Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.8% → 99.8%

Time: 4.9s

Alternatives: 17

Speedup: 1.4×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sinh.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-sinh.f64 N/A

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

   10. lift-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 2: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.001:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.001)
   (* (* 2.0 (sinh y)) (fma (* x x) -0.08333333333333333 0.5))
   (/
    (*
     (sin x)
     (*
      (fma
       (fma
        (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
        (* y y)
        0.16666666666666666)
       (* y y)
       1.0)
      y))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.001) {
		tmp = (2.0 * sinh(y)) * fma((x * x), -0.08333333333333333, 0.5);
	} else {
		tmp = (sin(x) * (fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0) * y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.001)
		tmp = Float64(Float64(2.0 * sinh(y)) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	else
		tmp = Float64(Float64(sin(x) * Float64(fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.001], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e-3

   1. Initial program 87.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) + \frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   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 78.1

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

   5. Applied rewrites 78.1%

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

   if 1e-3 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 92.1

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

   5. Applied rewrites 92.1%

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

Alternative 3: 91.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \sinh y\\ \mathbf{if}\;y \leq -0.00195:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;y \leq 0.022:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq 10^{+103}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 2.0 (sinh y))))
   (if (<= y -0.00195)
     (* t_0 0.5)
     (if (<= y 0.022)
       (* (/ (sin x) x) y)
       (if (<= y 1e+103)
         (* t_0 (fma (* x x) -0.08333333333333333 0.5))
         (/ (* (sin x) (* (fma (* y y) 0.16666666666666666 1.0) y)) x))))))

C

double code(double x, double y) {
	double t_0 = 2.0 * sinh(y);
	double tmp;
	if (y <= -0.00195) {
		tmp = t_0 * 0.5;
	} else if (y <= 0.022) {
		tmp = (sin(x) / x) * y;
	} else if (y <= 1e+103) {
		tmp = t_0 * fma((x * x), -0.08333333333333333, 0.5);
	} else {
		tmp = (sin(x) * (fma((y * y), 0.16666666666666666, 1.0) * y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 * sinh(y))
	tmp = 0.0
	if (y <= -0.00195)
		tmp = Float64(t_0 * 0.5);
	elseif (y <= 0.022)
		tmp = Float64(Float64(sin(x) / x) * y);
	elseif (y <= 1e+103)
		tmp = Float64(t_0 * fma(Float64(x * x), -0.08333333333333333, 0.5));
	else
		tmp = Float64(Float64(sin(x) * Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00195], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[y, 0.022], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1e+103], N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -0.0019499999999999999

   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 90.6

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

   5. Applied rewrites 90.6%

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

   if -0.0019499999999999999 < y < 0.021999999999999999

   1. Initial program 78.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 98.9

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

   5. Applied rewrites 98.9%

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

   if 0.021999999999999999 < y < 1e103

   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 80.0

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

   5. Applied rewrites 80.0%

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

   if 1e103 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 4: 78.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00162:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.00162)
   (* (* 2.0 (sinh y)) (fma (* x x) -0.08333333333333333 0.5))
   (/
    (*
     (sin x)
     (*
      (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0)
      y))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.00162) {
		tmp = (2.0 * sinh(y)) * fma((x * x), -0.08333333333333333, 0.5);
	} else {
		tmp = (sin(x) * (fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0) * y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.00162)
		tmp = Float64(Float64(2.0 * sinh(y)) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	else
		tmp = Float64(Float64(sin(x) * Float64(fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0) * y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.00162], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0016199999999999999

   1. Initial program 87.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) + \frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   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 78.1

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

   5. Applied rewrites 78.1%

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

   if 0.0016199999999999999 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

Alternative 5: 87.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \sinh y\\ \mathbf{if}\;y \leq -0.00195:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;y \leq 0.022:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 2.0 (sinh y))))
   (if (<= y -0.00195)
     (* t_0 0.5)
     (if (<= y 0.022)
       (* (/ (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 <= -0.00195) {
		tmp = t_0 * 0.5;
	} else if (y <= 0.022) {
		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 <= -0.00195)
		tmp = Float64(t_0 * 0.5);
	elseif (y <= 0.022)
		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, -0.00195], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[y, 0.022], 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 < -0.0019499999999999999

   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 90.6

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

   5. Applied rewrites 90.6%

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

   if -0.0019499999999999999 < y < 0.021999999999999999

   1. Initial program 78.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 98.9

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

   5. Applied rewrites 98.9%

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

   if 0.021999999999999999 < 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 81.0

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

   5. Applied rewrites 81.0%

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

Alternative 6: 86.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \sinh y\right) \cdot 0.5\\ \mathbf{if}\;y \leq -0.00195:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.022:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+90}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 2.0 (sinh y)) 0.5)))
   (if (<= y -0.00195)
     t_0
     (if (<= y 0.022)
       (* (/ (sin x) x) y)
       (if (<= y 4.5e+90)
         t_0
         (/
          (*
           (* (fma -0.16666666666666666 (* x x) 1.0) x)
           (* (fma (* y y) 0.16666666666666666 1.0) y))
          x))))))

C

double code(double x, double y) {
	double t_0 = (2.0 * sinh(y)) * 0.5;
	double tmp;
	if (y <= -0.00195) {
		tmp = t_0;
	} else if (y <= 0.022) {
		tmp = (sin(x) / x) * y;
	} else if (y <= 4.5e+90) {
		tmp = t_0;
	} else {
		tmp = ((fma(-0.16666666666666666, (x * x), 1.0) * x) * (fma((y * y), 0.16666666666666666, 1.0) * y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 * sinh(y)) * 0.5)
	tmp = 0.0
	if (y <= -0.00195)
		tmp = t_0;
	elseif (y <= 0.022)
		tmp = Float64(Float64(sin(x) / x) * y);
	elseif (y <= 4.5e+90)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x) * Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y)) / x);
	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, -0.00195], t$95$0, If[LessEqual[y, 0.022], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 4.5e+90], t$95$0, N[(N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0019499999999999999 or 0.021999999999999999 < y < 4.5e90

   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 86.2

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

   5. Applied rewrites 86.2%

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

   if -0.0019499999999999999 < y < 0.021999999999999999

   1. Initial program 78.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 98.9

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

   5. Applied rewrites 98.9%

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

   if 4.5e90 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 80.6

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

   8. Applied rewrites 80.6%

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

Alternative 7: 73.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ t_1 := \left(2 \cdot \sinh y\right) \cdot 0.5\\ \mathbf{if}\;y \leq -1 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.022:\\ \;\;\;\;x \cdot \frac{t\_0}{x}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+90}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot t\_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (* y y) 0.16666666666666666 1.0) y))
        (t_1 (* (* 2.0 (sinh y)) 0.5)))
   (if (<= y -1e-5)
     t_1
     (if (<= y 0.022)
       (* x (/ t_0 x))
       (if (<= y 4.5e+90)
         t_1
         (/ (* (* (fma -0.16666666666666666 (* x x) 1.0) x) t_0) x))))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), 0.16666666666666666, 1.0) * y;
	double t_1 = (2.0 * sinh(y)) * 0.5;
	double tmp;
	if (y <= -1e-5) {
		tmp = t_1;
	} else if (y <= 0.022) {
		tmp = x * (t_0 / x);
	} else if (y <= 4.5e+90) {
		tmp = t_1;
	} else {
		tmp = ((fma(-0.16666666666666666, (x * x), 1.0) * x) * t_0) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y)
	t_1 = Float64(Float64(2.0 * sinh(y)) * 0.5)
	tmp = 0.0
	if (y <= -1e-5)
		tmp = t_1;
	elseif (y <= 0.022)
		tmp = Float64(x * Float64(t_0 / x));
	elseif (y <= 4.5e+90)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x) * t_0) / x);
	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]}, Block[{t$95$1 = N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[y, -1e-5], t$95$1, If[LessEqual[y, 0.022], N[(x * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e+90], t$95$1, N[(N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * t$95$0), $MachinePrecision] / x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.00000000000000008e-5 or 0.021999999999999999 < y < 4.5e90

   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 86.2

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

   5. Applied rewrites 86.2%

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

   if -1.00000000000000008e-5 < y < 0.021999999999999999

   1. Initial program 78.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 78.3

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 35.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 72.9

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

   10. Applied rewrites 72.9%

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

   if 4.5e90 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 80.6

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

   8. Applied rewrites 80.6%

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

Alternative 8: 68.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.0001984126984126984, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+42}:\\ \;\;\;\;x \cdot \frac{t\_0}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot t\_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (* y y) 0.16666666666666666 1.0) y)))
   (if (<= y -3.6e+44)
     (*
      (fma
       (fma (* (* y y) 0.0001984126984126984) (* y y) 0.16666666666666666)
       (* y y)
       1.0)
      y)
     (if (<= y 7.2e+42)
       (* x (/ t_0 x))
       (/ (* (* (fma -0.16666666666666666 (* x x) 1.0) x) t_0) x)))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), 0.16666666666666666, 1.0) * y;
	double tmp;
	if (y <= -3.6e+44) {
		tmp = fma(fma(((y * y) * 0.0001984126984126984), (y * y), 0.16666666666666666), (y * y), 1.0) * y;
	} else if (y <= 7.2e+42) {
		tmp = x * (t_0 / x);
	} else {
		tmp = ((fma(-0.16666666666666666, (x * x), 1.0) * x) * t_0) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y)
	tmp = 0.0
	if (y <= -3.6e+44)
		tmp = Float64(fma(fma(Float64(Float64(y * y) * 0.0001984126984126984), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y);
	elseif (y <= 7.2e+42)
		tmp = Float64(x * Float64(t_0 / x));
	else
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x) * t_0) / x);
	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, -3.6e+44], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 7.2e+42], N[(x * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * t$95$0), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.6e44

   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 90.7

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

   5. Applied rewrites 90.7%

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

   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

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

   8. Applied rewrites 90.7%

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

   9. Taylor expanded in y around inf

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lift-*.f64 90.7

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

   11. Applied rewrites 90.7%

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

   if -3.6e44 < y < 7.2000000000000002e42

   1. Initial program 81.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 34.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 64.4

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

   10. Applied rewrites 64.4%

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

   if 7.2000000000000002e42 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 78.2

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 72.2

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

   8. Applied rewrites 72.2%

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

Alternative 9: 70.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+90}:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right) \cdot y, y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.5e+90)
   (*
    x
    (/
     (*
      (fma
       (fma
        (* (fma (* y y) 0.0001984126984126984 0.008333333333333333) y)
        y
        0.16666666666666666)
       (* y y)
       1.0)
      y)
     x))
   (/
    (*
     (* (fma -0.16666666666666666 (* x x) 1.0) x)
     (* (fma (* y y) 0.16666666666666666 1.0) y))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.5e+90) {
		tmp = x * ((fma(fma((fma((y * y), 0.0001984126984126984, 0.008333333333333333) * y), y, 0.16666666666666666), (y * y), 1.0) * y) / x);
	} else {
		tmp = ((fma(-0.16666666666666666, (x * x), 1.0) * x) * (fma((y * y), 0.16666666666666666, 1.0) * y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.5e+90)
		tmp = Float64(x * Float64(Float64(fma(fma(Float64(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333) * y), y, 0.16666666666666666), Float64(y * y), 1.0) * y) / x));
	else
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x) * Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.5e+90], N[(x * N[(N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * y), $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5e90

   1. Initial program 87.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 78.3

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 72.5

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

   10. Applied rewrites 72.5%

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

   if 4.5e90 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 80.6

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

   8. Applied rewrites 80.6%

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

Alternative 10: 64.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right) \cdot y, y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.15e+24)
   (*
    (fma
     (*
      (fma
       (* (fma (* y y) 0.0001984126984126984 0.008333333333333333) y)
       y
       0.16666666666666666)
      y)
     y
     1.0)
    y)
   (/ (* x (* (* (* y y) 0.16666666666666666) y)) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.15e+24) {
		tmp = fma((fma((fma((y * y), 0.0001984126984126984, 0.008333333333333333) * y), y, 0.16666666666666666) * y), y, 1.0) * y;
	} else {
		tmp = (x * (((y * y) * 0.16666666666666666) * y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.15e+24)
		tmp = Float64(fma(Float64(fma(Float64(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333) * y), y, 0.16666666666666666) * y), y, 1.0) * y);
	else
		tmp = Float64(Float64(x * Float64(Float64(Float64(y * y) * 0.16666666666666666) * y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.15e+24], N[(N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * y), $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(x * N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15e24

   1. Initial program 87.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 78.9

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

   8. Applied rewrites 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   10. Applied rewrites 71.8%

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

   if 1.15e24 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 29.9%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 46.6

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

   11. Applied rewrites 46.6%

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

Alternative 11: 63.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.0001984126984126984, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.15e+24)
   (*
    (fma
     (fma (* (* y y) 0.0001984126984126984) (* y y) 0.16666666666666666)
     (* y y)
     1.0)
    y)
   (/ (* x (* (* (* y y) 0.16666666666666666) y)) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.15e+24) {
		tmp = fma(fma(((y * y) * 0.0001984126984126984), (y * y), 0.16666666666666666), (y * y), 1.0) * y;
	} else {
		tmp = (x * (((y * y) * 0.16666666666666666) * y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.15e+24)
		tmp = Float64(fma(fma(Float64(Float64(y * y) * 0.0001984126984126984), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y);
	else
		tmp = Float64(Float64(x * Float64(Float64(Float64(y * y) * 0.16666666666666666) * y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.15e+24], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(x * N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15e24

   1. Initial program 87.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 78.9

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

   8. Applied rewrites 71.8%

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

   9. Taylor expanded in y around inf

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lift-*.f64 71.8

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

   11. Applied rewrites 71.8%

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

   if 1.15e24 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 29.9%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 46.6

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

   11. Applied rewrites 46.6%

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

Alternative 12: 62.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.15e+24)
   (*
    (fma (* (fma (* y y) 0.008333333333333333 0.16666666666666666) y) y 1.0)
    y)
   (/ (* x (* (* (* y y) 0.16666666666666666) y)) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.15e+24) {
		tmp = fma((fma((y * y), 0.008333333333333333, 0.16666666666666666) * y), y, 1.0) * y;
	} else {
		tmp = (x * (((y * y) * 0.16666666666666666) * y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.15e+24)
		tmp = Float64(fma(Float64(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666) * y), y, 1.0) * y);
	else
		tmp = Float64(Float64(x * Float64(Float64(Float64(y * y) * 0.16666666666666666) * y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.15e+24], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(x * N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15e24

   1. Initial program 87.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 78.9

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 70.3

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

   8. Applied rewrites 70.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. lift-*.f64 70.3

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

   10. Applied rewrites 70.3%

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

   if 1.15e24 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 29.9%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 46.6

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

   11. Applied rewrites 46.6%

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

Alternative 13: 61.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.15e+24)
   (*
    (fma (* (fma (* y y) 0.008333333333333333 0.16666666666666666) y) y 1.0)
    y)
   (* (* (* y y) 0.16666666666666666) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.15e+24) {
		tmp = fma((fma((y * y), 0.008333333333333333, 0.16666666666666666) * y), y, 1.0) * y;
	} else {
		tmp = ((y * y) * 0.16666666666666666) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.15e+24)
		tmp = Float64(fma(Float64(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666) * y), y, 1.0) * y);
	else
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.15e+24], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15e24

   1. Initial program 87.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 78.9

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 70.3

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

   8. Applied rewrites 70.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. lift-*.f64 70.3

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

   10. Applied rewrites 70.3%

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

   if 1.15e24 < x

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 33.2

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

   5. Applied rewrites 33.2%

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

   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. *-commutative N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 22.0

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

   8. Applied rewrites 22.0%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 38.7

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

   11. Applied rewrites 38.7%

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

Alternative 14: 61.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.15e+24)
   (* (fma (* (* y y) 0.008333333333333333) (* y y) 1.0) y)
   (* (* (* y y) 0.16666666666666666) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.15e+24) {
		tmp = fma(((y * y) * 0.008333333333333333), (y * y), 1.0) * y;
	} else {
		tmp = ((y * y) * 0.16666666666666666) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.15e+24)
		tmp = Float64(fma(Float64(Float64(y * y) * 0.008333333333333333), Float64(y * y), 1.0) * y);
	else
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.15e+24], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15e24

   1. Initial program 87.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 78.9

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 70.3

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

   8. Applied rewrites 70.3%

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

   9. Taylor expanded in y around inf

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lift-*.f64 70.2

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

   11. Applied rewrites 70.2%

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

   if 1.15e24 < x

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 33.2

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

   5. Applied rewrites 33.2%

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

   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. *-commutative N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 22.0

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

   8. Applied rewrites 22.0%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 38.7

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

   11. Applied rewrites 38.7%

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

Alternative 15: 51.8% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-7} \lor \neg \left(y \leq 6 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.3e-7) (not (<= y 6e-5)))
   (* (* (* y y) 0.16666666666666666) y)
   y))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.3e-7) || !(y <= 6e-5)) {
		tmp = ((y * y) * 0.16666666666666666) * y;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4.3d-7)) .or. (.not. (y <= 6d-5))) then
        tmp = ((y * y) * 0.16666666666666666d0) * y
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.3e-7) || !(y <= 6e-5)) {
		tmp = ((y * y) * 0.16666666666666666) * y;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.3e-7) or not (y <= 6e-5):
		tmp = ((y * y) * 0.16666666666666666) * y
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.3e-7) || !(y <= 6e-5))
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * y);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.3e-7) || ~((y <= 6e-5)))
		tmp = ((y * y) * 0.16666666666666666) * y;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.3e-7], N[Not[LessEqual[y, 6e-5]], $MachinePrecision]], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision], y]

Derivation

1. Split input into 2 regimes

2. if y < -4.3000000000000001e-7 or 6.00000000000000015e-5 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 76.1

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

   5. Applied rewrites 76.1%

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

   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. *-commutative N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 50.9

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

   8. Applied rewrites 50.9%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 50.9

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

   11. Applied rewrites 50.9%

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

   if -4.3000000000000001e-7 < y < 6.00000000000000015e-5

   1. Initial program 78.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 58.6

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 58.5%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-7} \lor \neg \left(y \leq 6 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 8.5e+23)
   (* (fma y (* y 0.16666666666666666) 1.0) y)
   (* (* (* y y) 0.16666666666666666) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 8.5e+23) {
		tmp = fma(y, (y * 0.16666666666666666), 1.0) * y;
	} else {
		tmp = ((y * y) * 0.16666666666666666) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 8.5e+23)
		tmp = Float64(fma(y, Float64(y * 0.16666666666666666), 1.0) * y);
	else
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 8.5e+23], N[(N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.5000000000000001e23

   1. Initial program 87.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 78.9

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 63.6

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

   8. Applied rewrites 63.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 63.6

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

   10. Applied rewrites 63.6%

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

   if 8.5000000000000001e23 < x

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 33.2

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

   5. Applied rewrites 33.2%

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

   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. *-commutative N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 22.0

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

   8. Applied rewrites 22.0%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 38.7

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

   11. Applied rewrites 38.7%

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

Alternative 17: 28.0% 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 90.5%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. rec-exp N/A

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

   4. sinh-undef N/A

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

   5. lower-*.f64 N/A

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

   6. lift-sinh.f64 68.6

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

5. Applied rewrites 68.6%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 27.3%

   \[\leadsto y \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025085 
(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))