Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 4.5s

Alternatives: 21

Speedup: 1.0×

• 48.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 80.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \sin x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (sin x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (*
       (fma
        (fma
         (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
         (* x x)
         -0.16666666666666666)
        (* x x)
        1.0)
       x)
      (* 0.16666666666666666 (* y y)))
     (if (<= t_1 1.0)
       (*
        (sin x)
        (fma
         (fma (* y y) 0.008333333333333333 0.16666666666666666)
         (* y y)
         1.0))
       (* x t_0)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = sin(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666), (x * x), 1.0) * x) * (0.16666666666666666 * (y * y));
	} else if (t_1 <= 1.0) {
		tmp = sin(x) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(sin(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * x) * Float64(0.16666666666666666 * Float64(y * y)));
	elseif (t_1 <= 1.0)
		tmp = Float64(sin(x) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 51.4

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 46.8%

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

   9. Taylor expanded in y around inf

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

       1. pow2 N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 46.8

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

   11. Applied rewrites 46.8%

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

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 1 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 78.6%

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

Alternative 3: 80.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \sin x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (sin x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (*
       (fma
        (fma
         (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
         (* x x)
         -0.16666666666666666)
        (* x x)
        1.0)
       x)
      (* 0.16666666666666666 (* y y)))
     (if (<= t_1 1.0)
       (* (sin x) (fma (* y y) 0.16666666666666666 1.0))
       (* x t_0)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = sin(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666), (x * x), 1.0) * x) * (0.16666666666666666 * (y * y));
	} else if (t_1 <= 1.0) {
		tmp = sin(x) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(sin(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * x) * Float64(0.16666666666666666 * Float64(y * y)));
	elseif (t_1 <= 1.0)
		tmp = Float64(sin(x) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 51.4

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 46.8%

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

   9. Taylor expanded in y around inf

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

       1. pow2 N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 46.8

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

   11. Applied rewrites 46.8%

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

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if 1 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 78.6%

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

Alternative 4: 80.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \sin x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (sin x) t_0)))
   (if (<= t_1 (- INFINITY))
     (*
      (*
       (fma
        (fma
         (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
         (* x x)
         -0.16666666666666666)
        (* x x)
        1.0)
       x)
      (* 0.16666666666666666 (* y y)))
     (if (<= t_1 1.0) (sin x) (* x t_0)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = sin(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666), (x * x), 1.0) * x) * (0.16666666666666666 * (y * y));
	} else if (t_1 <= 1.0) {
		tmp = sin(x);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(sin(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * x) * Float64(0.16666666666666666 * Float64(y * y)));
	elseif (t_1 <= 1.0)
		tmp = sin(x);
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[Sin[x], $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 51.4

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 46.8%

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

   9. Taylor expanded in y around inf

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

       1. pow2 N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 46.8

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

   11. Applied rewrites 46.8%

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

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lift-sin.f64 98.4

         \[\leadsto \sin x \]

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\sin x} \]

   if 1 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 78.6%

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

Alternative 5: 78.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y))))
   (if (<= t_0 (- INFINITY))
     (*
      (*
       (fma
        (fma
         (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
         (* x x)
         -0.16666666666666666)
        (* x x)
        1.0)
       x)
      (* 0.16666666666666666 (* y y)))
     (if (<= t_0 1.0)
       (sin x)
       (*
        (*
         (fma
          (fma (* x x) 0.008333333333333333 -0.16666666666666666)
          (* x x)
          1.0)
         x)
        (fma
         (fma
          (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
          (* y y)
          0.16666666666666666)
         (* y y)
         1.0))))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666), (x * x), 1.0) * x) * (0.16666666666666666 * (y * y));
	} else if (t_0 <= 1.0) {
		tmp = sin(x);
	} else {
		tmp = (fma(fma((x * x), 0.008333333333333333, -0.16666666666666666), (x * x), 1.0) * x) * fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * x) * Float64(0.16666666666666666 * Float64(y * y)));
	elseif (t_0 <= 1.0)
		tmp = sin(x);
	else
		tmp = Float64(Float64(fma(fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666), Float64(x * x), 1.0) * x) * fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[x], $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * 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]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 51.4

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 46.8%

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

   9. Taylor expanded in y around inf

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

       1. pow2 N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 46.8

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

   11. Applied rewrites 46.8%

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

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lift-sin.f64 98.4

         \[\leadsto \sin x \]

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\sin x} \]

   if 1 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 73.4

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

   8. Applied rewrites 73.4%

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

Alternative 6: 89.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;\sin x \cdot t\_0 \leq 1:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* (sin x) t_0) 1.0)
     (*
      (sin x)
      (fma
       (fma
        (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
        (* y y)
        0.16666666666666666)
       (* y y)
       1.0))
     (* x t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if ((sin(x) * t_0) <= 1.0) {
		tmp = sin(x) * fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (Float64(sin(x) * t_0) <= 1.0)
		tmp = Float64(sin(x) * fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision], 1.0], N[(N[Sin[x], $MachinePrecision] * 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]), $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 94.7

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

   5. Applied rewrites 94.7%

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

   if 1 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 78.6%

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

Alternative 7: 54.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.05)
   (*
    (*
     (fma
      (fma
       (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
       (* x x)
       -0.16666666666666666)
      (* x x)
      1.0)
     x)
    (fma (* y y) 0.16666666666666666 1.0))
   (*
    (*
     (fma (fma (* x x) 0.008333333333333333 -0.16666666666666666) (* x x) 1.0)
     x)
    (fma
     (fma
      (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
      (* y y)
      0.16666666666666666)
     (* y y)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.05) {
		tmp = (fma(fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666), (x * x), 1.0) * x) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = (fma(fma((x * x), 0.008333333333333333, -0.16666666666666666), (x * x), 1.0) * x) * fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * x) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(Float64(fma(fma(Float64(x * x), 0.008333333333333333, -0.16666666666666666), Float64(x * x), 1.0) * x) * fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * 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]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.5

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 29.3%

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

   if -0.050000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 66.5

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

   8. Applied rewrites 66.5%

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

Alternative 8: 54.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 5 \cdot 10^{-61}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \mathsf{fma}\left(\left(\left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right) \cdot y\right) \cdot y, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 5e-61)
   (*
    (*
     (fma
      (fma
       (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
       (* x x)
       -0.16666666666666666)
      (* x x)
      1.0)
     x)
    (fma (* y y) 0.16666666666666666 1.0))
   (*
    (*
     (fma (- (* (* x x) 0.008333333333333333) 0.16666666666666666) (* x x) 1.0)
     x)
    (fma (* (* (* (* y y) 0.0001984126984126984) y) y) (* y y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 5e-61) {
		tmp = (fma(fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666), (x * x), 1.0) * x) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = (fma((((x * x) * 0.008333333333333333) - 0.16666666666666666), (x * x), 1.0) * x) * fma(((((y * y) * 0.0001984126984126984) * y) * y), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 5e-61)
		tmp = Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * x) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(x * x) * 0.008333333333333333) - 0.16666666666666666), Float64(x * x), 1.0) * x) * fma(Float64(Float64(Float64(Float64(y * y) * 0.0001984126984126984) * y) * y), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 5e-61], N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 4.9999999999999999e-61

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.2

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 55.2%

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

   if 4.9999999999999999e-61 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.3

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{y}^{2}}\right) \cdot {y}^{4}, y \cdot y, 1\right) \]

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

   8. Applied rewrites 92.8%

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

   9. Taylor expanded in y around inf

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lift-*.f64 92.8

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

   11. Applied rewrites 92.8%

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

   12. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. pow2 N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. pow2 N/A

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

       8. *-commutative N/A

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

       9. lower--.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 47.5

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

   14. Applied rewrites 47.5%

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

Alternative 9: 54.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 2e-6)
   (*
    (*
     (fma
      (fma
       (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
       (* x x)
       -0.16666666666666666)
      (* x x)
      1.0)
     x)
    (fma (* y y) 0.16666666666666666 1.0))
   (*
    x
    (fma
     (* (* (fma (* y y) 0.0001984126984126984 0.008333333333333333) y) y)
     (* y y)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 2e-6) {
		tmp = (fma(fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666), (x * x), 1.0) * x) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = x * fma(((fma((y * y), 0.0001984126984126984, 0.008333333333333333) * y) * y), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 2e-6)
		tmp = Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * x) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(x * fma(Float64(Float64(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333) * y) * y), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 2e-6], N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1.99999999999999991e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 56.5%

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

   if 1.99999999999999991e-6 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{y}^{2}}\right) \cdot {y}^{4}, y \cdot y, 1\right) \]

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

   8. Applied rewrites 92.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 44.1%

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

Alternative 10: 58.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 2e-6)
   (*
    (* (fma -0.16666666666666666 (* x x) 1.0) x)
    (fma
     (fma
      (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
      (* y y)
      0.16666666666666666)
     (* y y)
     1.0))
   (*
    x
    (fma
     (* (* (fma (* y y) 0.0001984126984126984 0.008333333333333333) y) y)
     (* y y)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 2e-6) {
		tmp = (fma(-0.16666666666666666, (x * x), 1.0) * x) * fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = x * fma(((fma((y * y), 0.0001984126984126984, 0.008333333333333333) * y) * y), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 2e-6)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x) * fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(x * fma(Float64(Float64(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333) * y) * y), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 2e-6], N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * 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]), $MachinePrecision], N[(x * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1.99999999999999991e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.4

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 62.3

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

   8. Applied rewrites 62.3%

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

   if 1.99999999999999991e-6 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{y}^{2}}\right) \cdot {y}^{4}, y \cdot y, 1\right) \]

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

   8. Applied rewrites 92.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 44.1%

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

Alternative 11: 57.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 2e-6)
   (*
    (* (fma -0.16666666666666666 (* x x) 1.0) x)
    (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0))
   (*
    x
    (fma
     (* (* (fma (* y y) 0.0001984126984126984 0.008333333333333333) y) y)
     (* y y)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 2e-6) {
		tmp = (fma(-0.16666666666666666, (x * x), 1.0) * x) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = x * fma(((fma((y * y), 0.0001984126984126984, 0.008333333333333333) * y) * y), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 2e-6)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(x * fma(Float64(Float64(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333) * y) * y), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 2e-6], N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1.99999999999999991e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 91.4

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 61.0

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

   8. Applied rewrites 61.0%

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

   if 1.99999999999999991e-6 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{y}^{2}}\right) \cdot {y}^{4}, y \cdot y, 1\right) \]

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

   8. Applied rewrites 92.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 44.1%

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

Alternative 12: 53.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 2e-6)
   (*
    (* (fma -0.16666666666666666 (* x x) 1.0) x)
    (fma (* y y) 0.16666666666666666 1.0))
   (*
    x
    (fma
     (* (* (fma (* y y) 0.0001984126984126984 0.008333333333333333) y) y)
     (* y y)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 2e-6) {
		tmp = (fma(-0.16666666666666666, (x * x), 1.0) * x) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = x * fma(((fma((y * y), 0.0001984126984126984, 0.008333333333333333) * y) * y), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 2e-6)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(x * fma(Float64(Float64(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333) * y) * y), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 2e-6], N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1.99999999999999991e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 56.0

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

   8. Applied rewrites 56.0%

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

   if 1.99999999999999991e-6 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{y}^{2}}\right) \cdot {y}^{4}, y \cdot y, 1\right) \]

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

   8. Applied rewrites 92.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 44.1%

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

Alternative 13: 53.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\left(\left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right) \cdot y\right) \cdot y, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 2e-6)
   (*
    (* (fma -0.16666666666666666 (* x x) 1.0) x)
    (fma (* y y) 0.16666666666666666 1.0))
   (* x (fma (* (* (* (* y y) 0.0001984126984126984) y) y) (* y y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 2e-6) {
		tmp = (fma(-0.16666666666666666, (x * x), 1.0) * x) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = x * fma(((((y * y) * 0.0001984126984126984) * y) * y), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 2e-6)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(x * fma(Float64(Float64(Float64(Float64(y * y) * 0.0001984126984126984) * y) * y), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 2e-6], N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1.99999999999999991e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 56.0

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

   8. Applied rewrites 56.0%

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

   if 1.99999999999999991e-6 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 93.0

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

   5. Applied rewrites 93.0%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto \sin x \cdot \mathsf{fma}\left(\left(\frac{1}{5040} + \frac{1}{120} \cdot \frac{1}{{y}^{2}}\right) \cdot {y}^{4}, y \cdot y, 1\right) \]

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

   8. Applied rewrites 92.4%

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

   9. Taylor expanded in y around inf

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

       1. pow2 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lift-*.f64 92.4

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

   11. Applied rewrites 92.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 44.1%

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

Alternative 14: 52.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, 0.008333333333333333 \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.05)
   (*
    (* (fma -0.16666666666666666 (* x x) 1.0) x)
    (fma (* y y) 0.16666666666666666 1.0))
   (*
    x
    (fma (fma y (* 0.008333333333333333 y) 0.16666666666666666) (* y y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.05) {
		tmp = (fma(-0.16666666666666666, (x * x), 1.0) * x) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = x * fma(fma(y, (0.008333333333333333 * y), 0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(x * fma(fma(y, Float64(0.008333333333333333 * y), 0.16666666666666666), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * N[(0.008333333333333333 * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.5

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 28.4

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

   8. Applied rewrites 28.4%

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

   if -0.050000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 65.2

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

   8. Applied rewrites 65.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 64.7%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 64.7

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

   13. Applied rewrites 64.7%

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

Alternative 15: 44.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(\mathsf{fma}\left(y, 0.008333333333333333 \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.05)
   (* (fma -0.16666666666666666 (* x x) 1.0) x)
   (*
    x
    (fma (fma y (* 0.008333333333333333 y) 0.16666666666666666) (* y y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.05) {
		tmp = fma(-0.16666666666666666, (x * x), 1.0) * x;
	} else {
		tmp = x * fma(fma(y, (0.008333333333333333 * y), 0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x);
	else
		tmp = Float64(x * fma(fma(y, Float64(0.008333333333333333 * y), 0.16666666666666666), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(x * N[(N[(y * N[(0.008333333333333333 * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lift-sin.f64 40.8

         \[\leadsto \sin x \]

   5. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 8.2

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

   8. Applied rewrites 8.2%

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

   if -0.050000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 65.2

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

   8. Applied rewrites 65.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 64.7%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 64.7

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

   13. Applied rewrites 64.7%

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

Alternative 16: 44.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0.008333333333333333 \cdot \left(y \cdot y\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 2e-6)
   (* (fma -0.16666666666666666 (* x x) 1.0) x)
   (* x (fma (* 0.008333333333333333 (* y y)) (* y y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 2e-6) {
		tmp = fma(-0.16666666666666666, (x * x), 1.0) * x;
	} else {
		tmp = x * fma((0.008333333333333333 * (y * y)), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 2e-6)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x);
	else
		tmp = Float64(x * fma(Float64(0.008333333333333333 * Float64(y * y)), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 2e-6], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(x * N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1.99999999999999991e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lift-sin.f64 63.5

         \[\leadsto \sin x \]

   5. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 43.4

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

   8. Applied rewrites 43.4%

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

   if 1.99999999999999991e-6 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 89.9

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 42.7

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

   8. Applied rewrites 42.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 42.0%

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

   12. Taylor expanded in y around inf

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

       1. pow2 N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 42.0

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

   14. Applied rewrites 42.0%

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

Alternative 17: 44.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 2e-6)
   (* (fma -0.16666666666666666 (* x x) 1.0) x)
   (* x (* (* (fma (* y y) 0.008333333333333333 0.16666666666666666) y) y))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 2e-6) {
		tmp = fma(-0.16666666666666666, (x * x), 1.0) * x;
	} else {
		tmp = x * ((fma((y * y), 0.008333333333333333, 0.16666666666666666) * y) * y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 2e-6)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x);
	else
		tmp = Float64(x * Float64(Float64(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666) * y) * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 2e-6], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(x * N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1.99999999999999991e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lift-sin.f64 63.5

         \[\leadsto \sin x \]

   5. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 43.4

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

   8. Applied rewrites 43.4%

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

   if 1.99999999999999991e-6 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 89.9

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

   8. Applied rewrites 51.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 42.0%

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

Alternative 18: 41.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.05)
   (* (fma -0.16666666666666666 (* x x) 1.0) x)
   (* x (fma 0.16666666666666666 (* y y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.05) {
		tmp = fma(-0.16666666666666666, (x * x), 1.0) * x;
	} else {
		tmp = x * fma(0.16666666666666666, (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x);
	else
		tmp = Float64(x * fma(0.16666666666666666, Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(x * N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lift-sin.f64 40.8

         \[\leadsto \sin x \]

   5. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 8.2

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

   8. Applied rewrites 8.2%

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

   if -0.050000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 65.2

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

   8. Applied rewrites 65.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 64.7%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 57.8%

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

Alternative 19: 31.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\left(\left(-0.16666666666666666 \cdot x\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.05)
   (* (* (* -0.16666666666666666 x) x) x)
   x))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.05) {
		tmp = ((-0.16666666666666666 * x) * x) * x;
	} else {
		tmp = x;
	}
	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 ((sin(x) * (sinh(y) / y)) <= (-0.05d0)) then
        tmp = (((-0.16666666666666666d0) * x) * x) * x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sin(x) * (Math.sinh(y) / y)) <= -0.05) {
		tmp = ((-0.16666666666666666 * x) * x) * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sin(x) * (math.sinh(y) / y)) <= -0.05:
		tmp = ((-0.16666666666666666 * x) * x) * x
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * x) * x) * x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sin(x) * (sinh(y) / y)) <= -0.05)
		tmp = ((-0.16666666666666666 * x) * x) * x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lift-sin.f64 40.8

         \[\leadsto \sin x \]

   5. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 8.2

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

   8. Applied rewrites 8.2%

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

   9. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 7.8

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

   11. Applied rewrites 7.8%

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

   if -0.050000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lift-sin.f64 64.3

         \[\leadsto \sin x \]

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\sin x} \]

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 46.5%

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

   9. Taylor expanded in x around 0

      \[\leadsto x \]
   10. Step-by-step derivation

   11. Applied rewrites 40.8%

       \[\leadsto x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 35.5% accurate, 12.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. lift-sin.f64 55.4

      \[\leadsto \sin x \]

5. Applied rewrites 55.4%

   \[\leadsto \color{blue}{\sin x} \]

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 31.6

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

8. Applied rewrites 31.6%

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

Alternative 21: 27.4% accurate, 217.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. lift-sin.f64 55.4

      \[\leadsto \sin x \]

5. Applied rewrites 55.4%

   \[\leadsto \color{blue}{\sin x} \]

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

8. Applied rewrites 33.0%

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

9. Taylor expanded in x around 0

   \[\leadsto x \]
10. Step-by-step derivation

11. Applied rewrites 26.4%

    \[\leadsto x \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025051 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))