Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 3.8s

Alternatives: 16

Speedup: 1.0×

• 49.8% 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

Alternative 2: 79.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 75.2

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

   4. Applied rewrites 75.2%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 48.8

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

   7. Applied rewrites 48.8%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 48.8

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

   10. Applied rewrites 48.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. pow2 N/A

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

       4. *-commutative N/A

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

       5. pow2 N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 48.8

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

   12. Applied rewrites 48.8%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lift-sin.f64 98.3

         \[\leadsto \sin x \]

   4. Applied rewrites 98.3%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 73.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 73.6

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

   6. Applied rewrites 73.6%

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

Alternative 3: 62.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;\sin x \cdot t\_0 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(-0.16666666666666666 \cdot x\right), x, x\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* (sin x) t_0) 4e-6)
     (* (fma (* x (* -0.16666666666666666 x)) x x) t_0)
     (/ (* x (sinh y)) y))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if ((sin(x) * t_0) <= 4e-6) {
		tmp = fma((x * (-0.16666666666666666 * x)), x, x) * t_0;
	} else {
		tmp = (x * sinh(y)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (Float64(sin(x) * t_0) <= 4e-6)
		tmp = Float64(fma(Float64(x * Float64(-0.16666666666666666 * x)), x, x) * t_0);
	else
		tmp = Float64(Float64(x * sinh(y)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision], 4e-6], N[(N[(N[(x * N[(-0.16666666666666666 * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 3.99999999999999982e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 69.9

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

   4. Applied rewrites 69.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 69.9

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

   6. Applied rewrites 69.9%

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

   if 3.99999999999999982e-6 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 50.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 50.4

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

   6. Applied rewrites 50.4%

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

Alternative 4: 62.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;\sin x \cdot t\_0 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* (sin x) t_0) 4e-6)
     (* (* (fma -0.16666666666666666 (* x x) 1.0) x) t_0)
     (/ (* x (sinh y)) y))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if ((sin(x) * t_0) <= 4e-6) {
		tmp = (fma(-0.16666666666666666, (x * x), 1.0) * x) * t_0;
	} else {
		tmp = (x * sinh(y)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (Float64(sin(x) * t_0) <= 4e-6)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x) * t_0);
	else
		tmp = Float64(Float64(x * sinh(y)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision], 4e-6], N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 3.99999999999999982e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 69.9

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

   4. Applied rewrites 69.9%

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

   if 3.99999999999999982e-6 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 50.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 50.4

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

   6. Applied rewrites 50.4%

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

Alternative 5: 55.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 4e-6)
   (*
    (* (fma -0.16666666666666666 (* x x) 1.0) x)
    (fma (* y y) 0.16666666666666666 1.0))
   (/ (* x (sinh y)) y)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 4e-6) {
		tmp = (fma(-0.16666666666666666, (x * x), 1.0) * x) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = (x * sinh(y)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 4e-6)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(Float64(x * sinh(y)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 4e-6], N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 3.99999999999999982e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 69.9

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

   4. Applied rewrites 69.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 58.9

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

   7. Applied rewrites 58.9%

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

   if 3.99999999999999982e-6 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 50.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 50.4

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

   6. Applied rewrites 50.4%

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

Alternative 6: 55.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;\sin x \cdot t\_0 \leq -0.1:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \left(\left(0.16666666666666666 \cdot y\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* (sin x) t_0) -0.1)
     (*
      (* (fma -0.16666666666666666 (* x x) 1.0) x)
      (* (* 0.16666666666666666 y) y))
     (* x t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if ((sin(x) * t_0) <= -0.1) {
		tmp = (fma(-0.16666666666666666, (x * x), 1.0) * x) * ((0.16666666666666666 * y) * y);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (Float64(sin(x) * t_0) <= -0.1)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x) * Float64(Float64(0.16666666666666666 * y) * y));
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision], -0.1], N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(N[(0.16666666666666666 * y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.10000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 51.3

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

   4. Applied rewrites 51.3%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 33.6

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

   7. Applied rewrites 33.6%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 33.4

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

   10. Applied rewrites 33.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. pow2 N/A

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

       4. *-commutative N/A

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

       5. pow2 N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 33.3

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

   12. Applied rewrites 33.3%

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

   if -0.10000000000000001 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 68.8%

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

Alternative 7: 49.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.005:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.16666666666666666, x\right) \cdot y}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-46}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y))))
   (if (<= t_0 -0.005)
     (/ (* (fma (* (* x x) x) -0.16666666666666666 x) y) y)
     (if (<= t_0 5e-46)
       (* x (fma (* 0.16666666666666666 y) y 1.0))
       (/ (* x (sinh y)) y)))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.005) {
		tmp = (fma(((x * x) * x), -0.16666666666666666, x) * y) / y;
	} else if (t_0 <= 5e-46) {
		tmp = x * fma((0.16666666666666666 * y), y, 1.0);
	} else {
		tmp = (x * sinh(y)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.005)
		tmp = Float64(Float64(fma(Float64(Float64(x * x) * x), -0.16666666666666666, x) * y) / y);
	elseif (t_0 <= 5e-46)
		tmp = Float64(x * fma(Float64(0.16666666666666666 * y), y, 1.0));
	else
		tmp = Float64(Float64(x * sinh(y)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.005], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 5e-46], N[(x * N[(N[(0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0050000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 50.3

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

   6. Applied rewrites 50.3%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 12.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

   11. Applied rewrites 14.9%

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

   if -0.0050000000000000001 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 4.99999999999999992e-46

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 98.2

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

   10. Applied rewrites 98.2%

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

   if 4.99999999999999992e-46 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 52.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 52.1

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

   6. Applied rewrites 52.1%

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

Alternative 8: 48.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* (sin x) t_0) -0.005)
     (/ (* (fma (* (* x x) x) -0.16666666666666666 x) y) y)
     (* x t_0))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0050000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 50.3

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

   6. Applied rewrites 50.3%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 12.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

   11. Applied rewrites 14.9%

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

   if -0.0050000000000000001 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 69.6%

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

Alternative 9: 47.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.005:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.16666666666666666, x\right) \cdot y}{y}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-46}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.16666666666666666, y, y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y))))
   (if (<= t_0 -0.005)
     (/ (* (fma (* (* x x) x) -0.16666666666666666 x) y) y)
     (if (<= t_0 5e-46)
       (* x (fma (* 0.16666666666666666 y) y 1.0))
       (/ (* x (fma (* (* y y) 0.16666666666666666) y y)) y)))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.005) {
		tmp = (fma(((x * x) * x), -0.16666666666666666, x) * y) / y;
	} else if (t_0 <= 5e-46) {
		tmp = x * fma((0.16666666666666666 * y), y, 1.0);
	} else {
		tmp = (x * fma(((y * y) * 0.16666666666666666), y, y)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.005)
		tmp = Float64(Float64(fma(Float64(Float64(x * x) * x), -0.16666666666666666, x) * y) / y);
	elseif (t_0 <= 5e-46)
		tmp = Float64(x * fma(Float64(0.16666666666666666 * y), y, 1.0));
	else
		tmp = Float64(Float64(x * fma(Float64(Float64(y * y) * 0.16666666666666666), y, y)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.005], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 5e-46], N[(x * N[(N[(0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0050000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 50.3

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

   6. Applied rewrites 50.3%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 12.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

   11. Applied rewrites 14.9%

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

   if -0.0050000000000000001 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 4.99999999999999992e-46

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 98.2

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

   10. Applied rewrites 98.2%

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

   if 4.99999999999999992e-46 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 52.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 52.1

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

   6. Applied rewrites 52.1%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 40.7

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

   9. Applied rewrites 40.7%

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

Alternative 10: 47.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0050000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 50.3

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

   6. Applied rewrites 50.3%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 12.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

   11. Applied rewrites 14.9%

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

   if -0.0050000000000000001 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 69.6%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 61.4

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

   7. Applied rewrites 61.4%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 61.4

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

   9. Applied rewrites 61.4%

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

Alternative 11: 44.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.16666666666666666, x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.16666666666666666, y, y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 4e-6)
   (* (fma (* (* x x) x) -0.16666666666666666 x) 1.0)
   (/ (* x (fma (* (* y y) 0.16666666666666666) y y)) y)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 4e-6) {
		tmp = fma(((x * x) * x), -0.16666666666666666, x) * 1.0;
	} else {
		tmp = (x * fma(((y * y) * 0.16666666666666666), y, y)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 4e-6)
		tmp = Float64(fma(Float64(Float64(x * x) * x), -0.16666666666666666, x) * 1.0);
	else
		tmp = Float64(Float64(x * fma(Float64(Float64(y * y) * 0.16666666666666666), y, y)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 4e-6], N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(x * N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 3.99999999999999982e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 69.9

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

   4. Applied rewrites 69.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 46.2%

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

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 9.3

         \[\leadsto \left(\left(\left(-0.16666666666666666 \cdot x\right) \cdot x\right) \cdot x\right) \cdot 1 \]

   10. Applied rewrites 9.3%

       \[\leadsto \left(\left(\left(-0.16666666666666666 \cdot x\right) \cdot x\right) \cdot x\right) \cdot 1 \]

   11. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. distribute-lft1-in N/A

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

       4. *-commutative N/A

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

       5. +-commutative N/A

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

       6. pow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. distribute-lft1-in N/A

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

       10. associate-*l* N/A

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

       11. associate-*r* N/A

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

       12. pow2 N/A

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

       13. *-commutative N/A

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

       14. associate-*r* N/A

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

       15. pow2 N/A

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

       16. cube-mult N/A

           \[\leadsto \left({x}^{3} \cdot \frac{-1}{6} + x\right) \cdot 1 \]

       17. lower-fma.f64 N/A

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

       18. unpow3 N/A

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

       19. pow2 N/A

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

       20. lower-*.f64 N/A

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

       21. pow2 N/A

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

       22. lift-*.f64 46.2

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

   13. Applied rewrites 46.2%

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

   if 3.99999999999999982e-6 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 50.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 50.4

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

   6. Applied rewrites 50.4%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 38.6

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

   9. Applied rewrites 38.6%

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

Alternative 12: 44.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 0.4:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.16666666666666666, x\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(\left(y \cdot y\right) \cdot y\right) \cdot 0.16666666666666666}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 0.4)
   (* (fma (* (* x x) x) -0.16666666666666666 x) 1.0)
   (* x (/ (* (* (* y y) y) 0.16666666666666666) y))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 0.4) {
		tmp = fma(((x * x) * x), -0.16666666666666666, x) * 1.0;
	} else {
		tmp = x * ((((y * y) * y) * 0.16666666666666666) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 0.4)
		tmp = Float64(fma(Float64(Float64(x * x) * x), -0.16666666666666666, x) * 1.0);
	else
		tmp = Float64(x * Float64(Float64(Float64(Float64(y * y) * y) * 0.16666666666666666) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.4], N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision] * 1.0), $MachinePrecision], N[(x * N[(N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.40000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 66.7

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

   4. Applied rewrites 66.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 44.2%

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

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 8.9

         \[\leadsto \left(\left(\left(-0.16666666666666666 \cdot x\right) \cdot x\right) \cdot x\right) \cdot 1 \]

   10. Applied rewrites 8.9%

       \[\leadsto \left(\left(\left(-0.16666666666666666 \cdot x\right) \cdot x\right) \cdot x\right) \cdot 1 \]

   11. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. distribute-lft1-in N/A

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

       4. *-commutative N/A

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

       5. +-commutative N/A

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

       6. pow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. distribute-lft1-in N/A

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

       10. associate-*l* N/A

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

       11. associate-*r* N/A

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

       12. pow2 N/A

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

       13. *-commutative N/A

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

       14. associate-*r* N/A

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

       15. pow2 N/A

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

       16. cube-mult N/A

           \[\leadsto \left({x}^{3} \cdot \frac{-1}{6} + x\right) \cdot 1 \]

       17. lower-fma.f64 N/A

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

       18. unpow3 N/A

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

       19. pow2 N/A

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

       20. lower-*.f64 N/A

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

       21. pow2 N/A

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

       22. lift-*.f64 44.2

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

   13. Applied rewrites 44.2%

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

   if 0.40000000000000002 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 40.9

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

   7. Applied rewrites 40.9%

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

   8. Taylor expanded in y around inf

      \[\leadsto x \cdot \frac{\frac{1}{6} \cdot \color{blue}{{y}^{3}}}{y} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot \frac{{y}^{3} \cdot \frac{1}{6}}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot \frac{{y}^{3} \cdot \frac{1}{6}}{y} \]

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 41.0

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

   10. Applied rewrites 41.0%

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

Alternative 13: 43.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.005], N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * 1.0), $MachinePrecision], N[(x * N[(N[(0.16666666666666666 * y), $MachinePrecision] * y + 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. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 25.7

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

   4. Applied rewrites 25.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 17.5%

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

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lift-*.f64 17.3

         \[\leadsto \left(\left(\left(-0.16666666666666666 \cdot x\right) \cdot x\right) \cdot x\right) \cdot 1 \]

   10. Applied rewrites 17.3%

       \[\leadsto \left(\left(\left(-0.16666666666666666 \cdot x\right) \cdot x\right) \cdot x\right) \cdot 1 \]

   11. Taylor expanded in x around inf

       \[\leadsto \left(\frac{-1}{6} \cdot \color{blue}{{x}^{3}}\right) \cdot 1 \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {x}^{3}\right) \cdot 1 \]

       2. *-commutative N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {x}^{3}\right) \cdot 1 \]

       3. distribute-lft1-in N/A

          \[\leadsto \left(\frac{-1}{6} \cdot {\color{blue}{x}}^{3}\right) \cdot 1 \]

       4. *-commutative N/A

          \[\leadsto \left({x}^{3} \cdot \frac{-1}{6}\right) \cdot 1 \]

       5. lower-*.f64 N/A

          \[\leadsto \left({x}^{3} \cdot \frac{-1}{6}\right) \cdot 1 \]

       6. unpow3 N/A

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

       7. pow2 N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 17.3

           \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot -0.16666666666666666\right) \cdot 1 \]

   13. Applied rewrites 17.3%

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

   if -0.0050000000000000001 < (sin.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 74.6%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 62.6

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

   7. Applied rewrites 62.6%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 57.1

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

   10. Applied rewrites 57.1%

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

Alternative 14: 43.1% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 62.5%

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

5. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 52.2

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

7. Applied rewrites 52.2%

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

8. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. pow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 47.9

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

10. Applied rewrites 47.9%

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

Alternative 15: 29.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 5 \cdot 10^{-46}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 5e-46) (* x 1.0) (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 5e-46) {
		tmp = x * 1.0;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((sin(x) * (sinh(y) / y)) <= 5d-46) then
        tmp = x * 1.0d0
    else
        tmp = (x * y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sin(x) * (Math.sinh(y) / y)) <= 5e-46) {
		tmp = x * 1.0;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sin(x) * (math.sinh(y) / y)) <= 5e-46:
		tmp = x * 1.0
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 5e-46)
		tmp = Float64(x * 1.0);
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sin(x) * (sinh(y) / y)) <= 5e-46)
		tmp = x * 1.0;
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 5e-46], N[(x * 1.0), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 4.99999999999999992e-46

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 69.3%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 38.9%

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

   if 4.99999999999999992e-46 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 52.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 52.1

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

   6. Applied rewrites 52.1%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 15.2%

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

Alternative 16: 26.1% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x 1.0))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * 1.0d0
end function

Java

public static double code(double x, double y) {
	return x * 1.0;
}

Python

def code(x, y):
	return x * 1.0

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = x * 1.0;
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 62.5%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 26.1%

   \[\leadsto x \cdot \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025110 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))