Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 3.3s

Alternatives: 11

Speedup: 1.0×

• 49.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 0.9999999998717213:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (fma -0.5 (* x x) 1.0) t_0)
     (if (<= t_1 0.9999999998717213)
       (* (cos x) (fma (* y y) 0.16666666666666666 1.0))
       (/ (* 1.0 (sinh y)) y)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = cos(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-0.5, (x * x), 1.0) * t_0;
	} else if (t_1 <= 0.9999999998717213) {
		tmp = cos(x) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = (1.0 * sinh(y)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(cos(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(-0.5, Float64(x * x), 1.0) * t_0);
	elseif (t_1 <= 0.9999999998717213)
		tmp = Float64(cos(x) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(Float64(1.0 * sinh(y)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999998717213], N[(N[Cos[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 62.6

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

   4. Applied rewrites 62.6%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.99999999987172128

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 75.7

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

   4. Applied rewrites 75.7%

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

   if 0.99999999987172128 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 64.5%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{1 \cdot \frac{\sinh y}{y}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 \cdot \color{blue}{\frac{\sinh y}{y}} \]

      3. lift-sinh.f64 N/A

         \[\leadsto 1 \cdot \frac{\color{blue}{\sinh y}}{y} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \sinh y}{y}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \sinh y}{y}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{1 \cdot \sinh y}}{y} \]

      7. lift-sinh.f64 64.5

         \[\leadsto \frac{1 \cdot \color{blue}{\sinh y}}{y} \]

   6. Applied rewrites 64.5%

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

Alternative 3: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := \cos x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 0.9999999998717213:\\ \;\;\;\;\cos x \cdot \frac{y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (cos x) t_0)))
   (if (<= t_1 (- INFINITY))
     (* (fma -0.5 (* x x) 1.0) t_0)
     (if (<= t_1 0.9999999998717213)
       (* (cos x) (/ y y))
       (/ (* 1.0 (sinh y)) y)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = cos(x) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-0.5, (x * x), 1.0) * t_0;
	} else if (t_1 <= 0.9999999998717213) {
		tmp = cos(x) * (y / y);
	} else {
		tmp = (1.0 * sinh(y)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(cos(x) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(-0.5, Float64(x * x), 1.0) * t_0);
	elseif (t_1 <= 0.9999999998717213)
		tmp = Float64(cos(x) * Float64(y / y));
	else
		tmp = Float64(Float64(1.0 * sinh(y)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999998717213], N[(N[Cos[x], $MachinePrecision] * N[(y / y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 62.6

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

   4. Applied rewrites 62.6%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.99999999987172128

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 51.4%

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

   if 0.99999999987172128 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 64.5%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{1 \cdot \frac{\sinh y}{y}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 \cdot \color{blue}{\frac{\sinh y}{y}} \]

      3. lift-sinh.f64 N/A

         \[\leadsto 1 \cdot \frac{\color{blue}{\sinh y}}{y} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \sinh y}{y}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \sinh y}{y}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{1 \cdot \sinh y}}{y} \]

      7. lift-sinh.f64 64.5

         \[\leadsto \frac{1 \cdot \color{blue}{\sinh y}}{y} \]

   6. Applied rewrites 64.5%

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

Alternative 4: 77.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;\cos x \cdot t\_0 \leq -0.001:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* (cos x) t_0) -0.001)
     (* (fma -0.5 (* x x) 1.0) t_0)
     (/ (* 1.0 (sinh y)) y))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if ((cos(x) * t_0) <= -0.001) {
		tmp = fma(-0.5, (x * x), 1.0) * t_0;
	} else {
		tmp = (1.0 * sinh(y)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (Float64(cos(x) * t_0) <= -0.001)
		tmp = Float64(fma(-0.5, Float64(x * x), 1.0) * t_0);
	else
		tmp = Float64(Float64(1.0 * sinh(y)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision], -0.001], N[(N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(1.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -1e-3

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 62.6

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

   4. Applied rewrites 62.6%

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

   if -1e-3 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 64.5%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{1 \cdot \frac{\sinh y}{y}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 \cdot \color{blue}{\frac{\sinh y}{y}} \]

      3. lift-sinh.f64 N/A

         \[\leadsto 1 \cdot \frac{\color{blue}{\sinh y}}{y} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \sinh y}{y}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \sinh y}{y}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{1 \cdot \sinh y}}{y} \]

      7. lift-sinh.f64 64.5

         \[\leadsto \frac{1 \cdot \color{blue}{\sinh y}}{y} \]

   6. Applied rewrites 64.5%

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

Alternative 5: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.001:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.001)
   (* (fma -0.5 (* x x) 1.0) (fma (* y y) 0.16666666666666666 1.0))
   (/ (* 1.0 (sinh y)) y)))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.001) {
		tmp = fma(-0.5, (x * x), 1.0) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = (1.0 * sinh(y)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.001)
		tmp = Float64(fma(-0.5, Float64(x * x), 1.0) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(Float64(1.0 * sinh(y)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.001], N[(N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -1e-3

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 75.7

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

   4. Applied rewrites 75.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 49.3

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

   7. Applied rewrites 49.3%

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

   if -1e-3 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 64.5%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{1 \cdot \frac{\sinh y}{y}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 \cdot \color{blue}{\frac{\sinh y}{y}} \]

      3. lift-sinh.f64 N/A

         \[\leadsto 1 \cdot \frac{\color{blue}{\sinh y}}{y} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \sinh y}{y}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \sinh y}{y}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{1 \cdot \sinh y}}{y} \]

      7. lift-sinh.f64 64.5

         \[\leadsto \frac{1 \cdot \color{blue}{\sinh y}}{y} \]

   6. Applied rewrites 64.5%

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

Alternative 6: 71.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.001:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right) \cdot \frac{y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.001)
   (* (fma -0.5 (* x x) 1.0) (/ y y))
   (/ (* 1.0 (sinh y)) y)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.001) {
		tmp = fma(-0.5, (x * x), 1.0) * (y / y);
	} else {
		tmp = (1.0 * sinh(y)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.001)
		tmp = Float64(fma(-0.5, Float64(x * x), 1.0) * Float64(y / y));
	else
		tmp = Float64(Float64(1.0 * sinh(y)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.001], N[(N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y / y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -1e-3

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \color{blue}{1}\right) \cdot \frac{\sinh y}{y} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2} + 1\right) \cdot \frac{\sinh y}{y} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, \color{blue}{{x}^{2}}, 1\right) \cdot \frac{\sinh y}{y} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right) \cdot \frac{\sinh y}{y} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \cdot \frac{\sinh y}{y} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \cdot \frac{\sinh y}{y} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \cdot \frac{\sinh y}{y} \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot \color{blue}{x}, 1\right) \cdot \frac{\sinh y}{y} \]

      9. lower-*.f64 62.5

         \[\leadsto \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot \color{blue}{x}, 1\right) \cdot \frac{\sinh y}{y} \]

   4. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)} \cdot \frac{\sinh y}{y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) \cdot \frac{\color{blue}{y}}{y} \]
   6. Step-by-step derivation

   7. Applied rewrites 36.3%

      \[\leadsto \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) \cdot \frac{\color{blue}{y}}{y} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 33.1%

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

   if -1e-3 < (cos.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 64.5%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{1 \cdot \frac{\sinh y}{y}} \]

      2. lift-/.f64 N/A

         \[\leadsto 1 \cdot \color{blue}{\frac{\sinh y}{y}} \]

      3. lift-sinh.f64 N/A

         \[\leadsto 1 \cdot \frac{\color{blue}{\sinh y}}{y} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \sinh y}{y}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \sinh y}{y}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{1 \cdot \sinh y}}{y} \]

      7. lift-sinh.f64 64.5

         \[\leadsto \frac{1 \cdot \color{blue}{\sinh y}}{y} \]

   6. Applied rewrites 64.5%

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

Alternative 7: 62.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.001:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right) \cdot \frac{y}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.001)
   (* (fma -0.5 (* x x) 1.0) (/ y y))
   (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.001) {
		tmp = fma(-0.5, (x * x), 1.0) * (y / y);
	} else {
		tmp = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.001)
		tmp = Float64(fma(-0.5, Float64(x * x), 1.0) * Float64(y / y));
	else
		tmp = fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.001], N[(N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -1e-3

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \color{blue}{1}\right) \cdot \frac{\sinh y}{y} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2} + 1\right) \cdot \frac{\sinh y}{y} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, \color{blue}{{x}^{2}}, 1\right) \cdot \frac{\sinh y}{y} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right) \cdot \frac{\sinh y}{y} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \cdot \frac{\sinh y}{y} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \cdot \frac{\sinh y}{y} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \cdot \frac{\sinh y}{y} \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot \color{blue}{x}, 1\right) \cdot \frac{\sinh y}{y} \]

      9. lower-*.f64 62.5

         \[\leadsto \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot \color{blue}{x}, 1\right) \cdot \frac{\sinh y}{y} \]

   4. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)} \cdot \frac{\sinh y}{y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) \cdot \frac{\color{blue}{y}}{y} \]
   6. Step-by-step derivation

   7. Applied rewrites 36.3%

      \[\leadsto \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) \cdot \frac{\color{blue}{y}}{y} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 33.1%

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

   if -1e-3 < (cos.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-cos.f64 87.7

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 55.5

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

   7. Applied rewrites 55.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 53.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.001:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right) \cdot \frac{y}{y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(y, 0.16666666666666666 \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.001)
   (* (fma -0.5 (* x x) 1.0) (/ y y))
   (* 1.0 (fma y (* 0.16666666666666666 y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.001) {
		tmp = fma(-0.5, (x * x), 1.0) * (y / y);
	} else {
		tmp = 1.0 * fma(y, (0.16666666666666666 * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.001)
		tmp = Float64(fma(-0.5, Float64(x * x), 1.0) * Float64(y / y));
	else
		tmp = Float64(1.0 * fma(y, Float64(0.16666666666666666 * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.001], N[(N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(y / y), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(y * N[(0.16666666666666666 * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -1e-3

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + \color{blue}{1}\right) \cdot \frac{\sinh y}{y} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2} + 1\right) \cdot \frac{\sinh y}{y} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, \color{blue}{{x}^{2}}, 1\right) \cdot \frac{\sinh y}{y} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right) \cdot \frac{\sinh y}{y} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \cdot \frac{\sinh y}{y} \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \cdot \frac{\sinh y}{y} \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \cdot \frac{\sinh y}{y} \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot \color{blue}{x}, 1\right) \cdot \frac{\sinh y}{y} \]

      9. lower-*.f64 62.5

         \[\leadsto \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot \color{blue}{x}, 1\right) \cdot \frac{\sinh y}{y} \]

   4. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)} \cdot \frac{\sinh y}{y} \]

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) \cdot \frac{\color{blue}{y}}{y} \]
   6. Step-by-step derivation

   7. Applied rewrites 36.3%

      \[\leadsto \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) \cdot \frac{\color{blue}{y}}{y} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 33.1%

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

   if -1e-3 < (cos.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 64.5%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 28.9%

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

   8. Taylor expanded in y around 0

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

      1. sinh-def N/A

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

      2. sub-div N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 46.6

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

   10. Applied rewrites 46.6%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 46.6

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

   12. Applied rewrites 46.6%

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

Alternative 9: 46.6% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
3. Step-by-step derivation

4. Applied rewrites 64.5%

   \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

5. Taylor expanded in y around 0

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

7. Applied rewrites 28.9%

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

8. Taylor expanded in y around 0

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

   1. sinh-def N/A

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

   2. sub-div N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 46.6

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

10. Applied rewrites 46.6%

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

    1. lift-*.f64 N/A

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

    2. lift-fma.f64 N/A

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

    3. associate-*l* N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. lower-*.f64 46.6

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

12. Applied rewrites 46.6%

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

Alternative 10: 46.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq 2:\\ \;\;\;\;1 \cdot \frac{y}{y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) 2.0)
   (* 1.0 (/ y y))
   (* 1.0 (* (* y y) 0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= 2.0) {
		tmp = 1.0 * (y / y);
	} else {
		tmp = 1.0 * ((y * y) * 0.16666666666666666);
	}
	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 ((cos(x) * (sinh(y) / y)) <= 2.0d0) then
        tmp = 1.0d0 * (y / y)
    else
        tmp = 1.0d0 * ((y * y) * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.cos(x) * (Math.sinh(y) / y)) <= 2.0) {
		tmp = 1.0 * (y / y);
	} else {
		tmp = 1.0 * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.cos(x) * (math.sinh(y) / y)) <= 2.0:
		tmp = 1.0 * (y / y)
	else:
		tmp = 1.0 * ((y * y) * 0.16666666666666666)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= 2.0)
		tmp = Float64(1.0 * Float64(y / y));
	else
		tmp = Float64(1.0 * Float64(Float64(y * y) * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((cos(x) * (sinh(y) / y)) <= 2.0)
		tmp = 1.0 * (y / y);
	else
		tmp = 1.0 * ((y * y) * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 2.0], N[(1.0 * N[(y / y), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 64.5%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 28.9%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   3. Step-by-step derivation

   4. Applied rewrites 64.5%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 28.9%

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

   8. Taylor expanded in y around 0

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

      1. sinh-def N/A

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

      2. sub-div N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 46.6

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

   10. Applied rewrites 46.6%

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

   11. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 20.8

          \[\leadsto 1 \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \]

   13. Applied rewrites 20.8%

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

Alternative 11: 28.9% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
3. Step-by-step derivation

4. Applied rewrites 64.5%

   \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]

5. Taylor expanded in y around 0

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

7. Applied rewrites 28.9%

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

Reproduce

herbie shell --seed 2025139 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))