   1. Initial program 87.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 72.0

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   if 0.00640000000000000031 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, {y}^{2}, 1\right) \cdot y\right)}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}, {y}^{2}, 1\right) \cdot y\right)}{x} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left({y}^{2} \cdot \frac{1}{120} + \frac{1}{6}, {y}^{2}, 1\right) \cdot y\right)}{x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot y\right)}{x} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot y\right)}{x} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot y\right)}{x} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right), y \cdot y, 1\right) \cdot y\right)}{x} \]

      12. lower-*.f64 86.4

          \[\leadsto \frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right)}{x} \]

   5. Applied rewrites 86.4%

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

Alternative 8: 68.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{+62}:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left({y}^{6} \cdot 0.0001984126984126984\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.25e+62)
   (* (* 2.0 (sinh y)) 0.5)
   (* (* (pow y 6.0) 0.0001984126984126984) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.25e+62) {
		tmp = (2.0 * sinh(y)) * 0.5;
	} else {
		tmp = (pow(y, 6.0) * 0.0001984126984126984) * y;
	}
	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 (x <= 1.25d+62) then
        tmp = (2.0d0 * sinh(y)) * 0.5d0
    else
        tmp = ((y ** 6.0d0) * 0.0001984126984126984d0) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.25e+62) {
		tmp = (2.0 * Math.sinh(y)) * 0.5;
	} else {
		tmp = (Math.pow(y, 6.0) * 0.0001984126984126984) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.25e+62:
		tmp = (2.0 * math.sinh(y)) * 0.5
	else:
		tmp = (math.pow(y, 6.0) * 0.0001984126984126984) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.25e+62)
		tmp = Float64(Float64(2.0 * sinh(y)) * 0.5);
	else
		tmp = Float64(Float64((y ^ 6.0) * 0.0001984126984126984) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.25e+62)
		tmp = (2.0 * sinh(y)) * 0.5;
	else
		tmp = ((y ^ 6.0) * 0.0001984126984126984) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.25e+62], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Power[y, 6.0], $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25000000000000007e62

   1. Initial program 88.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 70.9

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 70.9%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   if 1.25000000000000007e62 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 30.0

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 30.0%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto y \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

   8. Applied rewrites 27.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \color{blue}{y} \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(\frac{1}{5040} \cdot {y}^{6}\right) \cdot y \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({y}^{6} \cdot \frac{1}{5040}\right) \cdot y \]

       2. metadata-eval N/A

          \[\leadsto \left({y}^{\left(3 + 3\right)} \cdot \frac{1}{5040}\right) \cdot y \]

       3. pow-prod-up N/A

          \[\leadsto \left(\left({y}^{3} \cdot {y}^{3}\right) \cdot \frac{1}{5040}\right) \cdot y \]

       4. pow-prod-down N/A

          \[\leadsto \left({\left(y \cdot y\right)}^{3} \cdot \frac{1}{5040}\right) \cdot y \]

       5. pow2 N/A

          \[\leadsto \left({\left({y}^{2}\right)}^{3} \cdot \frac{1}{5040}\right) \cdot y \]

       6. lower-*.f64 N/A

          \[\leadsto \left({\left({y}^{2}\right)}^{3} \cdot \frac{1}{5040}\right) \cdot y \]

       7. pow2 N/A

          \[\leadsto \left({\left(y \cdot y\right)}^{3} \cdot \frac{1}{5040}\right) \cdot y \]

       8. pow-prod-down N/A

          \[\leadsto \left(\left({y}^{3} \cdot {y}^{3}\right) \cdot \frac{1}{5040}\right) \cdot y \]

       9. pow-prod-up N/A

          \[\leadsto \left({y}^{\left(3 + 3\right)} \cdot \frac{1}{5040}\right) \cdot y \]

       10. metadata-eval N/A

           \[\leadsto \left({y}^{6} \cdot \frac{1}{5040}\right) \cdot y \]

       11. lower-pow.f64 50.6

           \[\leadsto \left({y}^{6} \cdot 0.0001984126984126984\right) \cdot y \]

   11. Applied rewrites 50.6%

       \[\leadsto \left({y}^{6} \cdot 0.0001984126984126984\right) \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 68.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{+62}:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;{y}^{7} \cdot 0.0001984126984126984\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.25e+62)
   (* (* 2.0 (sinh y)) 0.5)
   (* (pow y 7.0) 0.0001984126984126984)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.25e+62) {
		tmp = (2.0 * sinh(y)) * 0.5;
	} else {
		tmp = pow(y, 7.0) * 0.0001984126984126984;
	}
	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 (x <= 1.25d+62) then
        tmp = (2.0d0 * sinh(y)) * 0.5d0
    else
        tmp = (y ** 7.0d0) * 0.0001984126984126984d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.25e+62) {
		tmp = (2.0 * Math.sinh(y)) * 0.5;
	} else {
		tmp = Math.pow(y, 7.0) * 0.0001984126984126984;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.25e+62:
		tmp = (2.0 * math.sinh(y)) * 0.5
	else:
		tmp = math.pow(y, 7.0) * 0.0001984126984126984
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.25e+62)
		tmp = Float64(Float64(2.0 * sinh(y)) * 0.5);
	else
		tmp = Float64((y ^ 7.0) * 0.0001984126984126984);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.25e+62)
		tmp = (2.0 * sinh(y)) * 0.5;
	else
		tmp = (y ^ 7.0) * 0.0001984126984126984;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.25e+62], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Power[y, 7.0], $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25000000000000007e62

   1. Initial program 88.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 70.9

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 70.9%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   if 1.25000000000000007e62 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 30.0

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 30.0%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto y \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

   8. Applied rewrites 27.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \color{blue}{y} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{1}{5040} \cdot {y}^{\color{blue}{7}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {y}^{7} \cdot \frac{1}{5040} \]

       2. lower-*.f64 N/A

          \[\leadsto {y}^{7} \cdot \frac{1}{5040} \]

       3. lower-pow.f64 50.6

          \[\leadsto {y}^{7} \cdot 0.0001984126984126984 \]

   11. Applied rewrites 50.6%

       \[\leadsto {y}^{7} \cdot 0.0001984126984126984 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 64.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.1 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;{y}^{7} \cdot 0.0001984126984126984\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5.1e+61)
   (*
    (fma
     (*
      (fma
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       (* y y)
       0.16666666666666666)
      y)
     y
     1.0)
    y)
   (* (pow y 7.0) 0.0001984126984126984)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5.1e+61) {
		tmp = fma((fma(fma((y * y), 0.0001984126984126984, 0.008333333333333333), (y * y), 0.16666666666666666) * y), y, 1.0) * y;
	} else {
		tmp = pow(y, 7.0) * 0.0001984126984126984;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5.1e+61)
		tmp = Float64(fma(Float64(fma(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), Float64(y * y), 0.16666666666666666) * y), y, 1.0) * y);
	else
		tmp = Float64((y ^ 7.0) * 0.0001984126984126984);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5.1e+61], N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[Power[y, 7.0], $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.1000000000000001e61

   1. Initial program 88.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 70.9

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 70.9%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto y \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

   8. Applied rewrites 61.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \color{blue}{y} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right) \cdot y \]

      2. lift-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), y \cdot y, \frac{1}{6}\right) \cdot \left(y \cdot y\right) + 1\right) \cdot y \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), y \cdot y, \frac{1}{6}\right) \cdot \left(y \cdot y\right) + 1\right) \cdot y \]

      4. lift-fma.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right) \cdot \left(y \cdot y\right) + \frac{1}{6}\right) \cdot \left(y \cdot y\right) + 1\right) \cdot y \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right) \cdot \left(y \cdot y\right) + \frac{1}{6}\right) \cdot \left(y \cdot y\right) + 1\right) \cdot y \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(\left(\left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) \cdot \left(y \cdot y\right) + \frac{1}{6}\right) \cdot \left(y \cdot y\right) + 1\right) \cdot y \]

      7. associate-*r* N/A

         \[\leadsto \left(\left(\left(\left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) \cdot \left(y \cdot y\right) + \frac{1}{6}\right) \cdot y\right) \cdot y + 1\right) \cdot y \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) \cdot \left(y \cdot y\right) + \frac{1}{6}\right) \cdot y, y, 1\right) \cdot y \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) \cdot \left(y \cdot y\right) + \frac{1}{6}\right) \cdot y, y, 1\right) \cdot y \]

      10. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right) \cdot \left(y \cdot y\right) + \frac{1}{6}\right) \cdot y, y, 1\right) \cdot y \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right) \cdot \left(y \cdot y\right) + \frac{1}{6}\right) \cdot y, y, 1\right) \cdot y \]

      12. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), y \cdot y, \frac{1}{6}\right) \cdot y, y, 1\right) \cdot y \]

      13. lift-*.f64 61.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y \]

   10. Applied rewrites 61.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y \]

   if 5.1000000000000001e61 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 30.0

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 30.0%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto y \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

   8. Applied rewrites 27.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \color{blue}{y} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{1}{5040} \cdot {y}^{\color{blue}{7}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {y}^{7} \cdot \frac{1}{5040} \]

       2. lower-*.f64 N/A

          \[\leadsto {y}^{7} \cdot \frac{1}{5040} \]

       3. lower-pow.f64 50.6

          \[\leadsto {y}^{7} \cdot 0.0001984126984126984 \]

   11. Applied rewrites 50.6%

       \[\leadsto {y}^{7} \cdot 0.0001984126984126984 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 63.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot x\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.9e+26)
   (*
    (fma
     (*
      (fma
       (fma (* y y) 0.0001984126984126984 0.008333333333333333)
       (* y y)
       0.16666666666666666)
      y)
     y
     1.0)
    y)
   (/ (* (* (* (* y y) 0.16666666666666666) x) y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.9e+26) {
		tmp = fma((fma(fma((y * y), 0.0001984126984126984, 0.008333333333333333), (y * y), 0.16666666666666666) * y), y, 1.0) * y;
	} else {
		tmp = ((((y * y) * 0.16666666666666666) * x) * y) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.9e+26)
		tmp = Float64(fma(Float64(fma(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), Float64(y * y), 0.16666666666666666) * y), y, 1.0) * y);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(y * y) * 0.16666666666666666) * x) * y) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.9e+26], N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9000000000000001e26

   1. Initial program 87.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 71.4

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto y \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

   8. Applied rewrites 61.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \color{blue}{y} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right) \cdot y \]

      2. lift-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), y \cdot y, \frac{1}{6}\right) \cdot \left(y \cdot y\right) + 1\right) \cdot y \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), y \cdot y, \frac{1}{6}\right) \cdot \left(y \cdot y\right) + 1\right) \cdot y \]

      4. lift-fma.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right) \cdot \left(y \cdot y\right) + \frac{1}{6}\right) \cdot \left(y \cdot y\right) + 1\right) \cdot y \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right) \cdot \left(y \cdot y\right) + \frac{1}{6}\right) \cdot \left(y \cdot y\right) + 1\right) \cdot y \]

      6. lift-fma.f64 N/A

         \[\leadsto \left(\left(\left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) \cdot \left(y \cdot y\right) + \frac{1}{6}\right) \cdot \left(y \cdot y\right) + 1\right) \cdot y \]

      7. associate-*r* N/A

         \[\leadsto \left(\left(\left(\left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) \cdot \left(y \cdot y\right) + \frac{1}{6}\right) \cdot y\right) \cdot y + 1\right) \cdot y \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) \cdot \left(y \cdot y\right) + \frac{1}{6}\right) \cdot y, y, 1\right) \cdot y \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\left(y \cdot y\right) \cdot \frac{1}{5040} + \frac{1}{120}\right) \cdot \left(y \cdot y\right) + \frac{1}{6}\right) \cdot y, y, 1\right) \cdot y \]

      10. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right) \cdot \left(y \cdot y\right) + \frac{1}{6}\right) \cdot y, y, 1\right) \cdot y \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right) \cdot \left(y \cdot y\right) + \frac{1}{6}\right) \cdot y, y, 1\right) \cdot y \]

      12. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{5040}, \frac{1}{120}\right), y \cdot y, \frac{1}{6}\right) \cdot y, y, 1\right) \cdot y \]

      13. lift-*.f64 61.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y \]

   10. Applied rewrites 61.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y \]

   if 1.9000000000000001e26 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sin.f64 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\mathsf{fma}\left(y \cdot y, \frac{1}{6}, 1\right) \cdot x\right) \cdot y}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.9%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 48.4

          \[\leadsto \frac{\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot x\right) \cdot y}{x} \]

   11. Applied rewrites 48.4%

       \[\leadsto \frac{\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot x\right) \cdot y}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 63.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.0001984126984126984, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot x\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.9e+26)
   (*
    (fma
     (fma (* (* y y) 0.0001984126984126984) (* y y) 0.16666666666666666)
     (* y y)
     1.0)
    y)
   (/ (* (* (* (* y y) 0.16666666666666666) x) y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.9e+26) {
		tmp = fma(fma(((y * y) * 0.0001984126984126984), (y * y), 0.16666666666666666), (y * y), 1.0) * y;
	} else {
		tmp = ((((y * y) * 0.16666666666666666) * x) * y) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.9e+26)
		tmp = Float64(fma(fma(Float64(Float64(y * y) * 0.0001984126984126984), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(y * y) * 0.16666666666666666) * x) * y) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.9e+26], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9000000000000001e26

   1. Initial program 87.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 71.4

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto y \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

   8. Applied rewrites 61.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \color{blue}{y} \]

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040} \cdot {y}^{2}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right) \cdot y \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({y}^{2} \cdot \frac{1}{5040}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right) \cdot y \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({y}^{2} \cdot \frac{1}{5040}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right) \cdot y \]

       3. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \frac{1}{5040}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right) \cdot y \]

       4. lift-*.f64 61.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.0001984126984126984, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]

   11. Applied rewrites 61.6%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.0001984126984126984, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]

   if 1.9000000000000001e26 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sin.f64 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\mathsf{fma}\left(y \cdot y, \frac{1}{6}, 1\right) \cdot x\right) \cdot y}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.9%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 48.4

          \[\leadsto \frac{\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot x\right) \cdot y}{x} \]

   11. Applied rewrites 48.4%

       \[\leadsto \frac{\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot x\right) \cdot y}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 61.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot x\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.9e+26)
   (*
    (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0)
    y)
   (/ (* (* (* (* y y) 0.16666666666666666) x) y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.9e+26) {
		tmp = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0) * y;
	} else {
		tmp = ((((y * y) * 0.16666666666666666) * x) * y) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.9e+26)
		tmp = Float64(fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0) * y);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(y * y) * 0.16666666666666666) * x) * y) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.9e+26], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9000000000000001e26

   1. Initial program 87.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 71.4

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, {y}^{2}, 1\right) \cdot y \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}, {y}^{2}, 1\right) \cdot y \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left({y}^{2} \cdot \frac{1}{120} + \frac{1}{6}, {y}^{2}, 1\right) \cdot y \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot y \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot y \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot y \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right), y \cdot y, 1\right) \cdot y \]

      12. lift-*.f64 59.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]

   8. Applied rewrites 59.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \color{blue}{y} \]

   if 1.9000000000000001e26 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sin.f64 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\mathsf{fma}\left(y \cdot y, \frac{1}{6}, 1\right) \cdot x\right) \cdot y}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.9%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 48.4

          \[\leadsto \frac{\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot x\right) \cdot y}{x} \]

   11. Applied rewrites 48.4%

       \[\leadsto \frac{\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot x\right) \cdot y}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 60.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 8.5e+61)
   (*
    (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0)
    y)
   (* (* (* y y) 0.16666666666666666) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 8.5e+61) {
		tmp = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0) * y;
	} else {
		tmp = ((y * y) * 0.16666666666666666) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 8.5e+61)
		tmp = Float64(fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0) * y);
	else
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 8.5e+61], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.50000000000000035e61

   1. Initial program 88.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 70.9

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 70.9%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, {y}^{2}, 1\right) \cdot y \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}, {y}^{2}, 1\right) \cdot y \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left({y}^{2} \cdot \frac{1}{120} + \frac{1}{6}, {y}^{2}, 1\right) \cdot y \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({y}^{2}, \frac{1}{120}, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot y \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot y \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot y \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right), y \cdot y, 1\right) \cdot y \]

      12. lift-*.f64 59.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y \]

   8. Applied rewrites 59.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \color{blue}{y} \]

   if 8.50000000000000035e61 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 30.0

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 30.0%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 20.7

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

   8. Applied rewrites 20.7%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 40.9

          \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]

   11. Applied rewrites 40.9%

       \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 51.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.25 \cdot 10^{-9} \lor \neg \left(y \leq 2.4 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.25e-9) (not (<= y 2.4e-5)))
   (* (* (* y y) 0.16666666666666666) y)
   y))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.25e-9) || !(y <= 2.4e-5)) {
		tmp = ((y * y) * 0.16666666666666666) * y;
	} else {
		tmp = y;
	}
	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 ((y <= (-3.25d-9)) .or. (.not. (y <= 2.4d-5))) then
        tmp = ((y * y) * 0.16666666666666666d0) * y
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.25e-9) || !(y <= 2.4e-5)) {
		tmp = ((y * y) * 0.16666666666666666) * y;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.25e-9) or not (y <= 2.4e-5):
		tmp = ((y * y) * 0.16666666666666666) * y
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.25e-9) || !(y <= 2.4e-5))
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * y);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.25e-9) || ~((y <= 2.4e-5)))
		tmp = ((y * y) * 0.16666666666666666) * y;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.25e-9], N[Not[LessEqual[y, 2.4e-5]], $MachinePrecision]], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if y < -3.2500000000000002e-9 or 2.4000000000000001e-5 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 74.7

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 74.7%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 48.1

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

   8. Applied rewrites 48.1%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 48.1

          \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]

   11. Applied rewrites 48.1%

       \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]

   if -3.2500000000000002e-9 < y < 2.4000000000000001e-5

   1. Initial program 78.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 52.6

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto y \]
   7. Step-by-step derivation

   8. Applied rewrites 52.6%

      \[\leadsto y \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.25 \cdot 10^{-9} \lor \neg \left(y \leq 2.4 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 56.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.9e+26)
   (* (fma y (* y 0.16666666666666666) 1.0) y)
   (* (* (* y y) 0.16666666666666666) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.9e+26) {
		tmp = fma(y, (y * 0.16666666666666666), 1.0) * y;
	} else {
		tmp = ((y * y) * 0.16666666666666666) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.9e+26)
		tmp = Float64(fma(y, Float64(y * 0.16666666666666666), 1.0) * y);
	else
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.9e+26], N[(N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9000000000000001e26

   1. Initial program 87.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 71.4

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 56.1

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

   8. Applied rewrites 56.1%

      \[\leadsto \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \color{blue}{y} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 56.1

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

   10. Applied rewrites 56.1%

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

   if 1.9000000000000001e26 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 33.0

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 33.0%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 22.9

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

   8. Applied rewrites 22.9%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 40.4

          \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]

   11. Applied rewrites 40.4%

       \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 28.3% accurate, 217.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 y)

C

double code(double x, double y) {
	return 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 = y
end function

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 90.0%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. rec-exp N/A

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

   4. sinh-undef N/A

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

   5. lower-*.f64 N/A

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

   6. lift-sinh.f64 64.5

      \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

5. Applied rewrites 64.5%

   \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

6. Taylor expanded in y around 0

   \[\leadsto y \]
7. Step-by-step derivation

8. Applied rewrites 26.5%

   \[\leadsto y \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025084 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (* (sin x) (/ (sinh y) x)))

  (/ (* (sin x) (sinh y)) x))