Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 9.2s

Alternatives: 9

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: 83.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \frac{\sinh y}{y}\\ t_1 := \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\right) \cdot \left(x \cdot x\right), \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666, x\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right) \cdot y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666\right) \cdot x, x \cdot x, x\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y)))
        (t_1
         (fma
          (* (* (fma 0.0001984126984126984 (* y y) 0.008333333333333333) y) y)
          (* y y)
          1.0)))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (* (- x) (* x x))
       (- (* (* x x) 0.008333333333333333) 0.16666666666666666)
       x)
      t_1)
     (if (<= t_0 2.0)
       (*
        (sin x)
        (fma
         (* (fma 0.008333333333333333 (* y y) 0.16666666666666666) y)
         y
         1.0))
       (*
        (fma
         (* (- (* 0.008333333333333333 (* x x)) 0.16666666666666666) x)
         (* x x)
         x)
        t_1)))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (sinh(y) / y);
	double t_1 = fma(((fma(0.0001984126984126984, (y * y), 0.008333333333333333) * y) * y), (y * y), 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((-x * (x * x)), (((x * x) * 0.008333333333333333) - 0.16666666666666666), x) * t_1;
	} else if (t_0 <= 2.0) {
		tmp = sin(x) * fma((fma(0.008333333333333333, (y * y), 0.16666666666666666) * y), y, 1.0);
	} else {
		tmp = fma((((0.008333333333333333 * (x * x)) - 0.16666666666666666) * x), (x * x), x) * t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(sinh(y) / y))
	t_1 = fma(Float64(Float64(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333) * y) * y), Float64(y * y), 1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(-x) * Float64(x * x)), Float64(Float64(Float64(x * x) * 0.008333333333333333) - 0.16666666666666666), x) * t_1);
	elseif (t_0 <= 2.0)
		tmp = Float64(sin(x) * fma(Float64(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666) * y), y, 1.0));
	else
		tmp = Float64(fma(Float64(Float64(Float64(0.008333333333333333 * Float64(x * x)) - 0.16666666666666666) * x), Float64(x * x), x) * t_1);
	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]}, Block[{t$95$1 = N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[((-x) * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision] * t$95$1), $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

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

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

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

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

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in 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

   8. Applied rewrites 87.6%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. cube-mult N/A

          \[\leadsto \left(\color{blue}{{x}^{3}} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + x \cdot 1\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       6. *-rgt-identity N/A

          \[\leadsto \left({x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + \color{blue}{x}\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, x\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       9. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       13. lower-*.f64 68.0

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(x \cdot x\right)} \cdot 0.008333333333333333 - 0.16666666666666666, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

   11. Applied rewrites 68.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666, x\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 69.6%

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

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

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

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

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

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

   7. Applied rewrites 99.3%

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

   if 2 < (*.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. fp-cancel-sign-sub-inv N/A

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

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

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in 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

   8. Applied rewrites 82.2%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. cube-mult N/A

          \[\leadsto \left(\color{blue}{{x}^{3}} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + x \cdot 1\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       6. *-rgt-identity N/A

          \[\leadsto \left({x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + \color{blue}{x}\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, x\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       9. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       13. lower-*.f64 61.5

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(x \cdot x\right)} \cdot 0.008333333333333333 - 0.16666666666666666, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

   11. Applied rewrites 61.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666, x\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 61.5%

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

Alternative 3: 83.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \frac{\sinh y}{y}\\ t_1 := \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\right) \cdot \left(x \cdot x\right), \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666, x\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666\right) \cdot x, x \cdot x, x\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y)))
        (t_1
         (fma
          (* (* (fma 0.0001984126984126984 (* y y) 0.008333333333333333) y) y)
          (* y y)
          1.0)))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (* (- x) (* x x))
       (- (* (* x x) 0.008333333333333333) 0.16666666666666666)
       x)
      t_1)
     (if (<= t_0 2.0)
       (* (sin x) (fma (* 0.16666666666666666 y) y 1.0))
       (*
        (fma
         (* (- (* 0.008333333333333333 (* x x)) 0.16666666666666666) x)
         (* x x)
         x)
        t_1)))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (sinh(y) / y);
	double t_1 = fma(((fma(0.0001984126984126984, (y * y), 0.008333333333333333) * y) * y), (y * y), 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((-x * (x * x)), (((x * x) * 0.008333333333333333) - 0.16666666666666666), x) * t_1;
	} else if (t_0 <= 2.0) {
		tmp = sin(x) * fma((0.16666666666666666 * y), y, 1.0);
	} else {
		tmp = fma((((0.008333333333333333 * (x * x)) - 0.16666666666666666) * x), (x * x), x) * t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(sinh(y) / y))
	t_1 = fma(Float64(Float64(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333) * y) * y), Float64(y * y), 1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(-x) * Float64(x * x)), Float64(Float64(Float64(x * x) * 0.008333333333333333) - 0.16666666666666666), x) * t_1);
	elseif (t_0 <= 2.0)
		tmp = Float64(sin(x) * fma(Float64(0.16666666666666666 * y), y, 1.0));
	else
		tmp = Float64(fma(Float64(Float64(Float64(0.008333333333333333 * Float64(x * x)) - 0.16666666666666666) * x), Float64(x * x), x) * t_1);
	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]}, Block[{t$95$1 = N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[((-x) * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision] * t$95$1), $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

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

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

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

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

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in 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

   8. Applied rewrites 87.6%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. cube-mult N/A

          \[\leadsto \left(\color{blue}{{x}^{3}} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + x \cdot 1\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       6. *-rgt-identity N/A

          \[\leadsto \left({x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + \color{blue}{x}\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, x\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       9. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       13. lower-*.f64 68.0

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(x \cdot x\right)} \cdot 0.008333333333333333 - 0.16666666666666666, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

   11. Applied rewrites 68.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666, x\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 69.6%

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

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

   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.1

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.1%

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

   if 2 < (*.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. fp-cancel-sign-sub-inv N/A

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

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

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in 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

   8. Applied rewrites 82.2%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. cube-mult N/A

          \[\leadsto \left(\color{blue}{{x}^{3}} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + x \cdot 1\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       6. *-rgt-identity N/A

          \[\leadsto \left({x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + \color{blue}{x}\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, x\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       9. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       13. lower-*.f64 61.5

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(x \cdot x\right)} \cdot 0.008333333333333333 - 0.16666666666666666, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

   11. Applied rewrites 61.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666, x\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 61.5%

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

Alternative 4: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \frac{\sinh y}{y}\\ t_1 := \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\right) \cdot \left(x \cdot x\right), \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666, x\right) \cdot t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\sin x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666\right) \cdot x, x \cdot x, x\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y)))
        (t_1
         (fma
          (* (* (fma 0.0001984126984126984 (* y y) 0.008333333333333333) y) y)
          (* y y)
          1.0)))
   (if (<= t_0 (- INFINITY))
     (*
      (fma
       (* (- x) (* x x))
       (- (* (* x x) 0.008333333333333333) 0.16666666666666666)
       x)
      t_1)
     (if (<= t_0 2.0)
       (* (sin x) 1.0)
       (*
        (fma
         (* (- (* 0.008333333333333333 (* x x)) 0.16666666666666666) x)
         (* x x)
         x)
        t_1)))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (sinh(y) / y);
	double t_1 = fma(((fma(0.0001984126984126984, (y * y), 0.008333333333333333) * y) * y), (y * y), 1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((-x * (x * x)), (((x * x) * 0.008333333333333333) - 0.16666666666666666), x) * t_1;
	} else if (t_0 <= 2.0) {
		tmp = sin(x) * 1.0;
	} else {
		tmp = fma((((0.008333333333333333 * (x * x)) - 0.16666666666666666) * x), (x * x), x) * t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(sinh(y) / y))
	t_1 = fma(Float64(Float64(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333) * y) * y), Float64(y * y), 1.0)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(Float64(-x) * Float64(x * x)), Float64(Float64(Float64(x * x) * 0.008333333333333333) - 0.16666666666666666), x) * t_1);
	elseif (t_0 <= 2.0)
		tmp = Float64(sin(x) * 1.0);
	else
		tmp = Float64(fma(Float64(Float64(Float64(0.008333333333333333 * Float64(x * x)) - 0.16666666666666666) * x), Float64(x * x), x) * t_1);
	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]}, Block[{t$95$1 = N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[((-x) * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[Sin[x], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision] * t$95$1), $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

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

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

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

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

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 87.6%

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

   6. Taylor expanded in 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

   8. Applied rewrites 87.6%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. cube-mult N/A

          \[\leadsto \left(\color{blue}{{x}^{3}} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + x \cdot 1\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       6. *-rgt-identity N/A

          \[\leadsto \left({x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + \color{blue}{x}\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, x\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       9. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       13. lower-*.f64 68.0

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(x \cdot x\right)} \cdot 0.008333333333333333 - 0.16666666666666666, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

   11. Applied rewrites 68.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666, x\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 69.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.3%

      \[\leadsto \sin x \cdot \color{blue}{1} \]

   if 2 < (*.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. fp-cancel-sign-sub-inv N/A

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

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

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in 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

   8. Applied rewrites 82.2%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. cube-mult N/A

          \[\leadsto \left(\color{blue}{{x}^{3}} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + x \cdot 1\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       6. *-rgt-identity N/A

          \[\leadsto \left({x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + \color{blue}{x}\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, x\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       9. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       13. lower-*.f64 61.5

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(x \cdot x\right)} \cdot 0.008333333333333333 - 0.16666666666666666, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

   11. Applied rewrites 61.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666, x\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 61.5%

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

Alternative 5: 87.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right)\\ \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 2:\\ \;\;\;\;\sin x \cdot \mathsf{fma}\left(\mathsf{fma}\left(t\_0, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666\right) \cdot x, x \cdot x, x\right) \cdot \mathsf{fma}\left(\left(t\_0 \cdot y\right) \cdot y, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma 0.0001984126984126984 (* y y) 0.008333333333333333)))
   (if (<= (* (sin x) (/ (sinh y) y)) 2.0)
     (* (sin x) (fma (fma t_0 (* y y) 0.16666666666666666) (* y y) 1.0))
     (*
      (fma
       (* (- (* 0.008333333333333333 (* x x)) 0.16666666666666666) x)
       (* x x)
       x)
      (fma (* (* t_0 y) y) (* y y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = fma(0.0001984126984126984, (y * y), 0.008333333333333333);
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 2.0) {
		tmp = sin(x) * fma(fma(t_0, (y * y), 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = fma((((0.008333333333333333 * (x * x)) - 0.16666666666666666) * x), (x * x), x) * fma(((t_0 * y) * y), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 2.0)
		tmp = Float64(sin(x) * fma(fma(t_0, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(fma(Float64(Float64(Float64(0.008333333333333333 * Float64(x * x)) - 0.16666666666666666) * x), Float64(x * x), x) * fma(Float64(Float64(t_0 * y) * y), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]}, If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[Sin[x], $MachinePrecision] * N[(N[(t$95$0 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision] * N[(N[(N[(t$95$0 * 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)) < 2

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

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

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

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 95.8%

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

   if 2 < (*.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. fp-cancel-sign-sub-inv N/A

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

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

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in 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

   8. Applied rewrites 82.2%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. cube-mult N/A

          \[\leadsto \left(\color{blue}{{x}^{3}} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + x \cdot 1\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       6. *-rgt-identity N/A

          \[\leadsto \left({x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + \color{blue}{x}\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, x\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       9. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       13. lower-*.f64 61.5

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(x \cdot x\right)} \cdot 0.008333333333333333 - 0.16666666666666666, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

   11. Applied rewrites 61.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666, x\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 61.5%

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

Alternative 6: 58.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right)\\ \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\right) \cdot \left(x \cdot x\right), \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666, x\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666\right) \cdot x, x \cdot x, x\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (fma
          (* (* (fma 0.0001984126984126984 (* y y) 0.008333333333333333) y) y)
          (* y y)
          1.0)))
   (if (<= (sin x) -0.005)
     (*
      (fma
       (* (- x) (* x x))
       (- (* (* x x) 0.008333333333333333) 0.16666666666666666)
       x)
      t_0)
     (*
      (fma
       (* (- (* 0.008333333333333333 (* x x)) 0.16666666666666666) x)
       (* x x)
       x)
      t_0))))

C

double code(double x, double y) {
	double t_0 = fma(((fma(0.0001984126984126984, (y * y), 0.008333333333333333) * y) * y), (y * y), 1.0);
	double tmp;
	if (sin(x) <= -0.005) {
		tmp = fma((-x * (x * x)), (((x * x) * 0.008333333333333333) - 0.16666666666666666), x) * t_0;
	} else {
		tmp = fma((((0.008333333333333333 * (x * x)) - 0.16666666666666666) * x), (x * x), x) * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(Float64(Float64(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333) * y) * y), Float64(y * y), 1.0)
	tmp = 0.0
	if (sin(x) <= -0.005)
		tmp = Float64(fma(Float64(Float64(-x) * Float64(x * x)), Float64(Float64(Float64(x * x) * 0.008333333333333333) - 0.16666666666666666), x) * t_0);
	else
		tmp = Float64(fma(Float64(Float64(Float64(0.008333333333333333 * Float64(x * x)) - 0.16666666666666666) * x), Float64(x * x), x) * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Sin[x], $MachinePrecision], -0.005], N[(N[(N[((-x) * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0050000000000000001

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

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

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

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in 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

   8. Applied rewrites 92.2%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. cube-mult N/A

          \[\leadsto \left(\color{blue}{{x}^{3}} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + x \cdot 1\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       6. *-rgt-identity N/A

          \[\leadsto \left({x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + \color{blue}{x}\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, x\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       9. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       13. lower-*.f64 20.4

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(x \cdot x\right)} \cdot 0.008333333333333333 - 0.16666666666666666, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

   11. Applied rewrites 20.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666, x\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 21.6%

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

   if -0.0050000000000000001 < (sin.f64 x)

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

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

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

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 91.9%

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

   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

   8. Applied rewrites 91.9%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. cube-mult N/A

          \[\leadsto \left(\color{blue}{{x}^{3}} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + x \cdot 1\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       6. *-rgt-identity N/A

          \[\leadsto \left({x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + \color{blue}{x}\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, x\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       9. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       13. lower-*.f64 70.2

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(x \cdot x\right)} \cdot 0.008333333333333333 - 0.16666666666666666, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

   11. Applied rewrites 70.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666, x\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 70.2%

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

Alternative 7: 58.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666\right) \cdot x, x \cdot x, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.005)
   (*
    (fma (* x x) (* x -0.16666666666666666) x)
    (fma (* y y) 0.16666666666666666 1.0))
   (*
    (fma
     (* (- (* 0.008333333333333333 (* x x)) 0.16666666666666666) x)
     (* x x)
     x)
    (fma
     (* (* (fma 0.0001984126984126984 (* y y) 0.008333333333333333) y) y)
     (* y y)
     1.0))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.005) {
		tmp = fma((x * x), (x * -0.16666666666666666), x) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = fma((((0.008333333333333333 * (x * x)) - 0.16666666666666666) * x), (x * x), x) * fma(((fma(0.0001984126984126984, (y * y), 0.008333333333333333) * y) * y), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.005)
		tmp = Float64(fma(Float64(x * x), Float64(x * -0.16666666666666666), x) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(fma(Float64(Float64(Float64(0.008333333333333333 * Float64(x * x)) - 0.16666666666666666) * x), Float64(x * x), x) * fma(Float64(Float64(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333) * y) * y), Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.005], N[(N[(N[(x * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision] * N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0050000000000000001

   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

   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

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 21.7

          \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \]

   8. Applied rewrites 21.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 21.7%

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

   if -0.0050000000000000001 < (sin.f64 x)

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

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

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

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

      3. +-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. distribute-lft-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 91.9%

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

   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

   8. Applied rewrites 91.9%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. unpow2 N/A

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

       5. cube-mult N/A

          \[\leadsto \left(\color{blue}{{x}^{3}} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + x \cdot 1\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       6. *-rgt-identity N/A

          \[\leadsto \left({x}^{3} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + \color{blue}{x}\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       7. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, x\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       9. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{{x}^{2} \cdot \frac{1}{120}} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{120} - \frac{1}{6}, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{5040}, y \cdot y, \frac{1}{120}\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

       13. lower-*.f64 70.2

           \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\left(x \cdot x\right)} \cdot 0.008333333333333333 - 0.16666666666666666, x\right) \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]

   11. Applied rewrites 70.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666, x\right)} \cdot \mathsf{fma}\left(\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y\right) \cdot y, y \cdot y, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 70.2%

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

Alternative 8: 50.2% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 77.5

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

5. Applied rewrites 77.5%

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

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

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. pow-plus N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval 48.7

       \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, -0.16666666666666666, x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \]

8. Applied rewrites 48.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \]
9. Step-by-step derivation

10. Applied rewrites 48.7%

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

Alternative 9: 34.7% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

   \[\leadsto \sin x \cdot \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 53.1%

   \[\leadsto \sin x \cdot \color{blue}{1} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. pow-plus N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval 34.6

       \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, -0.16666666666666666, x\right) \cdot 1 \]

8. Applied rewrites 34.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)} \cdot 1 \]
9. Step-by-step derivation

10. Applied rewrites 34.6%

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

Reproduce

herbie shell --seed 2024359 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))