Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.2% → 99.8%

Time: 6.0s

Alternatives: 11

Speedup: 1.0×

• 49.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

3. Applied rewrites 99.8%

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

Alternative 2: 87.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.16666666666666666, x\right) \cdot \sinh y}{x}\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;\sin x \cdot \left(\frac{\mathsf{fma}\left(y \cdot 0.16666666666666666, y, 1\right)}{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sinh y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 (- INFINITY))
     (/ (* (fma (* (* x x) x) -0.16666666666666666 x) (sinh y)) x)
     (if (<= t_0 0.01)
       (* (sin x) (* (/ (fma (* y 0.16666666666666666) y 1.0) x) y))
       (/ (* x (sinh y)) x)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(((x * x) * x), -0.16666666666666666, x) * sinh(y)) / x;
	} else if (t_0 <= 0.01) {
		tmp = sin(x) * ((fma((y * 0.16666666666666666), y, 1.0) / x) * y);
	} else {
		tmp = (x * sinh(y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(Float64(x * x) * x), -0.16666666666666666, x) * sinh(y)) / x);
	elseif (t_0 <= 0.01)
		tmp = Float64(sin(x) * Float64(Float64(fma(Float64(y * 0.16666666666666666), y, 1.0) / x) * y));
	else
		tmp = Float64(Float64(x * sinh(y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(N[(y * 0.16666666666666666), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 89.2%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 52.6%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]

   6. Applied rewrites 52.6%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0100000000000000002

   1. Initial program 89.2%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 84.6%

      \[\leadsto \sin x \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{y}^{2}}{x}, \frac{1}{x}\right)\right)} \]

   8. Applied rewrites 84.6%

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

   if 0.0100000000000000002 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 89.2%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 52.5%

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

Alternative 3: 87.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.16666666666666666, x\right) \cdot \sinh y}{x}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-18}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sinh y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 (- INFINITY))
     (/ (* (fma (* (* x x) x) -0.16666666666666666 x) (sinh y)) x)
     (if (<= t_0 4e-18) (* y (/ (sin x) x)) (/ (* x (sinh y)) x)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(((x * x) * x), -0.16666666666666666, x) * sinh(y)) / x;
	} else if (t_0 <= 4e-18) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = (x * sinh(y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(Float64(Float64(x * x) * x), -0.16666666666666666, x) * sinh(y)) / x);
	elseif (t_0 <= 4e-18)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = Float64(Float64(x * sinh(y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 4e-18], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 89.2%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 52.6%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]

   6. Applied rewrites 52.6%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.0000000000000003e-18

   1. Initial program 89.2%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 80.8%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{y}^{2} \cdot \sin x}{x}, \frac{\sin x}{x}\right)} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 51.4%

      \[\leadsto y \cdot \left(1 + \color{blue}{0.16666666666666666 \cdot {y}^{2}}\right) \]

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 52.2%

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

   if 4.0000000000000003e-18 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 89.2%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 52.5%

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

Alternative 4: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.16666666666666666, x\right) \cdot \sinh y}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{-102}:\\ \;\;\;\;x \cdot \left(\frac{\mathsf{fma}\left(y \cdot 0.16666666666666666, y, 1\right)}{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sinh y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -2e-213)
     (/ (* (fma (* (* x x) x) -0.16666666666666666 x) (sinh y)) x)
     (if (<= t_0 1e-102)
       (* x (* (/ (fma (* y 0.16666666666666666) y 1.0) x) y))
       (/ (* x (sinh y)) x)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -2e-213) {
		tmp = (fma(((x * x) * x), -0.16666666666666666, x) * sinh(y)) / x;
	} else if (t_0 <= 1e-102) {
		tmp = x * ((fma((y * 0.16666666666666666), y, 1.0) / x) * y);
	} else {
		tmp = (x * sinh(y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -2e-213)
		tmp = Float64(Float64(fma(Float64(Float64(x * x) * x), -0.16666666666666666, x) * sinh(y)) / x);
	elseif (t_0 <= 1e-102)
		tmp = Float64(x * Float64(Float64(fma(Float64(y * 0.16666666666666666), y, 1.0) / x) * y));
	else
		tmp = Float64(Float64(x * sinh(y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-213], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1e-102], N[(x * N[(N[(N[(N[(y * 0.16666666666666666), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-213

   1. Initial program 89.2%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 52.6%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\right)} \cdot \sinh y}{x} \]

   6. Applied rewrites 52.6%

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

   if -1.9999999999999999e-213 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.99999999999999933e-103

   1. Initial program 89.2%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 84.6%

      \[\leadsto \sin x \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{y}^{2}}{x}, \frac{1}{x}\right)\right)} \]

   8. Applied rewrites 84.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 66.2%

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

   if 9.99999999999999933e-103 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 89.2%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 52.5%

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

Alternative 5: 60.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(x \cdot -0.16666666666666666, x, 1\right) \cdot x\right) \cdot y}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{-102}:\\ \;\;\;\;x \cdot \left(\frac{\mathsf{fma}\left(y \cdot 0.16666666666666666, y, 1\right)}{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sinh y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -2e-213)
     (/ (* (* (fma (* x -0.16666666666666666) x 1.0) x) y) x)
     (if (<= t_0 1e-102)
       (* x (* (/ (fma (* y 0.16666666666666666) y 1.0) x) y))
       (/ (* x (sinh y)) x)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -2e-213) {
		tmp = ((fma((x * -0.16666666666666666), x, 1.0) * x) * y) / x;
	} else if (t_0 <= 1e-102) {
		tmp = x * ((fma((y * 0.16666666666666666), y, 1.0) / x) * y);
	} else {
		tmp = (x * sinh(y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -2e-213)
		tmp = Float64(Float64(Float64(fma(Float64(x * -0.16666666666666666), x, 1.0) * x) * y) / x);
	elseif (t_0 <= 1e-102)
		tmp = Float64(x * Float64(Float64(fma(Float64(y * 0.16666666666666666), y, 1.0) / x) * y));
	else
		tmp = Float64(Float64(x * sinh(y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-213], N[(N[(N[(N[(N[(x * -0.16666666666666666), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1e-102], N[(x * N[(N[(N[(N[(y * 0.16666666666666666), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-213

   1. Initial program 89.2%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 26.4%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\right)} \cdot y}{x} \]

   9. Applied rewrites 26.4%

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

   if -1.9999999999999999e-213 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.99999999999999933e-103

   1. Initial program 89.2%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 84.6%

      \[\leadsto \sin x \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{y}^{2}}{x}, \frac{1}{x}\right)\right)} \]

   8. Applied rewrites 84.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 66.2%

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

   if 9.99999999999999933e-103 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 89.2%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 52.5%

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

Alternative 6: 57.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(x \cdot -0.16666666666666666, x, 1\right) \cdot x\right) \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\mathsf{fma}\left(y \cdot 0.16666666666666666, y, 1\right)}{x} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -2e-213)
   (/ (* (* (fma (* x -0.16666666666666666) x 1.0) x) y) x)
   (* x (* (/ (fma (* y 0.16666666666666666) y 1.0) x) y))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -2e-213) {
		tmp = ((fma((x * -0.16666666666666666), x, 1.0) * x) * y) / x;
	} else {
		tmp = x * ((fma((y * 0.16666666666666666), y, 1.0) / x) * y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -2e-213)
		tmp = Float64(Float64(Float64(fma(Float64(x * -0.16666666666666666), x, 1.0) * x) * y) / x);
	else
		tmp = Float64(x * Float64(Float64(fma(Float64(y * 0.16666666666666666), y, 1.0) / x) * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-213], N[(N[(N[(N[(N[(x * -0.16666666666666666), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(N[(N[(N[(y * 0.16666666666666666), $MachinePrecision] * y + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-213

   1. Initial program 89.2%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 26.4%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\right)} \cdot y}{x} \]

   9. Applied rewrites 26.4%

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

   if -1.9999999999999999e-213 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 89.2%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 84.6%

      \[\leadsto \sin x \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{y}^{2}}{x}, \frac{1}{x}\right)\right)} \]

   8. Applied rewrites 84.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 66.2%

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

Alternative 7: 51.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(x \cdot -0.16666666666666666, x, 1\right) \cdot x\right) \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(\frac{y}{x} \cdot y, x \cdot 0.16666666666666666, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -2e-213)
   (/ (* (* (fma (* x -0.16666666666666666) x 1.0) x) y) x)
   (* y (fma (* (/ y x) y) (* x 0.16666666666666666) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -2e-213) {
		tmp = ((fma((x * -0.16666666666666666), x, 1.0) * x) * y) / x;
	} else {
		tmp = y * fma(((y / x) * y), (x * 0.16666666666666666), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -2e-213)
		tmp = Float64(Float64(Float64(fma(Float64(x * -0.16666666666666666), x, 1.0) * x) * y) / x);
	else
		tmp = Float64(y * fma(Float64(Float64(y / x) * y), Float64(x * 0.16666666666666666), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-213], N[(N[(N[(N[(N[(x * -0.16666666666666666), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision], N[(y * N[(N[(N[(y / x), $MachinePrecision] * y), $MachinePrecision] * N[(x * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-213

   1. Initial program 89.2%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 26.4%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\right)} \cdot y}{x} \]

   9. Applied rewrites 26.4%

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

   if -1.9999999999999999e-213 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 89.2%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 80.8%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{y}^{2} \cdot \sin x}{x}, \frac{\sin x}{x}\right)} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 51.4%

      \[\leadsto y \cdot \left(1 + \color{blue}{0.16666666666666666 \cdot {y}^{2}}\right) \]

   9. Applied rewrites 54.2%

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

   11. Applied rewrites 54.2%

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

Alternative 8: 45.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(x \cdot -0.16666666666666666, x, 1\right) \cdot x\right) \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(x \cdot \left(y \cdot y\right), \frac{0.16666666666666666}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -2e-213)
   (/ (* (* (fma (* x -0.16666666666666666) x 1.0) x) y) x)
   (* y (fma (* x (* y y)) (/ 0.16666666666666666 x) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -2e-213) {
		tmp = ((fma((x * -0.16666666666666666), x, 1.0) * x) * y) / x;
	} else {
		tmp = y * fma((x * (y * y)), (0.16666666666666666 / x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -2e-213)
		tmp = Float64(Float64(Float64(fma(Float64(x * -0.16666666666666666), x, 1.0) * x) * y) / x);
	else
		tmp = Float64(y * fma(Float64(x * Float64(y * y)), Float64(0.16666666666666666 / x), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-213], N[(N[(N[(N[(N[(x * -0.16666666666666666), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision], N[(y * N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(0.16666666666666666 / x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-213

   1. Initial program 89.2%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 26.4%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\right)} \cdot y}{x} \]

   9. Applied rewrites 26.4%

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

   if -1.9999999999999999e-213 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 89.2%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 80.8%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{y}^{2} \cdot \sin x}{x}, \frac{\sin x}{x}\right)} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 51.4%

      \[\leadsto y \cdot \left(1 + \color{blue}{0.16666666666666666 \cdot {y}^{2}}\right) \]

   9. Applied rewrites 52.7%

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

Alternative 9: 44.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(x \cdot -0.16666666666666666, x, 1\right) \cdot x\right) \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -2e-213)
   (/ (* (* (fma (* x -0.16666666666666666) x 1.0) x) y) x)
   (* y (fma (* y y) 0.16666666666666666 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -2e-213) {
		tmp = ((fma((x * -0.16666666666666666), x, 1.0) * x) * y) / x;
	} else {
		tmp = y * fma((y * y), 0.16666666666666666, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -2e-213)
		tmp = Float64(Float64(Float64(fma(Float64(x * -0.16666666666666666), x, 1.0) * x) * y) / x);
	else
		tmp = Float64(y * fma(Float64(y * y), 0.16666666666666666, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -2e-213], N[(N[(N[(N[(N[(x * -0.16666666666666666), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision], N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-213

   1. Initial program 89.2%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 26.4%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\right)} \cdot y}{x} \]

   9. Applied rewrites 26.4%

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

   if -1.9999999999999999e-213 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 89.2%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 80.8%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{y}^{2} \cdot \sin x}{x}, \frac{\sin x}{x}\right)} \]

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 51.4%

      \[\leadsto y \cdot \left(1 + \color{blue}{0.16666666666666666 \cdot {y}^{2}}\right) \]

   9. Applied rewrites 51.4%

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

Alternative 10: 44.3% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

2. Taylor expanded in y around 0

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

4. Applied rewrites 80.8%

   \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(0.16666666666666666, \frac{{y}^{2} \cdot \sin x}{x}, \frac{\sin x}{x}\right)} \]

5. Taylor expanded in x around 0

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

7. Applied rewrites 51.4%

   \[\leadsto y \cdot \left(1 + \color{blue}{0.16666666666666666 \cdot {y}^{2}}\right) \]

9. Applied rewrites 51.4%

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

Alternative 11: 22.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot 1\right) \cdot y}{x} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return ((x * 1.0) * 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 = ((x * 1.0d0) * y) / x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

2. Taylor expanded in y around 0

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

4. Applied rewrites 41.5%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 26.4%

   \[\leadsto \frac{\color{blue}{\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\right)} \cdot y}{x} \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 22.5%

    \[\leadsto \frac{\left(x \cdot 1\right) \cdot y}{x} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025161 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64
  (/ (* (sin x) (sinh y)) x))