Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 3.2s

Alternatives: 13

Speedup: 0.9×

• 50.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot \cosh x\right) \cdot \frac{\frac{\sin y}{y}}{2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (* 2.0 (cosh x)) (/ (/ (sin y) y) 2.0)))

C

double code(double x, double y) {
	return (2.0 * cosh(x)) * ((sin(y) / y) / 2.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 * cosh(x)) * ((sin(y) / y) / 2.0d0)
end function

Java

public static double code(double x, double y) {
	return (2.0 * Math.cosh(x)) * ((Math.sin(y) / y) / 2.0);
}

Python

def code(x, y):
	return (2.0 * math.cosh(x)) * ((math.sin(y) / y) / 2.0)

Julia

function code(x, y)
	return Float64(Float64(2.0 * cosh(x)) * Float64(Float64(sin(y) / y) / 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (2.0 * cosh(x)) * ((sin(y) / y) / 2.0);
end

Wolfram

code[x_, y_] := N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. lift-cosh.f64 N/A

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

   2. cosh-def N/A

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

   3. rec-exp N/A

      \[\leadsto \frac{e^{x} + \color{blue}{\frac{1}{e^{x}}}}{2} \cdot \frac{\sin y}{y} \]

   4. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{e^{x} \cdot e^{x} - \frac{1}{e^{x}} \cdot \frac{1}{e^{x}}}{e^{x} - \frac{1}{e^{x}}}}}{2} \cdot \frac{\sin y}{y} \]

   5. associate-/l/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower--.f64 N/A

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

   8. prod-exp N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. rec-exp N/A

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

   12. rec-exp N/A

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

   13. exp-lft-sqr-rev N/A

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

   14. lower-exp.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-neg.f64 N/A

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

   17. lower-*.f64 N/A

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

3. Applied rewrites 4.2%

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

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(2 \cdot \cosh x\right) \cdot \frac{\frac{\sin y}{y}}{2}} \]
6. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 3: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999917532393764:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* (* y y) -0.16666666666666666))
     (if (<= t_1 0.9999917532393764)
       (* (fma (* x x) 0.5 1.0) t_0)
       (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_1 <= 0.9999917532393764) {
		tmp = fma((x * x), 0.5, 1.0) * t_0;
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * Float64(Float64(y * y) * -0.16666666666666666));
	elseif (t_1 <= 0.9999917532393764)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * t_0);
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999917532393764], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in y around inf

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 99.9

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

   10. Applied rewrites 99.9%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.999991753239376391

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 98.5

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \frac{\sin y}{y} \]

   4. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{\sin y}{y} \]

   if 0.999991753239376391 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 99.8%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999917532393764:\\ \;\;\;\;2 \cdot \frac{t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* (* y y) -0.16666666666666666))
     (if (<= t_1 0.9999917532393764) (* 2.0 (/ t_0 2.0)) (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_1 <= 0.9999917532393764) {
		tmp = 2.0 * (t_0 / 2.0);
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) / y;
	double t_1 = Math.cosh(x) * t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_1 <= 0.9999917532393764) {
		tmp = 2.0 * (t_0 / 2.0);
	} else {
		tmp = Math.cosh(x) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) / y
	t_1 = math.cosh(x) * t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = math.cosh(x) * ((y * y) * -0.16666666666666666)
	elif t_1 <= 0.9999917532393764:
		tmp = 2.0 * (t_0 / 2.0)
	else:
		tmp = math.cosh(x) * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * Float64(Float64(y * y) * -0.16666666666666666));
	elseif (t_1 <= 0.9999917532393764)
		tmp = Float64(2.0 * Float64(t_0 / 2.0));
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) / y;
	t_1 = cosh(x) * t_0;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = cosh(x) * ((y * y) * -0.16666666666666666);
	elseif (t_1 <= 0.9999917532393764)
		tmp = 2.0 * (t_0 / 2.0);
	else
		tmp = cosh(x) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999917532393764], N[(2.0 * N[(t$95$0 / 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in y around inf

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 99.9

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

   10. Applied rewrites 99.9%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.999991753239376391

   1. Initial program 99.6%

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

      1. lift-cosh.f64 N/A

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

      2. cosh-def N/A

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

      3. rec-exp N/A

         \[\leadsto \frac{e^{x} + \color{blue}{\frac{1}{e^{x}}}}{2} \cdot \frac{\sin y}{y} \]

      4. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{e^{x} \cdot e^{x} - \frac{1}{e^{x}} \cdot \frac{1}{e^{x}}}{e^{x} - \frac{1}{e^{x}}}}}{2} \cdot \frac{\sin y}{y} \]

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. prod-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. rec-exp N/A

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

      12. rec-exp N/A

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

      13. exp-lft-sqr-rev N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-*.f64 N/A

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

   3. Applied rewrites 9.5%

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

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\left(2 \cdot \cosh x\right) \cdot \frac{\frac{\sin y}{y}}{2}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2} \cdot \frac{\frac{\sin y}{y}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 97.8%

      \[\leadsto \color{blue}{2} \cdot \frac{\frac{\sin y}{y}}{2} \]

   if 0.999991753239376391 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 99.8%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ t_1 := \cosh x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999917532393764:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)) (t_1 (* (cosh x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (cosh x) (* (* y y) -0.16666666666666666))
     (if (<= t_1 0.9999917532393764) t_0 (* (cosh x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = sin(y) / y;
	double t_1 = cosh(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_1 <= 0.9999917532393764) {
		tmp = t_0;
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(y) / y;
	double t_1 = Math.cosh(x) * t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.cosh(x) * ((y * y) * -0.16666666666666666);
	} else if (t_1 <= 0.9999917532393764) {
		tmp = t_0;
	} else {
		tmp = Math.cosh(x) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(y) / y
	t_1 = math.cosh(x) * t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = math.cosh(x) * ((y * y) * -0.16666666666666666)
	elif t_1 <= 0.9999917532393764:
		tmp = t_0
	else:
		tmp = math.cosh(x) * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(y) / y)
	t_1 = Float64(cosh(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(x) * Float64(Float64(y * y) * -0.16666666666666666));
	elseif (t_1 <= 0.9999917532393764)
		tmp = t_0;
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(y) / y;
	t_1 = cosh(x) * t_0;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = cosh(x) * ((y * y) * -0.16666666666666666);
	elseif (t_1 <= 0.9999917532393764)
		tmp = t_0;
	else
		tmp = cosh(x) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cosh[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999917532393764], t$95$0, N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in y around inf

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 99.9

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

   10. Applied rewrites 99.9%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.999991753239376391

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 97.8

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

   4. Applied rewrites 97.8%

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

   if 0.999991753239376391 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 99.8%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 75.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\cosh x \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -2e-134)
   (* (cosh x) (fma -0.16666666666666666 (* y y) 1.0))
   (* (cosh x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -2e-134) {
		tmp = cosh(x) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -2e-134)
		tmp = Float64(cosh(x) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -2e-134], N[(N[Cosh[x], $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -2.00000000000000008e-134

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 71.8

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

   4. Applied rewrites 71.8%

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

   if -2.00000000000000008e-134 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 76.1%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\cosh x \cdot \left(\left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -2e-134)
   (* (cosh x) (* (* y y) -0.16666666666666666))
   (* (cosh x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -2e-134) {
		tmp = cosh(x) * ((y * y) * -0.16666666666666666);
	} else {
		tmp = cosh(x) * 1.0;
	}
	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) :: tmp
    if ((cosh(x) * (sin(y) / y)) <= (-2d-134)) then
        tmp = cosh(x) * ((y * y) * (-0.16666666666666666d0))
    else
        tmp = cosh(x) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.cosh(x) * (Math.sin(y) / y)) <= -2e-134) {
		tmp = Math.cosh(x) * ((y * y) * -0.16666666666666666);
	} else {
		tmp = Math.cosh(x) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.cosh(x) * (math.sin(y) / y)) <= -2e-134:
		tmp = math.cosh(x) * ((y * y) * -0.16666666666666666)
	else:
		tmp = math.cosh(x) * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -2e-134)
		tmp = Float64(cosh(x) * Float64(Float64(y * y) * -0.16666666666666666));
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((cosh(x) * (sin(y) / y)) <= -2e-134)
		tmp = cosh(x) * ((y * y) * -0.16666666666666666);
	else
		tmp = cosh(x) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -2e-134], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -2.00000000000000008e-134

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 71.8

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

   7. Applied rewrites 71.8%

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

   8. Taylor expanded in y around inf

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 71.8

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

   10. Applied rewrites 71.8%

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

   if -2.00000000000000008e-134 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 76.1%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 74.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -2e-134)
   (* (fma (* x x) 0.5 1.0) (fma (* y y) -0.16666666666666666 1.0))
   (* (cosh x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -2e-134) {
		tmp = fma((x * x), 0.5, 1.0) * fma((y * y), -0.16666666666666666, 1.0);
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -2e-134)
		tmp = Float64(fma(Float64(x * x), 0.5, 1.0) * fma(Float64(y * y), -0.16666666666666666, 1.0));
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -2e-134], N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -2.00000000000000008e-134

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 66.2

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \frac{\sin y}{y} \]

   4. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot \frac{\sin y}{y} \]

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 64.8

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \]

   7. Applied rewrites 64.8%

      \[\leadsto \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)} \]

   if -2.00000000000000008e-134 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 76.1%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 71.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) -2e-134)
   (/ (* (fma -0.16666666666666666 (* y y) 1.0) y) y)
   (* (cosh x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= -2e-134) {
		tmp = (fma(-0.16666666666666666, (y * y), 1.0) * y) / y;
	} else {
		tmp = cosh(x) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= -2e-134)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * y) / y);
	else
		tmp = Float64(cosh(x) * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -2e-134], N[(N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -2.00000000000000008e-134

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 30.2

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

   4. Applied rewrites 30.2%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 62.6%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 50.1%

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

   if -2.00000000000000008e-134 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 76.1%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 62.4% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y):
	return math.cosh(x) * 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around 0

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

4. Applied rewrites 62.4%

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

Alternative 11: 44.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{\sin y}{y} \leq 2:\\ \;\;\;\;1 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot 0.5\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cosh x) (/ (sin y) y)) 2.0) (* 1.0 1.0) (* (* (* x x) 0.5) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((cosh(x) * (sin(y) / y)) <= 2.0) {
		tmp = 1.0 * 1.0;
	} else {
		tmp = ((x * x) * 0.5) * 1.0;
	}
	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) :: tmp
    if ((cosh(x) * (sin(y) / y)) <= 2.0d0) then
        tmp = 1.0d0 * 1.0d0
    else
        tmp = ((x * x) * 0.5d0) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.cosh(x) * (Math.sin(y) / y)) <= 2.0) {
		tmp = 1.0 * 1.0;
	} else {
		tmp = ((x * x) * 0.5) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.cosh(x) * (math.sin(y) / y)) <= 2.0:
		tmp = 1.0 * 1.0
	else:
		tmp = ((x * x) * 0.5) * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(sin(y) / y)) <= 2.0)
		tmp = Float64(1.0 * 1.0);
	else
		tmp = Float64(Float64(Float64(x * x) * 0.5) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((cosh(x) * (sin(y) / y)) <= 2.0)
		tmp = 1.0 * 1.0;
	else
		tmp = ((x * x) * 0.5) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 2.0], N[(1.0 * 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 39.8%

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

   5. Taylor expanded in x around 0

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

      1. cosh-def 39.6

         \[\leadsto 1 \cdot 1 \]

      2. rec-exp 39.6

         \[\leadsto 1 \cdot 1 \]

      3. flip-+ 39.6

         \[\leadsto 1 \cdot 1 \]

      4. exp-sum 39.6

         \[\leadsto 1 \cdot 1 \]

      5. pow2 39.6

         \[\leadsto 1 \cdot 1 \]

      6. rec-exp 39.6

         \[\leadsto 1 \cdot 1 \]

      7. lift-neg.f64 N/A

         \[\leadsto 1 \cdot 1 \]

      8. exp-prod N/A

         \[\leadsto 1 \cdot 1 \]

      9. lift-neg.f64 39.6

         \[\leadsto 1 \cdot 1 \]

      10. rec-exp 39.6

          \[\leadsto 1 \cdot 1 \]

      11. sinh-undef-rev 39.6

          \[\leadsto 1 \cdot 1 \]

      12. associate-/r* 39.6

          \[\leadsto 1 \cdot 1 \]

   7. Applied rewrites 39.6%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. cosh-def N/A

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

      2. rec-exp N/A

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

      3. flip-+ N/A

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

      4. exp-sum N/A

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

      5. pow2 N/A

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

      6. rec-exp N/A

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

      7. lift-neg.f64 N/A

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

      8. exp-prod N/A

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

      9. lift-neg.f64 N/A

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

      10. rec-exp N/A

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

      11. sinh-undef-rev N/A

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

      12. associate-/r* N/A

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

   7. Applied rewrites 52.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot 1 \]

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 52.8

         \[\leadsto \left(\left(x \cdot x\right) \cdot 0.5\right) \cdot 1 \]

   10. Applied rewrites 52.8%

       \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{0.5}\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 44.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, 0.5, 1\right) \cdot 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (fma (* x x) 0.5 1.0) 1.0))

C

double code(double x, double y) {
	return fma((x * x), 0.5, 1.0) * 1.0;
}

Julia

function code(x, y)
	return Float64(fma(Float64(x * x), 0.5, 1.0) * 1.0)
end

Wolfram

code[x_, y_] := N[(N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around 0

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

4. Applied rewrites 62.4%

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

5. Taylor expanded in x around 0

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

   1. cosh-def N/A

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

   2. rec-exp N/A

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

   3. flip-+ N/A

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

   4. exp-sum N/A

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

   5. pow2 N/A

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

   6. rec-exp N/A

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

   7. lift-neg.f64 N/A

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

   8. exp-prod N/A

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

   9. lift-neg.f64 N/A

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

   10. rec-exp N/A

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

   11. sinh-undef-rev N/A

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

   12. associate-/r* N/A

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

7. Applied rewrites 44.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 0.5, 1\right)} \cdot 1 \]
8. Add Preprocessing

Alternative 13: 25.9% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around 0

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

4. Applied rewrites 62.4%

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

5. Taylor expanded in x around 0

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

   1. cosh-def 25.9

      \[\leadsto 1 \cdot 1 \]

   2. rec-exp 25.9

      \[\leadsto 1 \cdot 1 \]

   3. flip-+ 25.9

      \[\leadsto 1 \cdot 1 \]

   4. exp-sum 25.9

      \[\leadsto 1 \cdot 1 \]

   5. pow2 25.9

      \[\leadsto 1 \cdot 1 \]

   6. rec-exp 25.9

      \[\leadsto 1 \cdot 1 \]

   7. lift-neg.f64 N/A

      \[\leadsto 1 \cdot 1 \]

   8. exp-prod N/A

      \[\leadsto 1 \cdot 1 \]

   9. lift-neg.f64 25.9

      \[\leadsto 1 \cdot 1 \]

   10. rec-exp 25.9

       \[\leadsto 1 \cdot 1 \]

   11. sinh-undef-rev 25.9

       \[\leadsto 1 \cdot 1 \]

   12. associate-/r* 25.9

       \[\leadsto 1 \cdot 1 \]

7. Applied rewrites 25.9%

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

Reproduce

herbie shell --seed 2025119 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64
  (* (cosh x) (/ (sin y) y)))