Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.7% → 99.8%

Time: 4.6s

Alternatives: 13

Speedup: 1.0×

• 49.2% 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: 88.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

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

3. Applied rewrites 99.8%

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

Alternative 2: 95.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\right)}{x}\\ \mathbf{if}\;y \leq -9 \cdot 10^{+101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.95:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+103}:\\ \;\;\;\;\frac{\sinh y}{x} \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (* (fma (* y y) 0.16666666666666666 1.0) y)) x)))
   (if (<= y -9e+101)
     t_0
     (if (<= y -0.95)
       (* (* 2.0 (sinh y)) (fma (* x x) -0.08333333333333333 0.5))
       (if (<= y 2.2e-10)
         (* (/ (sin x) x) y)
         (if (<= y 1.02e+103)
           (* (/ (sinh y) x) (* (fma -0.16666666666666666 (* x x) 1.0) x))
           t_0))))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * (fma((y * y), 0.16666666666666666, 1.0) * y)) / x;
	double tmp;
	if (y <= -9e+101) {
		tmp = t_0;
	} else if (y <= -0.95) {
		tmp = (2.0 * sinh(y)) * fma((x * x), -0.08333333333333333, 0.5);
	} else if (y <= 2.2e-10) {
		tmp = (sin(x) / x) * y;
	} else if (y <= 1.02e+103) {
		tmp = (sinh(y) / x) * (fma(-0.16666666666666666, (x * x), 1.0) * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y)) / x)
	tmp = 0.0
	if (y <= -9e+101)
		tmp = t_0;
	elseif (y <= -0.95)
		tmp = Float64(Float64(2.0 * sinh(y)) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	elseif (y <= 2.2e-10)
		tmp = Float64(Float64(sin(x) / x) * y);
	elseif (y <= 1.02e+103)
		tmp = Float64(Float64(sinh(y) / x) * Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = 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, -9e+101], t$95$0, If[LessEqual[y, -0.95], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e-10], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.02e+103], N[(N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -9.0000000000000004e101 or 1.01999999999999991e103 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}}{x} \]
   3. 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 99.6

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

   4. Applied rewrites 99.6%

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

   if -9.0000000000000004e101 < y < -0.94999999999999996

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) + \frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   3. 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 76.5

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

   4. Applied rewrites 76.5%

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

   if -0.94999999999999996 < y < 2.1999999999999999e-10

   1. Initial program 77.3%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   3. 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 99.4

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

   4. Applied rewrites 99.4%

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

   if 2.1999999999999999e-10 < y < 1.01999999999999991e103

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 99.5

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 74.0

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

   6. Applied rewrites 74.0%

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

Alternative 3: 92.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\right) \cdot \sin x\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{+93}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.95:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+103}:\\ \;\;\;\;\frac{\sinh y}{x} \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* (/ (fma (* y y) 0.16666666666666666 1.0) x) y) (sin x))))
   (if (<= y -3.2e+93)
     t_0
     (if (<= y -0.95)
       (* (* 2.0 (sinh y)) (fma (* x x) -0.08333333333333333 0.5))
       (if (<= y 2.2e-10)
         (* (/ (sin x) x) y)
         (if (<= y 1.7e+103)
           (* (/ (sinh y) x) (* (fma -0.16666666666666666 (* x x) 1.0) x))
           t_0))))))

C

double code(double x, double y) {
	double t_0 = ((fma((y * y), 0.16666666666666666, 1.0) / x) * y) * sin(x);
	double tmp;
	if (y <= -3.2e+93) {
		tmp = t_0;
	} else if (y <= -0.95) {
		tmp = (2.0 * sinh(y)) * fma((x * x), -0.08333333333333333, 0.5);
	} else if (y <= 2.2e-10) {
		tmp = (sin(x) / x) * y;
	} else if (y <= 1.7e+103) {
		tmp = (sinh(y) / x) * (fma(-0.16666666666666666, (x * x), 1.0) * x);
	} 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) * sin(x))
	tmp = 0.0
	if (y <= -3.2e+93)
		tmp = t_0;
	elseif (y <= -0.95)
		tmp = Float64(Float64(2.0 * sinh(y)) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	elseif (y <= 2.2e-10)
		tmp = Float64(Float64(sin(x) / x) * y);
	elseif (y <= 1.7e+103)
		tmp = Float64(Float64(sinh(y) / x) * Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x));
	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] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.2e+93], t$95$0, If[LessEqual[y, -0.95], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e-10], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.7e+103], N[(N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.2000000000000001e93 or 1.6999999999999999e103 < y

   1. Initial program 100.0%

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

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2}}{x} + \frac{1}{x}\right)\right)} \cdot \sin x \]
   5. 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 90.5

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

   6. Applied rewrites 90.5%

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

   if -3.2000000000000001e93 < y < -0.94999999999999996

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) + \frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   3. 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 76.5

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

   4. Applied rewrites 76.5%

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

   if -0.94999999999999996 < y < 2.1999999999999999e-10

   1. Initial program 77.3%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   3. 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 99.4

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

   4. Applied rewrites 99.4%

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

   if 2.1999999999999999e-10 < y < 1.6999999999999999e103

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 99.5

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 73.9

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

   6. Applied rewrites 73.9%

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

Alternative 4: 87.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \sinh y\\ \mathbf{if}\;y \leq -6500000:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 2.0 (sinh y))))
   (if (<= y -6500000.0)
     (* t_0 0.5)
     (if (<= y 2.2e-10)
       (* (/ (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 <= -6500000.0) {
		tmp = t_0 * 0.5;
	} else if (y <= 2.2e-10) {
		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 <= -6500000.0)
		tmp = Float64(t_0 * 0.5);
	elseif (y <= 2.2e-10)
		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, -6500000.0], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[y, 2.2e-10], 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 < -6.5e6

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   3. 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

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

   4. Applied rewrites 74.8%

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

   if -6.5e6 < y < 2.1999999999999999e-10

   1. Initial program 77.5%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   3. 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 98.6

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

   4. Applied rewrites 98.6%

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

   if 2.1999999999999999e-10 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) + \frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   3. 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 75.8

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

   4. Applied rewrites 75.8%

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

Alternative 5: 74.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{x} \cdot x\\ \mathbf{if}\;x \leq 8.2 \cdot 10^{+260}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+289}:\\ \;\;\;\;\left(y + y\right) \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (sinh y) x) x)))
   (if (<= x 8.2e+260)
     t_0
     (if (<= x 3.7e+289) (* (+ y y) (* (* x x) -0.08333333333333333)) t_0))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) / x) * x;
	double tmp;
	if (x <= 8.2e+260) {
		tmp = t_0;
	} else if (x <= 3.7e+289) {
		tmp = (y + y) * ((x * x) * -0.08333333333333333);
	} 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 = (sinh(y) / x) * x
    if (x <= 8.2d+260) then
        tmp = t_0
    else if (x <= 3.7d+289) then
        tmp = (y + y) * ((x * x) * (-0.08333333333333333d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (Math.sinh(y) / x) * x;
	double tmp;
	if (x <= 8.2e+260) {
		tmp = t_0;
	} else if (x <= 3.7e+289) {
		tmp = (y + y) * ((x * x) * -0.08333333333333333);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sinh(y) / x) * x
	tmp = 0
	if x <= 8.2e+260:
		tmp = t_0
	elif x <= 3.7e+289:
		tmp = (y + y) * ((x * x) * -0.08333333333333333)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) / x) * x)
	tmp = 0.0
	if (x <= 8.2e+260)
		tmp = t_0;
	elseif (x <= 3.7e+289)
		tmp = Float64(Float64(y + y) * Float64(Float64(x * x) * -0.08333333333333333));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sinh(y) / x) * x;
	tmp = 0.0;
	if (x <= 8.2e+260)
		tmp = t_0;
	elseif (x <= 3.7e+289)
		tmp = (y + y) * ((x * x) * -0.08333333333333333);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, 8.2e+260], t$95$0, If[LessEqual[x, 3.7e+289], N[(N[(y + y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < 8.20000000000000051e260 or 3.70000000000000006e289 < x

   1. Initial program 88.4%

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

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 74.9%

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

   if 8.20000000000000051e260 < x < 3.70000000000000006e289

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) + \frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   3. 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 29.9

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

   4. Applied rewrites 29.9%

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

   5. Taylor expanded in y around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 29.9

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

   7. Applied rewrites 29.9%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 29.9

         \[\leadsto \left(y + y\right) \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333\right) \]

   10. Applied rewrites 29.9%

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

Alternative 6: 74.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+34}:\\ \;\;\;\;\frac{\sinh y}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.25e+34)
   (* (/ (sinh y) x) x)
   (* (* 2.0 (sinh y)) (fma (* x x) -0.08333333333333333 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.25e+34) {
		tmp = (sinh(y) / x) * x;
	} else {
		tmp = (2.0 * sinh(y)) * fma((x * x), -0.08333333333333333, 0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.25e+34)
		tmp = Float64(Float64(sinh(y) / x) * x);
	else
		tmp = Float64(Float64(2.0 * sinh(y)) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.25e+34], N[(N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision] * x), $MachinePrecision], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.25e34

   1. Initial program 85.4%

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

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 74.5%

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

   if 1.25e34 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) + \frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   3. 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 76.3

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

   4. Applied rewrites 76.3%

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

Alternative 7: 74.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+18}:\\ \;\;\;\;\frac{\sinh y}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \sinh y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5e+18)
   (* (/ (sinh y) x) x)
   (/ (* (* (fma -0.16666666666666666 (* x x) 1.0) x) (sinh y)) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5e+18) {
		tmp = (sinh(y) / x) * x;
	} else {
		tmp = ((fma(-0.16666666666666666, (x * x), 1.0) * x) * sinh(y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5e+18)
		tmp = Float64(Float64(sinh(y) / x) * x);
	else
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x) * sinh(y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5e+18], N[(N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5e18

   1. Initial program 85.2%

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

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 74.5%

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

   if 5e18 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 76.5

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

   4. Applied rewrites 76.5%

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

Alternative 8: 74.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{+81}:\\ \;\;\;\;\frac{\sinh y}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot 0.3333333333333333\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.6e+81)
   (* (/ (sinh y) x) x)
   (*
    (* (* (* y y) y) 0.3333333333333333)
    (fma (* x x) -0.08333333333333333 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.6e+81) {
		tmp = (sinh(y) / x) * x;
	} else {
		tmp = (((y * y) * y) * 0.3333333333333333) * fma((x * x), -0.08333333333333333, 0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.6e+81)
		tmp = Float64(Float64(sinh(y) / x) * x);
	else
		tmp = Float64(Float64(Float64(Float64(y * y) * y) * 0.3333333333333333) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.6e+81], N[(N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.5999999999999999e81

   1. Initial program 86.1%

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

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 74.5%

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

   if 5.5999999999999999e81 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) + \frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   3. 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 76.6

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + \frac{1}{3} \cdot {y}^{2}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{12}, \frac{1}{2}\right) \]
   6. 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 72.7

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

   7. Applied rewrites 72.7%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 72.8

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

   10. Applied rewrites 72.8%

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

Alternative 9: 67.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq 1.3 \cdot 10^{+106}:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+260}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+289}:\\ \;\;\;\;\left(y + y\right) \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* (/ (fma (* y y) 0.16666666666666666 1.0) x) y) x)))
   (if (<= x 1.3e+106)
     (* (* 2.0 (sinh y)) 0.5)
     (if (<= x 8.2e+260)
       t_0
       (if (<= x 3.7e+289)
         (* (+ y y) (* (* x x) -0.08333333333333333))
         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 <= 1.3e+106) {
		tmp = (2.0 * sinh(y)) * 0.5;
	} else if (x <= 8.2e+260) {
		tmp = t_0;
	} else if (x <= 3.7e+289) {
		tmp = (y + y) * ((x * x) * -0.08333333333333333);
	} 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 <= 1.3e+106)
		tmp = Float64(Float64(2.0 * sinh(y)) * 0.5);
	elseif (x <= 8.2e+260)
		tmp = t_0;
	elseif (x <= 3.7e+289)
		tmp = Float64(Float64(y + y) * Float64(Float64(x * x) * -0.08333333333333333));
	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, 1.3e+106], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 8.2e+260], t$95$0, If[LessEqual[x, 3.7e+289], N[(N[(y + y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.3000000000000001e106

   1. Initial program 86.6%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   3. 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 71.2

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

   4. Applied rewrites 71.2%

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

   if 1.3000000000000001e106 < x < 8.20000000000000051e260 or 3.70000000000000006e289 < x

   1. Initial program 99.8%

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

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2}}{x} + \frac{1}{x}\right)\right)} \cdot \sin x \]
   5. 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 76.8

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

   6. Applied rewrites 76.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 48.3%

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

   if 8.20000000000000051e260 < x < 3.70000000000000006e289

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) + \frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   3. 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 29.9

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

   4. Applied rewrites 29.9%

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

   5. Taylor expanded in y around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 29.9

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

   7. Applied rewrites 29.9%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 29.9

         \[\leadsto \left(y + y\right) \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333\right) \]

   10. Applied rewrites 29.9%

       \[\leadsto \left(y + y\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{-0.08333333333333333}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 66.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\right) \cdot x\\ \mathbf{if}\;x \leq 8.2 \cdot 10^{+260}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+289}:\\ \;\;\;\;\left(y + y\right) \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* (/ (fma (* y y) 0.16666666666666666 1.0) x) y) x)))
   (if (<= x 8.2e+260)
     t_0
     (if (<= x 3.7e+289) (* (+ y y) (* (* x x) -0.08333333333333333)) 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 <= 8.2e+260) {
		tmp = t_0;
	} else if (x <= 3.7e+289) {
		tmp = (y + y) * ((x * x) * -0.08333333333333333);
	} 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 <= 8.2e+260)
		tmp = t_0;
	elseif (x <= 3.7e+289)
		tmp = Float64(Float64(y + y) * Float64(Float64(x * x) * -0.08333333333333333));
	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, 8.2e+260], t$95$0, If[LessEqual[x, 3.7e+289], N[(N[(y + y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < 8.20000000000000051e260 or 3.70000000000000006e289 < x

   1. Initial program 88.4%

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

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot \frac{{y}^{2}}{x} + \frac{1}{x}\right)\right)} \cdot \sin x \]
   5. 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 85.2

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

   6. Applied rewrites 85.2%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 67.1%

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

   if 8.20000000000000051e260 < x < 3.70000000000000006e289

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) + \frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   3. 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 29.9

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

   4. Applied rewrites 29.9%

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

   5. Taylor expanded in y around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 29.9

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

   7. Applied rewrites 29.9%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 29.9

         \[\leadsto \left(y + y\right) \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333\right) \]

   10. Applied rewrites 29.9%

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

Alternative 11: 64.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \mathbf{if}\;y \leq -9 \cdot 10^{+101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (* y y) 0.16666666666666666 1.0) y)))
   (if (<= y -9e+101) t_0 (if (<= y 4.8e+102) (* (/ y x) x) t_0))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), 0.16666666666666666, 1.0) * y;
	double tmp;
	if (y <= -9e+101) {
		tmp = t_0;
	} else if (y <= 4.8e+102) {
		tmp = (y / x) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y)
	tmp = 0.0
	if (y <= -9e+101)
		tmp = t_0;
	elseif (y <= 4.8e+102)
		tmp = Float64(Float64(y / x) * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -9e+101], t$95$0, If[LessEqual[y, 4.8e+102], N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.0000000000000004e101 or 4.79999999999999989e102 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   3. 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

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

   4. Applied rewrites 75.6%

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

   5. Taylor expanded in y around 0

      \[\leadsto y \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
   6. 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 75.2

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

   7. Applied rewrites 75.2%

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

   if -9.0000000000000004e101 < y < 4.79999999999999989e102

   1. Initial program 83.0%

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

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 59.4

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

   6. Applied rewrites 59.4%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 74.2%

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

   10. Taylor expanded in y around 0

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

       1. lower-/.f64 58.7

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

   12. Applied rewrites 58.7%

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

Alternative 12: 51.5% accurate, 7.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (y / x) * 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 = (y / x) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 65.0

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

6. Applied rewrites 65.0%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 74.6%

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

10. Taylor expanded in y around 0

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

    1. lower-/.f64 51.5

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

12. Applied rewrites 51.5%

    \[\leadsto \color{blue}{\frac{y}{x}} \cdot x \]
13. Add Preprocessing

Alternative 13: 28.8% accurate, 51.3× 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 88.7%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
3. 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 64.1

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

4. Applied rewrites 64.1%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 28.8%

   \[\leadsto y \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025112 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64
  (/ (* (sin x) (sinh y)) x))