Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.3% → 99.9%

Time: 6.3s

Alternatives: 14

Speedup: 1.0×

• 49.7% 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: 89.3% 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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (sinh(y) / x) * sin(x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.2%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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. lift-sin.f64 N/A

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

   4. lift-sinh.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-sinh.f64 N/A

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

   10. lift-sin.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 87.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \sinh y\\ \mathbf{if}\;y \leq -9.8 \cdot 10^{-8}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;y \leq 0.00012:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 2.0 (sinh y))))
   (if (<= y -9.8e-8)
     (* t_0 0.5)
     (if (<= y 0.00012)
       (* (/ (sin x) x) y)
       (* t_0 (fma (* x x) -0.08333333333333333 0.5))))))

C

double code(double x, double y) {
	double t_0 = 2.0 * sinh(y);
	double tmp;
	if (y <= -9.8e-8) {
		tmp = t_0 * 0.5;
	} else if (y <= 0.00012) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = t_0 * fma((x * x), -0.08333333333333333, 0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 * sinh(y))
	tmp = 0.0
	if (y <= -9.8e-8)
		tmp = Float64(t_0 * 0.5);
	elseif (y <= 0.00012)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = Float64(t_0 * fma(Float64(x * x), -0.08333333333333333, 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9.8e-8], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[y, 0.00012], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.8000000000000004e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 81.5

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

   5. Applied rewrites 81.5%

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

   if -9.8000000000000004e-8 < y < 1.20000000000000003e-4

   1. Initial program 80.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. lift-sin.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 1.20000000000000003e-4 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

Alternative 3: 86.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \sinh y\right) \cdot 0.5\\ \mathbf{if}\;y \leq -9.8 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 21000000000000:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+103}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 2.0 (sinh y)) 0.5)))
   (if (<= y -9.8e-8)
     t_0
     (if (<= y 21000000000000.0)
       (* (/ (sin x) x) y)
       (if (<= y 4.4e+103)
         t_0
         (*
          (* (* (* y y) 0.3333333333333333) y)
          (fma (* x x) -0.08333333333333333 0.5)))))))

C

double code(double x, double y) {
	double t_0 = (2.0 * sinh(y)) * 0.5;
	double tmp;
	if (y <= -9.8e-8) {
		tmp = t_0;
	} else if (y <= 21000000000000.0) {
		tmp = (sin(x) / x) * y;
	} else if (y <= 4.4e+103) {
		tmp = t_0;
	} else {
		tmp = (((y * y) * 0.3333333333333333) * y) * fma((x * x), -0.08333333333333333, 0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 * sinh(y)) * 0.5)
	tmp = 0.0
	if (y <= -9.8e-8)
		tmp = t_0;
	elseif (y <= 21000000000000.0)
		tmp = Float64(Float64(sin(x) / x) * y);
	elseif (y <= 4.4e+103)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(y * y) * 0.3333333333333333) * y) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[y, -9.8e-8], t$95$0, If[LessEqual[y, 21000000000000.0], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 4.4e+103], t$95$0, N[(N[(N[(N[(y * y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.8000000000000004e-8 or 2.1e13 < y < 4.39999999999999985e103

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 79.2

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

   5. Applied rewrites 79.2%

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

   if -9.8000000000000004e-8 < y < 2.1e13

   1. Initial program 81.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if 4.39999999999999985e103 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + \frac{1}{3} \cdot {y}^{2}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{12}, \frac{1}{2}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(2 + \frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot \color{blue}{x}, \frac{-1}{12}, \frac{1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(2 + \frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot \color{blue}{x}, \frac{-1}{12}, \frac{1}{2}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{1}{3} \cdot {y}^{2} + 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3}, {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3}, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      6. lower-*.f64 82.4

         \[\leadsto \left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   8. Applied rewrites 82.4%

      \[\leadsto \left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.08333333333333333, 0.5\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(\left(\frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{1}{3}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{1}{3}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

       3. pow2 N/A

          \[\leadsto \left(\left(\left(y \cdot y\right) \cdot \frac{1}{3}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

       4. lift-*.f64 82.4

          \[\leadsto \left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   11. Applied rewrites 82.4%

       \[\leadsto \left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 71.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+103}:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.4e+103)
   (*
    x
    (/
     (*
      (fma
       (*
        (fma
         (fma (* y y) 0.0001984126984126984 0.008333333333333333)
         (* y y)
         0.16666666666666666)
        y)
       y
       1.0)
      y)
     x))
   (*
    (* (* (* y y) 0.3333333333333333) y)
    (fma (* x x) -0.08333333333333333 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.4e+103) {
		tmp = x * ((fma((fma(fma((y * y), 0.0001984126984126984, 0.008333333333333333), (y * y), 0.16666666666666666) * y), y, 1.0) * y) / x);
	} else {
		tmp = (((y * y) * 0.3333333333333333) * y) * fma((x * x), -0.08333333333333333, 0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.4e+103)
		tmp = Float64(x * Float64(Float64(fma(Float64(fma(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), Float64(y * y), 0.16666666666666666) * y), y, 1.0) * y) / x));
	else
		tmp = Float64(Float64(Float64(Float64(y * y) * 0.3333333333333333) * y) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.4e+103], N[(x * N[(N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.39999999999999985e103

   1. Initial program 87.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 53.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 73.5

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

   7. Applied rewrites 73.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 56.9%

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

   11. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

   13. Applied rewrites 73.7%

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

   if 4.39999999999999985e103 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + \frac{1}{3} \cdot {y}^{2}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{12}, \frac{1}{2}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(2 + \frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot \color{blue}{x}, \frac{-1}{12}, \frac{1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(2 + \frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot \color{blue}{x}, \frac{-1}{12}, \frac{1}{2}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{1}{3} \cdot {y}^{2} + 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3}, {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3}, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      6. lower-*.f64 82.4

         \[\leadsto \left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   8. Applied rewrites 82.4%

      \[\leadsto \left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.08333333333333333, 0.5\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(\left(\frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{1}{3}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{1}{3}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

       3. pow2 N/A

          \[\leadsto \left(\left(\left(y \cdot y\right) \cdot \frac{1}{3}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

       4. lift-*.f64 82.4

          \[\leadsto \left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   11. Applied rewrites 82.4%

       \[\leadsto \left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 69.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+103}:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.4e+103)
   (*
    x
    (/
     (*
      (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0)
      y)
     x))
   (*
    (* (* (* y y) 0.3333333333333333) y)
    (fma (* x x) -0.08333333333333333 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.4e+103) {
		tmp = x * ((fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0) * y) / x);
	} else {
		tmp = (((y * y) * 0.3333333333333333) * y) * fma((x * x), -0.08333333333333333, 0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.4e+103)
		tmp = Float64(x * Float64(Float64(fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0) * y) / x));
	else
		tmp = Float64(Float64(Float64(Float64(y * y) * 0.3333333333333333) * y) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.4e+103], N[(x * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.39999999999999985e103

   1. Initial program 87.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 53.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 73.5

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

   7. Applied rewrites 73.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 56.9%

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

   11. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 71.4

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

   13. Applied rewrites 71.4%

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

   if 4.39999999999999985e103 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + \frac{1}{3} \cdot {y}^{2}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{12}, \frac{1}{2}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(2 + \frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot \color{blue}{x}, \frac{-1}{12}, \frac{1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(2 + \frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot \color{blue}{x}, \frac{-1}{12}, \frac{1}{2}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{1}{3} \cdot {y}^{2} + 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3}, {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3}, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      6. lower-*.f64 82.4

         \[\leadsto \left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   8. Applied rewrites 82.4%

      \[\leadsto \left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.08333333333333333, 0.5\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(\left(\frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{1}{3}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{1}{3}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

       3. pow2 N/A

          \[\leadsto \left(\left(\left(y \cdot y\right) \cdot \frac{1}{3}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

       4. lift-*.f64 82.4

          \[\leadsto \left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   11. Applied rewrites 82.4%

       \[\leadsto \left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 66.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \mathbf{if}\;y \leq -9.2 \cdot 10^{-101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-26}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (* (fma 0.008333333333333333 (* y y) 0.16666666666666666) y)
           y
           1.0)
          y)))
   (if (<= y -9.2e-101)
     t_0
     (if (<= y 1.35e-26)
       (* x (/ y x))
       (if (<= y 9.5e+78) (* (fma -0.16666666666666666 (* x x) 1.0) y) t_0)))))

C

double code(double x, double y) {
	double t_0 = fma((fma(0.008333333333333333, (y * y), 0.16666666666666666) * y), y, 1.0) * y;
	double tmp;
	if (y <= -9.2e-101) {
		tmp = t_0;
	} else if (y <= 1.35e-26) {
		tmp = x * (y / x);
	} else if (y <= 9.5e+78) {
		tmp = fma(-0.16666666666666666, (x * x), 1.0) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666) * y), y, 1.0) * y)
	tmp = 0.0
	if (y <= -9.2e-101)
		tmp = t_0;
	elseif (y <= 1.35e-26)
		tmp = Float64(x * Float64(y / x));
	elseif (y <= 9.5e+78)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -9.2e-101], t$95$0, If[LessEqual[y, 1.35e-26], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.5e+78], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.1999999999999998e-101 or 9.5000000000000006e78 < y

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 72.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 72.0%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 69.9%

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

   if -9.1999999999999998e-101 < y < 1.34999999999999991e-26

   1. Initial program 78.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 78.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 77.8%

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

   if 1.34999999999999991e-26 < y < 9.5000000000000006e78

   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. lift-sin.f64 18.9

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

   5. Applied rewrites 18.9%

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

   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. pow2 N/A

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

      4. lift-*.f64 40.6

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

   8. Applied rewrites 40.6%

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

Alternative 7: 65.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-26}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.2e-101)
   (*
    (fma (* (fma 0.008333333333333333 (* y y) 0.16666666666666666) y) y 1.0)
    y)
   (if (<= y 1.35e-26)
     (* x (/ y x))
     (*
      (* (fma 0.3333333333333333 (* y y) 2.0) y)
      (fma (* x x) -0.08333333333333333 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.2e-101) {
		tmp = fma((fma(0.008333333333333333, (y * y), 0.16666666666666666) * y), y, 1.0) * y;
	} else if (y <= 1.35e-26) {
		tmp = x * (y / x);
	} else {
		tmp = (fma(0.3333333333333333, (y * y), 2.0) * y) * fma((x * x), -0.08333333333333333, 0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.2e-101)
		tmp = Float64(fma(Float64(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666) * y), y, 1.0) * y);
	elseif (y <= 1.35e-26)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(Float64(fma(0.3333333333333333, Float64(y * y), 2.0) * y) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.2e-101], N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.35e-26], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.3333333333333333 * N[(y * y), $MachinePrecision] + 2.0), $MachinePrecision] * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.1999999999999998e-101

   1. Initial program 96.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 75.8

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 71.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 71.1%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 67.6%

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

   if -9.1999999999999998e-101 < y < 1.34999999999999991e-26

   1. Initial program 78.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 78.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 77.8%

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

   if 1.34999999999999991e-26 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + \frac{1}{3} \cdot {y}^{2}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{12}, \frac{1}{2}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(2 + \frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot \color{blue}{x}, \frac{-1}{12}, \frac{1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(2 + \frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot \color{blue}{x}, \frac{-1}{12}, \frac{1}{2}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{1}{3} \cdot {y}^{2} + 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3}, {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3}, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      6. lower-*.f64 67.3

         \[\leadsto \left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   8. Applied rewrites 67.3%

      \[\leadsto \left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.08333333333333333, 0.5\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 65.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \mathbf{elif}\;y \leq 820:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.2e-101)
   (*
    (fma (* (fma 0.008333333333333333 (* y y) 0.16666666666666666) y) y 1.0)
    y)
   (if (<= y 820.0)
     (* x (/ y x))
     (*
      (* (* (* y y) 0.3333333333333333) y)
      (fma (* x x) -0.08333333333333333 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.2e-101) {
		tmp = fma((fma(0.008333333333333333, (y * y), 0.16666666666666666) * y), y, 1.0) * y;
	} else if (y <= 820.0) {
		tmp = x * (y / x);
	} else {
		tmp = (((y * y) * 0.3333333333333333) * y) * fma((x * x), -0.08333333333333333, 0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.2e-101)
		tmp = Float64(fma(Float64(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666) * y), y, 1.0) * y);
	elseif (y <= 820.0)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(Float64(Float64(Float64(y * y) * 0.3333333333333333) * y) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.2e-101], N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 820.0], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.1999999999999998e-101

   1. Initial program 96.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 75.8

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 71.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 71.1%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 67.6%

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

   if -9.1999999999999998e-101 < y < 820

   1. Initial program 78.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 78.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 76.6%

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

   if 820 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + \frac{1}{3} \cdot {y}^{2}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{12}, \frac{1}{2}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(2 + \frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot \color{blue}{x}, \frac{-1}{12}, \frac{1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(2 + \frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot \color{blue}{x}, \frac{-1}{12}, \frac{1}{2}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{1}{3} \cdot {y}^{2} + 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3}, {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3}, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      6. lower-*.f64 68.6

         \[\leadsto \left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   8. Applied rewrites 68.6%

      \[\leadsto \left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.08333333333333333, 0.5\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(\left(\frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{1}{3}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{1}{3}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

       3. pow2 N/A

          \[\leadsto \left(\left(\left(y \cdot y\right) \cdot \frac{1}{3}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

       4. lift-*.f64 68.6

          \[\leadsto \left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   11. Applied rewrites 68.6%

       \[\leadsto \left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 66.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+103}:\\ \;\;\;\;x \cdot \left(\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.4e+103)
   (* x (* (/ (fma (* y y) 0.16666666666666666 1.0) x) y))
   (*
    (* (* (* y y) 0.3333333333333333) y)
    (fma (* x x) -0.08333333333333333 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.4e+103) {
		tmp = x * ((fma((y * y), 0.16666666666666666, 1.0) / x) * y);
	} else {
		tmp = (((y * y) * 0.3333333333333333) * y) * fma((x * x), -0.08333333333333333, 0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.4e+103)
		tmp = Float64(x * Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) / x) * y));
	else
		tmp = Float64(Float64(Float64(Float64(y * y) * 0.3333333333333333) * y) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.4e+103], N[(x * N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.39999999999999985e103

   1. Initial program 87.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 53.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 73.5

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

   7. Applied rewrites 73.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 56.9%

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

   11. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. associate-*r/ N/A

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

       4. div-add-rev N/A

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

       5. +-commutative N/A

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

       6. lower-/.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 69.5

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

   13. Applied rewrites 69.5%

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

   if 4.39999999999999985e103 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + \frac{1}{3} \cdot {y}^{2}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{12}, \frac{1}{2}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(2 + \frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot \color{blue}{x}, \frac{-1}{12}, \frac{1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(2 + \frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot \color{blue}{x}, \frac{-1}{12}, \frac{1}{2}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{1}{3} \cdot {y}^{2} + 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3}, {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3}, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      6. lower-*.f64 82.4

         \[\leadsto \left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   8. Applied rewrites 82.4%

      \[\leadsto \left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.08333333333333333, 0.5\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(\left(\frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{1}{3}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left({y}^{2} \cdot \frac{1}{3}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

       3. pow2 N/A

          \[\leadsto \left(\left(\left(y \cdot y\right) \cdot \frac{1}{3}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

       4. lift-*.f64 82.4

          \[\leadsto \left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   11. Applied rewrites 82.4%

       \[\leadsto \left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 62.4% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \mathbf{if}\;y \leq -9.2 \cdot 10^{-101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-26}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot y, x \cdot x, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (* y y) 0.16666666666666666 1.0) y)))
   (if (<= y -9.2e-101)
     t_0
     (if (<= y 1.35e-26)
       (* x (/ y x))
       (if (<= y 8e+109) (fma (* -0.16666666666666666 y) (* x x) y) t_0)))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), 0.16666666666666666, 1.0) * y;
	double tmp;
	if (y <= -9.2e-101) {
		tmp = t_0;
	} else if (y <= 1.35e-26) {
		tmp = x * (y / x);
	} else if (y <= 8e+109) {
		tmp = fma((-0.16666666666666666 * y), (x * x), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y)
	tmp = 0.0
	if (y <= -9.2e-101)
		tmp = t_0;
	elseif (y <= 1.35e-26)
		tmp = Float64(x * Float64(y / x));
	elseif (y <= 8e+109)
		tmp = fma(Float64(-0.16666666666666666 * y), Float64(x * x), y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -9.2e-101], t$95$0, If[LessEqual[y, 1.35e-26], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+109], N[(N[(-0.16666666666666666 * y), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.1999999999999998e-101 or 7.99999999999999985e109 < y

   1. Initial program 97.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 66.6

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

   8. Applied rewrites 66.6%

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

   if -9.1999999999999998e-101 < y < 1.34999999999999991e-26

   1. Initial program 78.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 78.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 77.8%

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

   if 1.34999999999999991e-26 < y < 7.99999999999999985e109

   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. lift-sin.f64 14.2

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

   5. Applied rewrites 14.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 36.7%

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

   9. Taylor expanded in x around 0

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

       1. lift-*.f64 36.7

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

   11. Applied rewrites 36.7%

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

Alternative 11: 52.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{+127} \lor \neg \left(x \leq 4 \cdot 10^{+220}\right):\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot y, x \cdot x, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x 5.5e+127) (not (<= x 4e+220)))
   (* (fma (* y y) 0.16666666666666666 1.0) y)
   (fma (* -0.16666666666666666 y) (* x x) y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= 5.5e+127) || !(x <= 4e+220)) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * y;
	} else {
		tmp = fma((-0.16666666666666666 * y), (x * x), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= 5.5e+127) || !(x <= 4e+220))
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y);
	else
		tmp = fma(Float64(-0.16666666666666666 * y), Float64(x * x), y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, 5.5e+127], N[Not[LessEqual[x, 4e+220]], $MachinePrecision]], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(-0.16666666666666666 * y), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000041e127 or 4e220 < x

   1. Initial program 89.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 65.0

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

   5. Applied rewrites 65.0%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.8

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

   8. Applied rewrites 55.8%

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

   if 5.50000000000000041e127 < x < 4e220

   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. lift-sin.f64 44.4

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 43.5%

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

   9. Taylor expanded in x around 0

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

       1. lift-*.f64 43.6

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

   11. Applied rewrites 43.6%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{+127} \lor \neg \left(x \leq 4 \cdot 10^{+220}\right):\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot y, x \cdot x, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{+127} \lor \neg \left(x \leq 4 \cdot 10^{+220}\right):\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot x\right) \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x 5.5e+127) (not (<= x 4e+220)))
   (* (fma (* y y) 0.16666666666666666 1.0) y)
   (* (* (* -0.16666666666666666 x) x) y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= 5.5e+127) || !(x <= 4e+220)) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * y;
	} else {
		tmp = ((-0.16666666666666666 * x) * x) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= 5.5e+127) || !(x <= 4e+220))
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y);
	else
		tmp = Float64(Float64(Float64(-0.16666666666666666 * x) * x) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, 5.5e+127], N[Not[LessEqual[x, 4e+220]], $MachinePrecision]], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000041e127 or 4e220 < x

   1. Initial program 89.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 65.0

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

   5. Applied rewrites 65.0%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.8

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

   8. Applied rewrites 55.8%

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

   if 5.50000000000000041e127 < x < 4e220

   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. lift-sin.f64 44.4

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

   5. Applied rewrites 44.4%

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

   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. pow2 N/A

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

      4. lift-*.f64 43.6

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

   8. Applied rewrites 43.6%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 43.6

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

   11. Applied rewrites 43.6%

       \[\leadsto \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot y \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. pow2 N/A

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

       4. *-commutative N/A

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

       5. pow2 N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 43.6

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

   13. Applied rewrites 43.6%

       \[\leadsto \left(\left(-0.16666666666666666 \cdot x\right) \cdot x\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{+127} \lor \neg \left(x \leq 4 \cdot 10^{+220}\right):\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot x\right) \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 32.5% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{+18}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot x\right) \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.1e+18) y (* (* (* -0.16666666666666666 x) x) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.1e+18) {
		tmp = y;
	} else {
		tmp = ((-0.16666666666666666 * 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 (x <= 1.1d+18) then
        tmp = y
    else
        tmp = (((-0.16666666666666666d0) * x) * x) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.1e+18) {
		tmp = y;
	} else {
		tmp = ((-0.16666666666666666 * x) * x) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.1e+18:
		tmp = y
	else:
		tmp = ((-0.16666666666666666 * x) * x) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.1e+18)
		tmp = y;
	else
		tmp = Float64(Float64(Float64(-0.16666666666666666 * x) * x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.1e+18)
		tmp = y;
	else
		tmp = ((-0.16666666666666666 * x) * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.1e+18], y, N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1e18

   1. Initial program 87.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 55.7

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 33.4%

      \[\leadsto y \]

   if 1.1e18 < 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. lift-sin.f64 46.3

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

   5. Applied rewrites 46.3%

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

   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. pow2 N/A

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

      4. lift-*.f64 34.4

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

   8. Applied rewrites 34.4%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 34.4

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

   11. Applied rewrites 34.4%

       \[\leadsto \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot y \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. pow2 N/A

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

       4. *-commutative N/A

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

       5. pow2 N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 34.4

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

   13. Applied rewrites 34.4%

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

Alternative 14: 28.6% accurate, 217.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 y)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y
end function

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 90.2%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lift-sin.f64 53.7

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

5. Applied rewrites 53.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 27.1%

   \[\leadsto y \]
9. Add Preprocessing

Developer Target 1: 99.9% 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 2025037 
(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))