Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.2% → 99.9%

Time: 4.8s

Alternatives: 17

Speedup: 1.4×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sinh.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-sinh.f64 N/A

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

   10. lift-sin.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 69.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\\ \mathbf{if}\;\sinh y \leq -5 \cdot 10^{-8}:\\ \;\;\;\;t\_0 \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\right)\\ \mathbf{elif}\;\sinh y \leq 5 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right) \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (fma (* y y) 0.16666666666666666 1.0) x) y)))
   (if (<= (sinh y) -5e-8)
     (*
      t_0
      (*
       (fma
        (- (* 0.008333333333333333 (* x x)) 0.16666666666666666)
        (* x x)
        1.0)
       x))
     (if (<= (sinh y) 5e-13)
       (* x (/ y x))
       (*
        t_0
        (*
         (fma
          (-
           (*
            (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
            (* x x))
           0.16666666666666666)
          (* x x)
          1.0)
         x))))))

C

double code(double x, double y) {
	double t_0 = (fma((y * y), 0.16666666666666666, 1.0) / x) * y;
	double tmp;
	if (sinh(y) <= -5e-8) {
		tmp = t_0 * (fma(((0.008333333333333333 * (x * x)) - 0.16666666666666666), (x * x), 1.0) * x);
	} else if (sinh(y) <= 5e-13) {
		tmp = x * (y / x);
	} else {
		tmp = t_0 * (fma(((fma(-0.0001984126984126984, (x * x), 0.008333333333333333) * (x * x)) - 0.16666666666666666), (x * x), 1.0) * x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) / x) * y)
	tmp = 0.0
	if (sinh(y) <= -5e-8)
		tmp = Float64(t_0 * Float64(fma(Float64(Float64(0.008333333333333333 * Float64(x * x)) - 0.16666666666666666), Float64(x * x), 1.0) * x));
	elseif (sinh(y) <= 5e-13)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(t_0 * Float64(fma(Float64(Float64(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333) * Float64(x * x)) - 0.16666666666666666), Float64(x * x), 1.0) * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[Sinh[y], $MachinePrecision], -5e-8], N[(t$95$0 * N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 5e-13], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sinh.f64 y) < -4.9999999999999998e-8

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 71.9

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

   7. Applied rewrites 71.9%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 66.0

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

   10. Applied rewrites 66.0%

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

   if -4.9999999999999998e-8 < (sinh.f64 y) < 4.9999999999999999e-13

   1. Initial program 77.4%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 29.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 73.3

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

   10. Applied rewrites 73.3%

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

   if 4.9999999999999999e-13 < (sinh.f64 y)

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 71.5

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

   7. Applied rewrites 71.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 65.6%

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

Alternative 3: 69.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\\ \mathbf{if}\;\sinh y \leq -5 \cdot 10^{-8}:\\ \;\;\;\;t\_0 \cdot \left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\right)\\ \mathbf{elif}\;\sinh y \leq 5 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (fma (* y y) 0.16666666666666666 1.0) x) y)))
   (if (<= (sinh y) -5e-8)
     (*
      t_0
      (*
       (fma
        (- (* 0.008333333333333333 (* x x)) 0.16666666666666666)
        (* x x)
        1.0)
       x))
     (if (<= (sinh y) 5e-13)
       (* x (/ y x))
       (* t_0 (* (fma -0.16666666666666666 (* x x) 1.0) x))))))

C

double code(double x, double y) {
	double t_0 = (fma((y * y), 0.16666666666666666, 1.0) / x) * y;
	double tmp;
	if (sinh(y) <= -5e-8) {
		tmp = t_0 * (fma(((0.008333333333333333 * (x * x)) - 0.16666666666666666), (x * x), 1.0) * x);
	} else if (sinh(y) <= 5e-13) {
		tmp = x * (y / x);
	} else {
		tmp = t_0 * (fma(-0.16666666666666666, (x * x), 1.0) * x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) / x) * y)
	tmp = 0.0
	if (sinh(y) <= -5e-8)
		tmp = Float64(t_0 * Float64(fma(Float64(Float64(0.008333333333333333 * Float64(x * x)) - 0.16666666666666666), Float64(x * x), 1.0) * x));
	elseif (sinh(y) <= 5e-13)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(t_0 * Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[Sinh[y], $MachinePrecision], -5e-8], N[(t$95$0 * N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 5e-13], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sinh.f64 y) < -4.9999999999999998e-8

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 71.9

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

   7. Applied rewrites 71.9%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 66.0

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

   10. Applied rewrites 66.0%

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

   if -4.9999999999999998e-8 < (sinh.f64 y) < 4.9999999999999999e-13

   1. Initial program 77.4%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 29.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 73.3

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

   10. Applied rewrites 73.3%

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

   if 4.9999999999999999e-13 < (sinh.f64 y)

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 71.5

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

   7. Applied rewrites 71.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 64.6

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

   10. Applied rewrites 64.6%

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

Alternative 4: 95.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \sinh y\\ t_1 := \frac{\sin x \cdot \left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\right)}{x}\\ \mathbf{if}\;y \leq -2 \cdot 10^{+103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-8}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;y \leq 0.0005:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq 10^{+103}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 2.0 (sinh y)))
        (t_1 (/ (* (sin x) (* (fma (* y y) 0.16666666666666666 1.0) y)) x)))
   (if (<= y -2e+103)
     t_1
     (if (<= y -7.2e-8)
       (* t_0 0.5)
       (if (<= y 0.0005)
         (* (/ (sin x) x) y)
         (if (<= y 1e+103)
           (* t_0 (fma (* x x) -0.08333333333333333 0.5))
           t_1))))))

C

double code(double x, double y) {
	double t_0 = 2.0 * sinh(y);
	double t_1 = (sin(x) * (fma((y * y), 0.16666666666666666, 1.0) * y)) / x;
	double tmp;
	if (y <= -2e+103) {
		tmp = t_1;
	} else if (y <= -7.2e-8) {
		tmp = t_0 * 0.5;
	} else if (y <= 0.0005) {
		tmp = (sin(x) / x) * y;
	} else if (y <= 1e+103) {
		tmp = t_0 * fma((x * x), -0.08333333333333333, 0.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 * sinh(y))
	t_1 = Float64(Float64(sin(x) * Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y)) / x)
	tmp = 0.0
	if (y <= -2e+103)
		tmp = t_1;
	elseif (y <= -7.2e-8)
		tmp = Float64(t_0 * 0.5);
	elseif (y <= 0.0005)
		tmp = Float64(Float64(sin(x) / x) * y);
	elseif (y <= 1e+103)
		tmp = Float64(t_0 * fma(Float64(x * x), -0.08333333333333333, 0.5));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, -2e+103], t$95$1, If[LessEqual[y, -7.2e-8], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[y, 0.0005], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1e+103], N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2e103 or 1e103 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2e103 < y < -7.19999999999999962e-8

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 73.4

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

   5. Applied rewrites 73.4%

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

   if -7.19999999999999962e-8 < y < 5.0000000000000001e-4

   1. Initial program 77.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 5.0000000000000001e-4 < y < 1e103

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

Alternative 5: 93.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \sinh y\\ t_1 := \left(\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\right) \cdot \sin x\\ \mathbf{if}\;y \leq -1.02 \cdot 10^{+133}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-8}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;y \leq 0.0005:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+153}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 2.0 (sinh y)))
        (t_1 (* (* (/ (fma (* y y) 0.16666666666666666 1.0) x) y) (sin x))))
   (if (<= y -1.02e+133)
     t_1
     (if (<= y -7.2e-8)
       (* t_0 0.5)
       (if (<= y 0.0005)
         (* (/ (sin x) x) y)
         (if (<= y 3.8e+153)
           (* t_0 (fma (* x x) -0.08333333333333333 0.5))
           t_1))))))

C

double code(double x, double y) {
	double t_0 = 2.0 * sinh(y);
	double t_1 = ((fma((y * y), 0.16666666666666666, 1.0) / x) * y) * sin(x);
	double tmp;
	if (y <= -1.02e+133) {
		tmp = t_1;
	} else if (y <= -7.2e-8) {
		tmp = t_0 * 0.5;
	} else if (y <= 0.0005) {
		tmp = (sin(x) / x) * y;
	} else if (y <= 3.8e+153) {
		tmp = t_0 * fma((x * x), -0.08333333333333333, 0.5);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 * sinh(y))
	t_1 = Float64(Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) / x) * y) * sin(x))
	tmp = 0.0
	if (y <= -1.02e+133)
		tmp = t_1;
	elseif (y <= -7.2e-8)
		tmp = Float64(t_0 * 0.5);
	elseif (y <= 0.0005)
		tmp = Float64(Float64(sin(x) / x) * y);
	elseif (y <= 3.8e+153)
		tmp = Float64(t_0 * fma(Float64(x * x), -0.08333333333333333, 0.5));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.02e+133], t$95$1, If[LessEqual[y, -7.2e-8], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[y, 0.0005], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 3.8e+153], N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.02e133 or 3.79999999999999966e153 < y

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 98.0

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

   7. Applied rewrites 98.0%

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

   if -1.02e133 < y < -7.19999999999999962e-8

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 74.1

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

   5. Applied rewrites 74.1%

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

   if -7.19999999999999962e-8 < y < 5.0000000000000001e-4

   1. Initial program 77.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 5.0000000000000001e-4 < y < 3.79999999999999966e153

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

Alternative 6: 67.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\\ \mathbf{if}\;\sinh y \leq 2 \cdot 10^{-59}:\\ \;\;\;\;t\_0 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (fma (* y y) 0.16666666666666666 1.0) x) y)))
   (if (<= (sinh y) 2e-59)
     (* t_0 x)
     (* t_0 (* (fma -0.16666666666666666 (* x x) 1.0) x)))))

C

double code(double x, double y) {
	double t_0 = (fma((y * y), 0.16666666666666666, 1.0) / x) * y;
	double tmp;
	if (sinh(y) <= 2e-59) {
		tmp = t_0 * x;
	} else {
		tmp = t_0 * (fma(-0.16666666666666666, (x * x), 1.0) * x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) / x) * y)
	tmp = 0.0
	if (sinh(y) <= 2e-59)
		tmp = Float64(t_0 * x);
	else
		tmp = Float64(t_0 * Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[Sinh[y], $MachinePrecision], 2e-59], N[(t$95$0 * x), $MachinePrecision], N[(t$95$0 * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 2.0000000000000001e-59

   1. Initial program 84.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 89.4

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

   7. Applied rewrites 89.4%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 57.1

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

   10. Applied rewrites 57.1%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 69.3%

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

   if 2.0000000000000001e-59 < (sinh.f64 y)

   1. Initial program 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 74.7

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

   7. Applied rewrites 74.7%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 63.1

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

   10. Applied rewrites 63.1%

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

Alternative 7: 87.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \sinh y\\ \mathbf{if}\;y \leq -7.2 \cdot 10^{-8}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;y \leq 0.0005:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 2.0 (sinh y))))
   (if (<= y -7.2e-8)
     (* t_0 0.5)
     (if (<= y 0.0005)
       (* (/ (sin x) x) y)
       (* t_0 (fma (* x x) -0.08333333333333333 0.5))))))

C

double code(double x, double y) {
	double t_0 = 2.0 * sinh(y);
	double tmp;
	if (y <= -7.2e-8) {
		tmp = t_0 * 0.5;
	} else if (y <= 0.0005) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = t_0 * fma((x * x), -0.08333333333333333, 0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(2.0 * sinh(y))
	tmp = 0.0
	if (y <= -7.2e-8)
		tmp = Float64(t_0 * 0.5);
	elseif (y <= 0.0005)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = Float64(t_0 * fma(Float64(x * x), -0.08333333333333333, 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.2e-8], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[y, 0.0005], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.19999999999999962e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 74.8

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

   5. Applied rewrites 74.8%

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

   if -7.19999999999999962e-8 < y < 5.0000000000000001e-4

   1. Initial program 77.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 5.0000000000000001e-4 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

Alternative 8: 87.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \sinh y\right) \cdot 0.5\\ \mathbf{if}\;y \leq -7.2 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 11000:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq 10^{+66}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 2.0 (sinh y)) 0.5)))
   (if (<= y -7.2e-8)
     t_0
     (if (<= y 11000.0)
       (* (/ (sin x) x) y)
       (if (<= y 1e+66)
         t_0
         (*
          (*
           (fma
            (fma 0.016666666666666666 (* y y) 0.3333333333333333)
            (* y y)
            2.0)
           y)
          (fma (* x x) -0.08333333333333333 0.5)))))))

C

double code(double x, double y) {
	double t_0 = (2.0 * sinh(y)) * 0.5;
	double tmp;
	if (y <= -7.2e-8) {
		tmp = t_0;
	} else if (y <= 11000.0) {
		tmp = (sin(x) / x) * y;
	} else if (y <= 1e+66) {
		tmp = t_0;
	} else {
		tmp = (fma(fma(0.016666666666666666, (y * y), 0.3333333333333333), (y * y), 2.0) * y) * fma((x * x), -0.08333333333333333, 0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 * sinh(y)) * 0.5)
	tmp = 0.0
	if (y <= -7.2e-8)
		tmp = t_0;
	elseif (y <= 11000.0)
		tmp = Float64(Float64(sin(x) / x) * y);
	elseif (y <= 1e+66)
		tmp = t_0;
	else
		tmp = Float64(Float64(fma(fma(0.016666666666666666, Float64(y * y), 0.3333333333333333), Float64(y * y), 2.0) * y) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[y, -7.2e-8], t$95$0, If[LessEqual[y, 11000.0], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1e+66], t$95$0, N[(N[(N[(N[(0.016666666666666666 * N[(y * y), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 2.0), $MachinePrecision] * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.19999999999999962e-8 or 11000 < y < 9.99999999999999945e65

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 75.6

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

   5. Applied rewrites 75.6%

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

   if -7.19999999999999962e-8 < y < 11000

   1. Initial program 78.1%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 9.99999999999999945e65 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + {y}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{12}, \frac{1}{2}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(2 + {y}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right)\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot \color{blue}{x}, \frac{-1}{12}, \frac{1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(2 + {y}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right)\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot \color{blue}{x}, \frac{-1}{12}, \frac{1}{2}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left({y}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right) + 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right) \cdot {y}^{2} + 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}, {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{60} \cdot {y}^{2} + \frac{1}{3}, {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, {y}^{2}, \frac{1}{3}\right), {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, y \cdot y, \frac{1}{3}\right), {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, y \cdot y, \frac{1}{3}\right), {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, y \cdot y, \frac{1}{3}\right), y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 76.2

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   8. Applied rewrites 76.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.08333333333333333, 0.5\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 74.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \sinh y\right) \cdot 0.5\\ \mathbf{if}\;y \leq -2 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 10^{-41}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq 10^{+66}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 2.0 (sinh y)) 0.5)))
   (if (<= y -2e-8)
     t_0
     (if (<= y 1e-41)
       (* x (/ y x))
       (if (<= y 1e+66)
         t_0
         (*
          (*
           (fma
            (fma 0.016666666666666666 (* y y) 0.3333333333333333)
            (* y y)
            2.0)
           y)
          (fma (* x x) -0.08333333333333333 0.5)))))))

C

double code(double x, double y) {
	double t_0 = (2.0 * sinh(y)) * 0.5;
	double tmp;
	if (y <= -2e-8) {
		tmp = t_0;
	} else if (y <= 1e-41) {
		tmp = x * (y / x);
	} else if (y <= 1e+66) {
		tmp = t_0;
	} else {
		tmp = (fma(fma(0.016666666666666666, (y * y), 0.3333333333333333), (y * y), 2.0) * y) * fma((x * x), -0.08333333333333333, 0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 * sinh(y)) * 0.5)
	tmp = 0.0
	if (y <= -2e-8)
		tmp = t_0;
	elseif (y <= 1e-41)
		tmp = Float64(x * Float64(y / x));
	elseif (y <= 1e+66)
		tmp = t_0;
	else
		tmp = Float64(Float64(fma(fma(0.016666666666666666, Float64(y * y), 0.3333333333333333), Float64(y * y), 2.0) * y) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[y, -2e-8], t$95$0, If[LessEqual[y, 1e-41], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+66], t$95$0, N[(N[(N[(N[(0.016666666666666666 * N[(y * y), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 2.0), $MachinePrecision] * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2e-8 or 1.00000000000000001e-41 < y < 9.99999999999999945e65

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 73.7

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

   5. Applied rewrites 73.7%

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

   if -2e-8 < y < 1.00000000000000001e-41

   1. Initial program 76.6%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 76.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 28.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 74.0

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

   10. Applied rewrites 74.0%

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

   if 9.99999999999999945e65 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + {y}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{12}, \frac{1}{2}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(2 + {y}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right)\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot \color{blue}{x}, \frac{-1}{12}, \frac{1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(2 + {y}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right)\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot \color{blue}{x}, \frac{-1}{12}, \frac{1}{2}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left({y}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right) + 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right) \cdot {y}^{2} + 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}, {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{60} \cdot {y}^{2} + \frac{1}{3}, {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, {y}^{2}, \frac{1}{3}\right), {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, y \cdot y, \frac{1}{3}\right), {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, y \cdot y, \frac{1}{3}\right), {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, y \cdot y, \frac{1}{3}\right), y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 76.2

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   8. Applied rewrites 76.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.08333333333333333, 0.5\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 67.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{+62}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.55e+62)
   (* (* (/ (fma (* y y) 0.16666666666666666 1.0) x) y) x)
   (*
    (*
     (fma (fma 0.016666666666666666 (* y y) 0.3333333333333333) (* y y) 2.0)
     y)
    (fma (* x x) -0.08333333333333333 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.55e+62) {
		tmp = ((fma((y * y), 0.16666666666666666, 1.0) / x) * y) * x;
	} else {
		tmp = (fma(fma(0.016666666666666666, (y * y), 0.3333333333333333), (y * y), 2.0) * y) * fma((x * x), -0.08333333333333333, 0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.55e+62)
		tmp = Float64(Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) / x) * y) * x);
	else
		tmp = Float64(Float64(fma(fma(0.016666666666666666, Float64(y * y), 0.3333333333333333), Float64(y * y), 2.0) * y) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.55e+62], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(0.016666666666666666 * N[(y * y), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 2.0), $MachinePrecision] * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.55000000000000007e62

   1. Initial program 86.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 85.1

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

   7. Applied rewrites 85.1%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 55.1

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

   10. Applied rewrites 55.1%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 65.1%

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

   if 1.55000000000000007e62 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + {y}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{12}, \frac{1}{2}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(2 + {y}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right)\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot \color{blue}{x}, \frac{-1}{12}, \frac{1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(2 + {y}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right)\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot \color{blue}{x}, \frac{-1}{12}, \frac{1}{2}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left({y}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right) + 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}\right) \cdot {y}^{2} + 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3} + \frac{1}{60} \cdot {y}^{2}, {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{60} \cdot {y}^{2} + \frac{1}{3}, {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, {y}^{2}, \frac{1}{3}\right), {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, y \cdot y, \frac{1}{3}\right), {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, y \cdot y, \frac{1}{3}\right), {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, y \cdot y, \frac{1}{3}\right), y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 76.0

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   8. Applied rewrites 76.0%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, y \cdot y, 0.3333333333333333\right), y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.08333333333333333, 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 62.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+178}:\\ \;\;\;\;\left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.3e+98)
   (*
    (fma
     (fma
      (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
      (* y y)
      0.16666666666666666)
     (* y y)
     1.0)
    y)
   (if (<= x 1.3e+178)
     (*
      (* (* (* y y) 0.3333333333333333) y)
      (fma (* x x) -0.08333333333333333 0.5))
     (* (* (/ (fma (* y y) 0.16666666666666666 1.0) x) y) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.3e+98) {
		tmp = fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0) * y;
	} else if (x <= 1.3e+178) {
		tmp = (((y * y) * 0.3333333333333333) * y) * fma((x * x), -0.08333333333333333, 0.5);
	} else {
		tmp = ((fma((y * y), 0.16666666666666666, 1.0) / x) * y) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.3e+98)
		tmp = Float64(fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y);
	elseif (x <= 1.3e+178)
		tmp = Float64(Float64(Float64(Float64(y * y) * 0.3333333333333333) * y) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	else
		tmp = Float64(Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) / x) * y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.3e+98], N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 1.3e+178], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.30000000000000013e98

   1. Initial program 87.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 70.5

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 65.1%

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

   if 2.30000000000000013e98 < x < 1.3e178

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 24.5

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

   5. Applied rewrites 24.5%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 23.9

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

   8. Applied rewrites 23.9%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 38.1

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

   11. Applied rewrites 38.1%

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

   if 1.3e178 < x

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 77.8

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

   7. Applied rewrites 77.8%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 24.7

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

   10. Applied rewrites 24.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 55.2%

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

Alternative 12: 66.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq 2.3 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+178}:\\ \;\;\;\;\left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* (/ (fma (* y y) 0.16666666666666666 1.0) x) y) x)))
   (if (<= x 2.3e+98)
     t_0
     (if (<= x 1.3e+178)
       (*
        (* (* (* y y) 0.3333333333333333) y)
        (fma (* x x) -0.08333333333333333 0.5))
       t_0))))

C

double code(double x, double y) {
	double t_0 = ((fma((y * y), 0.16666666666666666, 1.0) / x) * y) * x;
	double tmp;
	if (x <= 2.3e+98) {
		tmp = t_0;
	} else if (x <= 1.3e+178) {
		tmp = (((y * y) * 0.3333333333333333) * y) * fma((x * x), -0.08333333333333333, 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) / x) * y) * x)
	tmp = 0.0
	if (x <= 2.3e+98)
		tmp = t_0;
	elseif (x <= 1.3e+178)
		tmp = Float64(Float64(Float64(Float64(y * y) * 0.3333333333333333) * y) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, 2.3e+98], t$95$0, If[LessEqual[x, 1.3e+178], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.30000000000000013e98 or 1.3e178 < x

   1. Initial program 88.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 85.4

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

   7. Applied rewrites 85.4%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 61.2

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

   10. Applied rewrites 61.2%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 67.9%

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

   if 2.30000000000000013e98 < x < 1.3e178

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 24.5

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

   5. Applied rewrites 24.5%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 23.9

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

   8. Applied rewrites 23.9%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 38.1

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

   11. Applied rewrites 38.1%

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

Alternative 13: 65.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.95 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+62}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.95e-8)
   (*
    (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0)
    y)
   (if (<= y 1.55e+62)
     (* x (/ y x))
     (*
      (* (* (* y y) 0.3333333333333333) y)
      (fma (* x x) -0.08333333333333333 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.95e-8) {
		tmp = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0) * y;
	} else if (y <= 1.55e+62) {
		tmp = x * (y / x);
	} else {
		tmp = (((y * y) * 0.3333333333333333) * y) * fma((x * x), -0.08333333333333333, 0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.95e-8)
		tmp = Float64(fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0) * y);
	elseif (y <= 1.55e+62)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(Float64(Float64(Float64(y * y) * 0.3333333333333333) * y) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.95e-8], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.55e+62], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.9499999999999999e-8

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 74.8

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.5

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

   8. Applied rewrites 61.5%

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

   if -2.9499999999999999e-8 < y < 1.55000000000000007e62

   1. Initial program 80.2%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 69.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 27.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 66.2

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

   10. Applied rewrites 66.2%

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

   if 1.55000000000000007e62 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 68.3

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

   8. Applied rewrites 68.3%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 68.3

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

   11. Applied rewrites 68.3%

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

Alternative 14: 66.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \mathbf{if}\;y \leq -2.95 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+56}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (fma
           (fma (* y y) 0.008333333333333333 0.16666666666666666)
           (* y y)
           1.0)
          y)))
   (if (<= y -2.95e-8) t_0 (if (<= y 2.5e+56) (* x (/ y x)) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0) * y;
	double tmp;
	if (y <= -2.95e-8) {
		tmp = t_0;
	} else if (y <= 2.5e+56) {
		tmp = x * (y / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0) * y)
	tmp = 0.0
	if (y <= -2.95e-8)
		tmp = t_0;
	elseif (y <= 2.5e+56)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -2.95e-8], t$95$0, If[LessEqual[y, 2.5e+56], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9499999999999999e-8 or 2.50000000000000012e56 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 66.4

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

   8. Applied rewrites 66.4%

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

   if -2.9499999999999999e-8 < y < 2.50000000000000012e56

   1. Initial program 79.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 27.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 66.9

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

   10. Applied rewrites 66.9%

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

Alternative 15: 51.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \mathbf{if}\;y \leq -2.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-8}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* (* y y) 0.16666666666666666) y)))
   (if (<= y -2.4) t_0 (if (<= y 9.2e-8) y t_0))))

C

double code(double x, double y) {
	double t_0 = ((y * y) * 0.16666666666666666) * y;
	double tmp;
	if (y <= -2.4) {
		tmp = t_0;
	} else if (y <= 9.2e-8) {
		tmp = y;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = ((y * y) * 0.16666666666666666d0) * y
    if (y <= (-2.4d0)) then
        tmp = t_0
    else if (y <= 9.2d-8) then
        tmp = y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = ((y * y) * 0.16666666666666666) * y;
	double tmp;
	if (y <= -2.4) {
		tmp = t_0;
	} else if (y <= 9.2e-8) {
		tmp = y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((y * y) * 0.16666666666666666) * y
	tmp = 0
	if y <= -2.4:
		tmp = t_0
	elif y <= 9.2e-8:
		tmp = y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(y * y) * 0.16666666666666666) * y)
	tmp = 0.0
	if (y <= -2.4)
		tmp = t_0;
	elseif (y <= 9.2e-8)
		tmp = y;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((y * y) * 0.16666666666666666) * y;
	tmp = 0.0;
	if (y <= -2.4)
		tmp = t_0;
	elseif (y <= 9.2e-8)
		tmp = y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -2.4], t$95$0, If[LessEqual[y, 9.2e-8], y, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.39999999999999991 or 9.2000000000000003e-8 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.6

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

   8. Applied rewrites 51.6%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 51.1

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

   11. Applied rewrites 51.1%

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

   if -2.39999999999999991 < y < 9.2000000000000003e-8

   1. Initial program 77.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.4%

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

Alternative 16: 56.5% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 320000:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 320000.0)
   (* (fma y (* y 0.16666666666666666) 1.0) y)
   (* (* (* y y) 0.16666666666666666) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 320000.0) {
		tmp = fma(y, (y * 0.16666666666666666), 1.0) * y;
	} else {
		tmp = ((y * y) * 0.16666666666666666) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 320000.0)
		tmp = Float64(fma(y, Float64(y * 0.16666666666666666), 1.0) * y);
	else
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 320000.0], N[(N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2e5

   1. Initial program 85.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 61.4

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

   8. Applied rewrites 61.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

         \[\leadsto \left(\left(y \cdot y\right) \cdot \frac{1}{6} + 1\right) \cdot y \]

      3. associate-*l* N/A

         \[\leadsto \left(y \cdot \left(y \cdot \frac{1}{6}\right) + 1\right) \cdot y \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \frac{1}{6}, 1\right) \cdot y \]

      5. lower-*.f64 61.4

         \[\leadsto \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot y \]

   10. Applied rewrites 61.4%

       \[\leadsto \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot y \]

   if 3.2e5 < x

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \frac{1}{2} \]

      4. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \frac{1}{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \frac{1}{2} \]

      6. lift-sinh.f64 28.0

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 28.0%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto y \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot y \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot y \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot y \]

      4. *-commutative N/A

         \[\leadsto \left({y}^{2} \cdot \frac{1}{6} + 1\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({y}^{2}, \frac{1}{6}, 1\right) \cdot y \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(y \cdot y, \frac{1}{6}, 1\right) \cdot y \]

      7. lower-*.f64 20.7

         \[\leadsto \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y \]

   8. Applied rewrites 20.7%

      \[\leadsto \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \color{blue}{y} \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot y \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({y}^{2} \cdot \frac{1}{6}\right) \cdot y \]

       2. lower-*.f64 N/A

          \[\leadsto \left({y}^{2} \cdot \frac{1}{6}\right) \cdot y \]

       3. pow2 N/A

          \[\leadsto \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right) \cdot y \]

       4. lift-*.f64 41.2

          \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]

   11. Applied rewrites 41.2%

       \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 27.3% accurate, 217.0× speedup

Math

\[\begin{array}{l} \\ y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 y)

C

double code(double x, double y) {
	return y;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y
end function

Java

public static double code(double x, double y) {
	return y;
}

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

function tmp = code(x, y)
	tmp = y;
end

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 89.2%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\sin x \cdot y}{x} \]

   2. associate-*l/ N/A

      \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{y} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\sin x}{x} \cdot \color{blue}{y} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\sin x}{x} \cdot y \]

   5. lift-sin.f64 51.0

      \[\leadsto \frac{\sin x}{x} \cdot y \]

5. Applied rewrites 51.0%

   \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot y} \]

6. Taylor expanded in x around 0

   \[\leadsto y \]
7. Step-by-step derivation

8. Applied rewrites 27.3%

   \[\leadsto y \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \frac{\sinh y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) x)))

C

double code(double x, double y) {
	return sin(x) * (sinh(y) / x);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) * (sinh(y) / x)
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) * (Math.sinh(y) / x);
}

Python

def code(x, y):
	return math.sin(x) * (math.sinh(y) / x)

Julia

function code(x, y)
	return Float64(sin(x) * Float64(sinh(y) / x))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) * (sinh(y) / x);
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025089 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (* (sin x) (/ (sinh y) x)))

  (/ (* (sin x) (sinh y)) x))