Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.0% → 99.8%

Time: 6.3s

Alternatives: 15

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: 89.0% 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 88.9%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 69.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-67}:\\ \;\;\;\;\left(x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)}{x}\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-133}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -2e-67)
     (*
      (*
       x
       (/
        (fma
         (fma 0.008333333333333333 (* y y) 0.16666666666666666)
         (* y y)
         1.0)
        x))
      y)
     (if (<= t_0 5e-133)
       (* (/ y x) x)
       (*
        (*
         (fma
          (fma 0.008333333333333333 (* x x) -0.16666666666666666)
          (* x x)
          1.0)
         (fma
          (fma (* y y) 0.008333333333333333 0.16666666666666666)
          (* y y)
          1.0))
        y)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -2e-67) {
		tmp = (x * (fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0) / x)) * y;
	} else if (t_0 <= 5e-133) {
		tmp = (y / x) * x;
	} else {
		tmp = (fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * x), 1.0) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0)) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -2e-67)
		tmp = Float64(Float64(x * Float64(fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) / x)) * y);
	elseif (t_0 <= 5e-133)
		tmp = Float64(Float64(y / x) * x);
	else
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0)) * y);
	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]}, If[LessEqual[t$95$0, -2e-67], N[(N[(x * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 5e-133], N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999989e-67

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 76.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 84.5

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 84.5

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

   7. Applied rewrites 84.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \left(x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)}{x}\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 73.7%

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

   if -1.99999999999999989e-67 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.9999999999999999e-133

   1. Initial program 72.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 24.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 70.3

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

   10. Applied rewrites 70.3%

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

   if 4.9999999999999999e-133 < (/.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 y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 67.0

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

   8. Applied rewrites 67.0%

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

Alternative 3: 69.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-67} \lor \neg \left(t\_0 \leq 5 \cdot 10^{-250}\right):\\ \;\;\;\;\left(x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)}{x}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (or (<= t_0 -2e-67) (not (<= t_0 5e-250)))
     (*
      (*
       x
       (/
        (fma
         (fma 0.008333333333333333 (* y y) 0.16666666666666666)
         (* y y)
         1.0)
        x))
      y)
     (* (/ y x) x))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if ((t_0 <= -2e-67) || !(t_0 <= 5e-250)) {
		tmp = (x * (fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0) / x)) * y;
	} else {
		tmp = (y / x) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if ((t_0 <= -2e-67) || !(t_0 <= 5e-250))
		tmp = Float64(Float64(x * Float64(fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) / x)) * y);
	else
		tmp = Float64(Float64(y / x) * x);
	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]}, If[Or[LessEqual[t$95$0, -2e-67], N[Not[LessEqual[t$95$0, 5e-250]], $MachinePrecision]], N[(N[(x * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999989e-67 or 5.00000000000000027e-250 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 80.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 84.2

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 84.2

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

   7. Applied rewrites 84.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \left(x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)}{x}\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 65.6%

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

   if -1.99999999999999989e-67 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 5.00000000000000027e-250

   1. Initial program 68.3%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 22.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.6

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

   10. Applied rewrites 75.6%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-67} \lor \neg \left(\frac{\sin x \cdot \sinh y}{x} \leq 5 \cdot 10^{-250}\right):\\ \;\;\;\;\left(x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)}{x}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-319} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.008333333333333333 \cdot y, y, 0.16666666666666666\right), 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (or (<= t_0 -1e-319) (not (<= t_0 0.0)))
     (*
      (fma (* y y) (fma (* 0.008333333333333333 y) y 0.16666666666666666) 1.0)
      y)
     (* (/ y x) x))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if ((t_0 <= -1e-319) || !(t_0 <= 0.0)) {
		tmp = fma((y * y), fma((0.008333333333333333 * y), y, 0.16666666666666666), 1.0) * y;
	} else {
		tmp = (y / x) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if ((t_0 <= -1e-319) || !(t_0 <= 0.0))
		tmp = Float64(fma(Float64(y * y), fma(Float64(0.008333333333333333 * y), y, 0.16666666666666666), 1.0) * y);
	else
		tmp = Float64(Float64(y / x) * x);
	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]}, If[Or[LessEqual[t$95$0, -1e-319], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(y * y), $MachinePrecision] * N[(N[(0.008333333333333333 * y), $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.99989e-320 or 0.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 83.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

   8. Applied rewrites 58.7%

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

   if -9.99989e-320 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 44.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 44.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 6.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 98.0

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

   10. Applied rewrites 98.0%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -1 \cdot 10^{-319} \lor \neg \left(\frac{\sin x \cdot \sinh y}{x} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.008333333333333333 \cdot y, y, 0.16666666666666666\right), 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \sinh y}{x}\\ t_1 := \frac{\left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x\right) \cdot y}{x}\\ \mathbf{if}\;y \leq -1.16 \cdot 10^{+103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -0.000105:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.56:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+101}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (sinh y)) x))
        (t_1 (/ (* (* (fma (* y y) 0.16666666666666666 1.0) (sin x)) y) x)))
   (if (<= y -1.16e+103)
     t_1
     (if (<= y -0.000105)
       t_0
       (if (<= y 0.56) (* (/ (sin x) x) y) (if (<= y 1.7e+101) t_0 t_1))))))

C

double code(double x, double y) {
	double t_0 = (x * sinh(y)) / x;
	double t_1 = ((fma((y * y), 0.16666666666666666, 1.0) * sin(x)) * y) / x;
	double tmp;
	if (y <= -1.16e+103) {
		tmp = t_1;
	} else if (y <= -0.000105) {
		tmp = t_0;
	} else if (y <= 0.56) {
		tmp = (sin(x) / x) * y;
	} else if (y <= 1.7e+101) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * sinh(y)) / x)
	t_1 = Float64(Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * sin(x)) * y) / x)
	tmp = 0.0
	if (y <= -1.16e+103)
		tmp = t_1;
	elseif (y <= -0.000105)
		tmp = t_0;
	elseif (y <= 0.56)
		tmp = Float64(Float64(sin(x) / x) * y);
	elseif (y <= 1.7e+101)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, -1.16e+103], t$95$1, If[LessEqual[y, -0.000105], t$95$0, If[LessEqual[y, 0.56], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.7e+101], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1600000000000001e103 or 1.70000000000000009e101 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 74.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{y \cdot \left(\sin x + \frac{1}{6} \cdot \left({y}^{2} \cdot \sin x\right)\right)}}{x} \]
   7. 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. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 98.9%

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

   if -1.1600000000000001e103 < y < -1.05e-4 or 0.56000000000000005 < y < 1.70000000000000009e101

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 76.7%

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

   if -1.05e-4 < y < 0.56000000000000005

   1. Initial program 77.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. lower-sin.f64 98.7

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

   5. Applied rewrites 98.7%

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

Alternative 6: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \sinh y}{x}\\ t_1 := \left(\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\right) \cdot \sin x\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -0.000105:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.56:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+109}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* x (sinh y)) x))
        (t_1 (* (* (/ (fma (* y y) 0.16666666666666666 1.0) x) y) (sin x))))
   (if (<= y -2.2e+103)
     t_1
     (if (<= y -0.000105)
       t_0
       (if (<= y 0.56) (* (/ (sin x) x) y) (if (<= y 2.3e+109) t_0 t_1))))))

C

double code(double x, double y) {
	double t_0 = (x * sinh(y)) / x;
	double t_1 = ((fma((y * y), 0.16666666666666666, 1.0) / x) * y) * sin(x);
	double tmp;
	if (y <= -2.2e+103) {
		tmp = t_1;
	} else if (y <= -0.000105) {
		tmp = t_0;
	} else if (y <= 0.56) {
		tmp = (sin(x) / x) * y;
	} else if (y <= 2.3e+109) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * sinh(y)) / x)
	t_1 = Float64(Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) / x) * y) * sin(x))
	tmp = 0.0
	if (y <= -2.2e+103)
		tmp = t_1;
	elseif (y <= -0.000105)
		tmp = t_0;
	elseif (y <= 0.56)
		tmp = Float64(Float64(sin(x) / x) * y);
	elseif (y <= 2.3e+109)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $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, -2.2e+103], t$95$1, If[LessEqual[y, -0.000105], t$95$0, If[LessEqual[y, 0.56], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 2.3e+109], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.19999999999999992e103 or 2.3000000000000001e109 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 94.4

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

   10. Applied rewrites 94.4%

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

   if -2.19999999999999992e103 < y < -1.05e-4 or 0.56000000000000005 < y < 2.3000000000000001e109

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 77.1%

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

   if -1.05e-4 < y < 0.56000000000000005

   1. Initial program 77.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. lower-sin.f64 98.7

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

   5. Applied rewrites 98.7%

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

Alternative 7: 93.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.56 \lor \neg \left(y \leq 2.2 \cdot 10^{+77}\right):\\ \;\;\;\;\left(\sin x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)}{x}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sinh y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 0.56) (not (<= y 2.2e+77)))
   (*
    (*
     (sin x)
     (/
      (fma (fma 0.008333333333333333 (* y y) 0.16666666666666666) (* y y) 1.0)
      x))
    y)
   (/ (* x (sinh y)) x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 0.56) || !(y <= 2.2e+77)) {
		tmp = (sin(x) * (fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0) / x)) * y;
	} else {
		tmp = (x * sinh(y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 0.56) || !(y <= 2.2e+77))
		tmp = Float64(Float64(sin(x) * Float64(fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) / x)) * y);
	else
		tmp = Float64(Float64(x * sinh(y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 0.56], N[Not[LessEqual[y, 2.2e+77]], $MachinePrecision]], N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.56000000000000005 or 2.2e77 < y

   1. Initial program 88.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 92.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 94.8

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 94.8

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

   7. Applied rewrites 94.8%

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

   if 0.56000000000000005 < y < 2.2e77

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 75.0%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.56 \lor \neg \left(y \leq 2.2 \cdot 10^{+77}\right):\\ \;\;\;\;\left(\sin x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)}{x}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sinh y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-130}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.008333333333333333 \cdot y, y, 0.16666666666666666\right), 1\right)}{x} \cdot y\right) \cdot \sin x\\ \mathbf{elif}\;x \leq 0.0112:\\ \;\;\;\;\frac{x \cdot \sinh y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.8e-130)
   (*
    (*
     (/
      (fma (* y y) (fma (* 0.008333333333333333 y) y 0.16666666666666666) 1.0)
      x)
     y)
    (sin x))
   (if (<= x 0.0112)
     (/ (* x (sinh y)) x)
     (/
      (*
       (*
        (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 <= 2.8e-130) {
		tmp = ((fma((y * y), fma((0.008333333333333333 * y), y, 0.16666666666666666), 1.0) / x) * y) * sin(x);
	} else if (x <= 0.0112) {
		tmp = (x * sinh(y)) / x;
	} 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 <= 2.8e-130)
		tmp = Float64(Float64(Float64(fma(Float64(y * y), fma(Float64(0.008333333333333333 * y), y, 0.16666666666666666), 1.0) / x) * y) * sin(x));
	elseif (x <= 0.0112)
		tmp = Float64(Float64(x * sinh(y)) / x);
	else
		tmp = Float64(Float64(Float64(sin(x) * 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, 2.8e-130], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(0.008333333333333333 * y), $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0112], N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.80000000000000016e-130

   1. Initial program 83.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 93.2%

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

   if 2.80000000000000016e-130 < x < 0.0111999999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.9%

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

   if 0.0111999999999999999 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.6%

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

Alternative 9: 93.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.56:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.008333333333333333 \cdot y, y, 0.16666666666666666\right), 1\right)}{x} \cdot y\right) \cdot \sin x\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{x \cdot \sinh y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)}{x}\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.56)
   (*
    (*
     (/
      (fma (* y y) (fma (* 0.008333333333333333 y) y 0.16666666666666666) 1.0)
      x)
     y)
    (sin x))
   (if (<= y 2.2e+77)
     (/ (* x (sinh y)) x)
     (*
      (*
       (sin x)
       (/
        (fma
         (fma 0.008333333333333333 (* y y) 0.16666666666666666)
         (* y y)
         1.0)
        x))
      y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.56) {
		tmp = ((fma((y * y), fma((0.008333333333333333 * y), y, 0.16666666666666666), 1.0) / x) * y) * sin(x);
	} else if (y <= 2.2e+77) {
		tmp = (x * sinh(y)) / x;
	} else {
		tmp = (sin(x) * (fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0) / x)) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.56)
		tmp = Float64(Float64(Float64(fma(Float64(y * y), fma(Float64(0.008333333333333333 * y), y, 0.16666666666666666), 1.0) / x) * y) * sin(x));
	elseif (y <= 2.2e+77)
		tmp = Float64(Float64(x * sinh(y)) / x);
	else
		tmp = Float64(Float64(sin(x) * Float64(fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) / x)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.56], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(0.008333333333333333 * y), $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e+77], N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.56000000000000005

   1. Initial program 85.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 93.9%

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

   if 0.56000000000000005 < y < 2.2e77

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 75.0%

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

   if 2.2e77 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

Alternative 10: 86.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.000105 \lor \neg \left(y \leq 0.56\right):\\ \;\;\;\;\frac{x \cdot \sinh y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.000105) (not (<= y 0.56)))
   (/ (* x (sinh y)) x)
   (* (/ (sin x) x) y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.000105) || !(y <= 0.56)) {
		tmp = (x * sinh(y)) / x;
	} 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) :: tmp
    if ((y <= (-0.000105d0)) .or. (.not. (y <= 0.56d0))) then
        tmp = (x * sinh(y)) / x
    else
        tmp = (sin(x) / x) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -0.000105) || !(y <= 0.56)) {
		tmp = (x * Math.sinh(y)) / x;
	} else {
		tmp = (Math.sin(x) / x) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -0.000105) or not (y <= 0.56):
		tmp = (x * math.sinh(y)) / x
	else:
		tmp = (math.sin(x) / x) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.000105) || !(y <= 0.56))
		tmp = Float64(Float64(x * sinh(y)) / x);
	else
		tmp = Float64(Float64(sin(x) / x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -0.000105) || ~((y <= 0.56)))
		tmp = (x * sinh(y)) / x;
	else
		tmp = (sin(x) / x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.000105], N[Not[LessEqual[y, 0.56]], $MachinePrecision]], N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.05e-4 or 0.56000000000000005 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 75.2%

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

   if -1.05e-4 < y < 0.56000000000000005

   1. Initial program 77.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. lower-sin.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.000105 \lor \neg \left(y \leq 0.56\right):\\ \;\;\;\;\frac{x \cdot \sinh y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+17}:\\ \;\;\;\;\left(x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)}{x}\right) \cdot y\\ \mathbf{elif}\;y \leq 1.22:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.1e+17)
   (*
    (*
     x
     (/
      (fma (fma 0.008333333333333333 (* y y) 0.16666666666666666) (* y y) 1.0)
      x))
    y)
   (if (<= y 1.22)
     (* (/ (sin x) x) y)
     (*
      (*
       (fma
        (fma 0.008333333333333333 (* x x) -0.16666666666666666)
        (* x x)
        1.0)
       (fma
        (fma (* y y) 0.008333333333333333 0.16666666666666666)
        (* y y)
        1.0))
      y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.1e+17) {
		tmp = (x * (fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0) / x)) * y;
	} else if (y <= 1.22) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = (fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * x), 1.0) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0)) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.1e+17)
		tmp = Float64(Float64(x * Float64(fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) / x)) * y);
	elseif (y <= 1.22)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.1e+17], N[(N[(x * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.22], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1e17

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 85.7

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 85.7

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

   7. Applied rewrites 85.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \left(x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right)}{x}\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 62.5%

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

   if -1.1e17 < y < 1.21999999999999997

   1. Initial program 77.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. lower-sin.f64 97.2

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

   5. Applied rewrites 97.2%

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

   if 1.21999999999999997 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 73.2

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

   8. Applied rewrites 73.2%

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

Alternative 12: 51.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{+15} \lor \neg \left(x \leq 4.1 \cdot 10^{+223}\right):\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.008333333333333333, x \cdot x, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x 5.2e+15) (not (<= x 4.1e+223)))
   (* (/ y x) x)
   (* (fma (* (* x x) 0.008333333333333333) (* x x) 1.0) y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= 5.2e+15) || !(x <= 4.1e+223)) {
		tmp = (y / x) * x;
	} else {
		tmp = fma(((x * x) * 0.008333333333333333), (x * x), 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= 5.2e+15) || !(x <= 4.1e+223))
		tmp = Float64(Float64(y / x) * x);
	else
		tmp = Float64(fma(Float64(Float64(x * x) * 0.008333333333333333), Float64(x * x), 1.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, 5.2e+15], N[Not[LessEqual[x, 4.1e+223]], $MachinePrecision]], N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.2e15 or 4.1e223 < x

   1. Initial program 86.6%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 38.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 26.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 56.9

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

   10. Applied rewrites 56.9%

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

   if 5.2e15 < x < 4.1e223

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. lower-sin.f64 39.7

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

   5. Applied rewrites 39.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 37.5

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

   8. Applied rewrites 37.5%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2}, x \cdot x, 1\right) \cdot y \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 37.5

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

   11. Applied rewrites 37.5%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{+15} \lor \neg \left(x \leq 4.1 \cdot 10^{+223}\right):\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.008333333333333333, x \cdot x, 1\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.95 \cdot 10^{+130}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.95e+130)
   (* (/ y x) x)
   (* (fma -0.16666666666666666 (* x x) 1.0) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.95e+130) {
		tmp = (y / x) * x;
	} else {
		tmp = fma(-0.16666666666666666, (x * x), 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.95e+130)
		tmp = Float64(Float64(y / x) * x);
	else
		tmp = Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.95e+130], N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9500000000000001e130

   1. Initial program 87.2%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 39.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 27.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 53.9

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

   10. Applied rewrites 53.9%

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

   if 1.9500000000000001e130 < x

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. lower-sin.f64 35.2

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

   5. Applied rewrites 35.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 36.1

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

   8. Applied rewrites 36.1%

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

Alternative 14: 36.9% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (fma -0.16666666666666666 (* x x) 1.0) y))

C

double code(double x, double y) {
	return fma(-0.16666666666666666, (x * x), 1.0) * y;
}

Julia

function code(x, y)
	return Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * y)
end

Wolfram

code[x_, y_] := N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 88.9%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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. lower-sin.f64 49.8

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

5. Applied rewrites 49.8%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 39.3

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

8. Applied rewrites 39.3%

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot y \]
9. Add Preprocessing

Alternative 15: 28.5% 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 88.9%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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. lower-sin.f64 49.8

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

5. Applied rewrites 49.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 30.4%

   \[\leadsto y \]
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 2025026 
(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))