Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 3.1s

Alternatives: 15

Speedup: 1.0×

• 49.6% 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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\left(\left(x \cdot x\right) \cdot -0.5\right) \cdot \sinh y}{y}\\ \mathbf{elif}\;t\_0 \leq 0.9976192908615417:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 (- INFINITY))
     (/ (* (* (* x x) -0.5) (sinh y)) y)
     (if (<= t_0 0.9976192908615417)
       (* (cos x) (fma (* y y) 0.16666666666666666 1.0))
       (/ (* 1.0 (sinh y)) y)))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (((x * x) * -0.5) * sinh(y)) / y;
	} else if (t_0 <= 0.9976192908615417) {
		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(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(x * x) * -0.5) * sinh(y)) / y);
	elseif (t_0 <= 0.9976192908615417)
		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[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 0.9976192908615417], 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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 100.0

         \[\leadsto \left(\left(x \cdot x\right) \cdot -0.5\right) \cdot \frac{\sinh y}{y} \]

   7. Applied rewrites 100.0%

      \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{-0.5}\right) \cdot \frac{\sinh y}{y} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 100.0

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

   9. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.997619290861541663

   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 99.2

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

   4. Applied rewrites 99.2%

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

   if 0.997619290861541663 < (*.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 98.9%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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 98.9

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

   6. Applied rewrites 98.9%

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

Alternative 3: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\left(\left(x \cdot x\right) \cdot -0.5\right) \cdot \sinh y}{y}\\ \mathbf{elif}\;t\_0 \leq 0.9976192908615417:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 (- INFINITY))
     (/ (* (* (* x x) -0.5) (sinh y)) y)
     (if (<= t_0 0.9976192908615417) (cos x) (/ (* 1.0 (sinh y)) y)))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (((x * x) * -0.5) * sinh(y)) / y;
	} else if (t_0 <= 0.9976192908615417) {
		tmp = cos(x);
	} else {
		tmp = (1.0 * sinh(y)) / y;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) * (Math.sinh(y) / y);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (((x * x) * -0.5) * Math.sinh(y)) / y;
	} else if (t_0 <= 0.9976192908615417) {
		tmp = Math.cos(x);
	} else {
		tmp = (1.0 * Math.sinh(y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) * (math.sinh(y) / y)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (((x * x) * -0.5) * math.sinh(y)) / y
	elif t_0 <= 0.9976192908615417:
		tmp = math.cos(x)
	else:
		tmp = (1.0 * math.sinh(y)) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(x * x) * -0.5) * sinh(y)) / y);
	elseif (t_0 <= 0.9976192908615417)
		tmp = cos(x);
	else
		tmp = Float64(Float64(1.0 * sinh(y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) * (sinh(y) / y);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (((x * x) * -0.5) * sinh(y)) / y;
	elseif (t_0 <= 0.9976192908615417)
		tmp = cos(x);
	else
		tmp = (1.0 * sinh(y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 0.9976192908615417], N[Cos[x], $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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 100.0

         \[\leadsto \left(\left(x \cdot x\right) \cdot -0.5\right) \cdot \frac{\sinh y}{y} \]

   7. Applied rewrites 100.0%

      \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{-0.5}\right) \cdot \frac{\sinh y}{y} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 100.0

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

   9. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.997619290861541663

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lift-cos.f64 98.5

         \[\leadsto \cos x \]

   4. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\cos x} \]

   if 0.997619290861541663 < (*.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 98.9%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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 98.9

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

   6. Applied rewrites 98.9%

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

Alternative 4: 77.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;\cos x \cdot t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* (cos x) t_0) -0.01)
     (* (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) <= -0.01) {
		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) <= -0.01)
		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], -0.01], 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)) < -0.0100000000000000002

   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 52.4

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

   4. Applied rewrites 52.4%

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

   if -0.0100000000000000002 < (*.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 86.1%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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 86.1

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

   6. Applied rewrites 86.1%

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

Alternative 5: 77.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.01:\\ \;\;\;\;\frac{\left(\left(x \cdot x\right) \cdot -0.5\right) \cdot \sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.01)
   (/ (* (* (* x x) -0.5) (sinh y)) y)
   (/ (* 1.0 (sinh y)) y)))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.01) {
		tmp = (((x * x) * -0.5) * sinh(y)) / y;
	} else {
		tmp = (1.0 * sinh(y)) / y;
	}
	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)) <= (-0.01d0)) then
        tmp = (((x * x) * (-0.5d0)) * sinh(y)) / y
    else
        tmp = (1.0d0 * sinh(y)) / y
    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)) <= -0.01) {
		tmp = (((x * x) * -0.5) * Math.sinh(y)) / y;
	} else {
		tmp = (1.0 * Math.sinh(y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.cos(x) * (math.sinh(y) / y)) <= -0.01:
		tmp = (((x * x) * -0.5) * math.sinh(y)) / y
	else:
		tmp = (1.0 * math.sinh(y)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.01)
		tmp = Float64(Float64(Float64(Float64(x * x) * -0.5) * sinh(y)) / y);
	else
		tmp = Float64(Float64(1.0 * sinh(y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((cos(x) * (sinh(y) / y)) <= -0.01)
		tmp = (((x * x) * -0.5) * sinh(y)) / y;
	else
		tmp = (1.0 * sinh(y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $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)) < -0.0100000000000000002

   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 52.4

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

   4. Applied rewrites 52.4%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 52.4

         \[\leadsto \left(\left(x \cdot x\right) \cdot -0.5\right) \cdot \frac{\sinh y}{y} \]

   7. Applied rewrites 52.4%

      \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{-0.5}\right) \cdot \frac{\sinh y}{y} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 52.4

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

   9. Applied rewrites 52.4%

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

   if -0.0100000000000000002 < (*.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 86.1%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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 86.1

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

   6. Applied rewrites 86.1%

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

Alternative 6: 76.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.01:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot -0.5\right) \cdot \frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.01)
   (* (* (* x x) -0.5) (/ (* (fma (* y y) 0.16666666666666666 1.0) y) y))
   (/ (* 1.0 (sinh y)) y)))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.01) {
		tmp = ((x * x) * -0.5) * ((fma((y * y), 0.16666666666666666, 1.0) * y) / y);
	} 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)) <= -0.01)
		tmp = Float64(Float64(Float64(x * x) * -0.5) * Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y) / y));
	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], -0.01], N[(N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision] * N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $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)) < -0.0100000000000000002

   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 52.4

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

   4. Applied rewrites 52.4%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 52.4

         \[\leadsto \left(\left(x \cdot x\right) \cdot -0.5\right) \cdot \frac{\sinh y}{y} \]

   7. Applied rewrites 52.4%

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

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 47.5

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

   10. Applied rewrites 47.5%

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

   if -0.0100000000000000002 < (*.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 86.1%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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 86.1

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

   6. Applied rewrites 86.1%

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

Alternative 7: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot -0.001388888888888889, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.01)
   (fma (* (* (* x x) (* x x)) -0.001388888888888889) (* x x) 1.0)
   (/ (* 1.0 (sinh y)) y)))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.01) {
		tmp = fma((((x * x) * (x * x)) * -0.001388888888888889), (x * x), 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)) <= -0.01)
		tmp = fma(Float64(Float64(Float64(x * x) * Float64(x * x)) * -0.001388888888888889), Float64(x * x), 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], -0.01], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * -0.001388888888888889), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $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)) < -0.0100000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lift-cos.f64 50.4

         \[\leadsto \cos x \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\cos x} \]

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{\color{blue}{2}}, 1\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {x}^{2}, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) \]

      14. lift-*.f64 44.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) \]

   7. Applied rewrites 44.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1}{720} \cdot {x}^{4}, x \cdot x, 1\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left({x}^{4} \cdot \frac{-1}{720}, x \cdot x, 1\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{4} \cdot \frac{-1}{720}, x \cdot x, 1\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\left(2 \cdot 2\right)} \cdot \frac{-1}{720}, x \cdot x, 1\right) \]

      4. pow-sqr N/A

         \[\leadsto \mathsf{fma}\left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \frac{-1}{720}, x \cdot x, 1\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \frac{-1}{720}, x \cdot x, 1\right) \]

      6. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot {x}^{2}\right) \cdot \frac{-1}{720}, x \cdot x, 1\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot {x}^{2}\right) \cdot \frac{-1}{720}, x \cdot x, 1\right) \]

      8. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{720}, x \cdot x, 1\right) \]

      9. lift-*.f64 44.3

         \[\leadsto \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot -0.001388888888888889, x \cdot x, 1\right) \]

   10. Applied rewrites 44.3%

       \[\leadsto \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot -0.001388888888888889, x \cdot x, 1\right) \]

   if -0.0100000000000000002 < (*.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 86.1%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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 86.1

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

   6. Applied rewrites 86.1%

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

Alternative 8: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.01:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot -0.001388888888888889\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (if (<= (* (cos x) (/ (sinh y) y)) -0.01)
     (* (* t_0 t_0) -0.001388888888888889)
     (/ (* 1.0 (sinh y)) y))))

C

double code(double x, double y) {
	double t_0 = (x * x) * x;
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.01) {
		tmp = (t_0 * t_0) * -0.001388888888888889;
	} else {
		tmp = (1.0 * sinh(y)) / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) * x
    if ((cos(x) * (sinh(y) / y)) <= (-0.01d0)) then
        tmp = (t_0 * t_0) * (-0.001388888888888889d0)
    else
        tmp = (1.0d0 * sinh(y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * x) * x;
	double tmp;
	if ((Math.cos(x) * (Math.sinh(y) / y)) <= -0.01) {
		tmp = (t_0 * t_0) * -0.001388888888888889;
	} else {
		tmp = (1.0 * Math.sinh(y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * x) * x
	tmp = 0
	if (math.cos(x) * (math.sinh(y) / y)) <= -0.01:
		tmp = (t_0 * t_0) * -0.001388888888888889
	else:
		tmp = (1.0 * math.sinh(y)) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.01)
		tmp = Float64(Float64(t_0 * t_0) * -0.001388888888888889);
	else
		tmp = Float64(Float64(1.0 * sinh(y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * x) * x;
	tmp = 0.0;
	if ((cos(x) * (sinh(y) / y)) <= -0.01)
		tmp = (t_0 * t_0) * -0.001388888888888889;
	else
		tmp = (1.0 * sinh(y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * -0.001388888888888889), $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)) < -0.0100000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lift-cos.f64 50.4

         \[\leadsto \cos x \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\cos x} \]

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{\color{blue}{2}}, 1\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {x}^{2}, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) \]

      14. lift-*.f64 44.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) \]

   7. Applied rewrites 44.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{-1}{720} \cdot {x}^{\color{blue}{6}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {x}^{6} \cdot \frac{-1}{720} \]

      2. metadata-eval N/A

         \[\leadsto {x}^{\left(3 + 3\right)} \cdot \frac{-1}{720} \]

      3. pow-prod-up N/A

         \[\leadsto \left({x}^{3} \cdot {x}^{3}\right) \cdot \frac{-1}{720} \]

      4. pow-prod-down N/A

         \[\leadsto {\left(x \cdot x\right)}^{3} \cdot \frac{-1}{720} \]

      5. pow2 N/A

         \[\leadsto {\left({x}^{2}\right)}^{3} \cdot \frac{-1}{720} \]

      6. lower-*.f64 N/A

         \[\leadsto {\left({x}^{2}\right)}^{3} \cdot \frac{-1}{720} \]

      7. pow2 N/A

         \[\leadsto {\left(x \cdot x\right)}^{3} \cdot \frac{-1}{720} \]

      8. pow-prod-down N/A

         \[\leadsto \left({x}^{3} \cdot {x}^{3}\right) \cdot \frac{-1}{720} \]

      9. metadata-eval N/A

         \[\leadsto \left({x}^{\left(\frac{6}{2}\right)} \cdot {x}^{3}\right) \cdot \frac{-1}{720} \]

      10. metadata-eval N/A

          \[\leadsto \left({x}^{\left(\frac{6}{2}\right)} \cdot {x}^{\left(\frac{6}{2}\right)}\right) \cdot \frac{-1}{720} \]

      11. lower-*.f64 N/A

          \[\leadsto \left({x}^{\left(\frac{6}{2}\right)} \cdot {x}^{\left(\frac{6}{2}\right)}\right) \cdot \frac{-1}{720} \]

      12. metadata-eval N/A

          \[\leadsto \left({x}^{3} \cdot {x}^{\left(\frac{6}{2}\right)}\right) \cdot \frac{-1}{720} \]

      13. unpow3 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot {x}^{\left(\frac{6}{2}\right)}\right) \cdot \frac{-1}{720} \]

      14. pow2 N/A

          \[\leadsto \left(\left({x}^{2} \cdot x\right) \cdot {x}^{\left(\frac{6}{2}\right)}\right) \cdot \frac{-1}{720} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\left({x}^{2} \cdot x\right) \cdot {x}^{\left(\frac{6}{2}\right)}\right) \cdot \frac{-1}{720} \]

      16. pow2 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot {x}^{\left(\frac{6}{2}\right)}\right) \cdot \frac{-1}{720} \]

      17. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot {x}^{\left(\frac{6}{2}\right)}\right) \cdot \frac{-1}{720} \]

      18. metadata-eval N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot {x}^{3}\right) \cdot \frac{-1}{720} \]

      19. unpow3 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{-1}{720} \]

      20. pow2 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left({x}^{2} \cdot x\right)\right) \cdot \frac{-1}{720} \]

      21. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left({x}^{2} \cdot x\right)\right) \cdot \frac{-1}{720} \]

      22. pow2 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{-1}{720} \]

      23. lift-*.f64 44.3

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot -0.001388888888888889 \]

   10. Applied rewrites 44.3%

       \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot -0.001388888888888889 \]

   if -0.0100000000000000002 < (*.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 86.1%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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 86.1

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

   6. Applied rewrites 86.1%

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

Alternative 9: 66.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.01:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot -0.001388888888888889\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot y, y, 0.16666666666666666\right) \cdot y, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (if (<= (* (cos x) (/ (sinh y) y)) -0.01)
     (* (* t_0 t_0) -0.001388888888888889)
     (fma
      (* (fma (* 0.008333333333333333 y) y 0.16666666666666666) y)
      y
      1.0))))

C

double code(double x, double y) {
	double t_0 = (x * x) * x;
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.01) {
		tmp = (t_0 * t_0) * -0.001388888888888889;
	} else {
		tmp = fma((fma((0.008333333333333333 * y), y, 0.16666666666666666) * y), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.01)
		tmp = Float64(Float64(t_0 * t_0) * -0.001388888888888889);
	else
		tmp = fma(Float64(fma(Float64(0.008333333333333333 * y), y, 0.16666666666666666) * y), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.01], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * -0.001388888888888889), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * y), $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0100000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lift-cos.f64 50.4

         \[\leadsto \cos x \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\cos x} \]

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{\color{blue}{2}}, 1\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {x}^{2}, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) \]

      14. lift-*.f64 44.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) \]

   7. Applied rewrites 44.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{-1}{720} \cdot {x}^{\color{blue}{6}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {x}^{6} \cdot \frac{-1}{720} \]

      2. metadata-eval N/A

         \[\leadsto {x}^{\left(3 + 3\right)} \cdot \frac{-1}{720} \]

      3. pow-prod-up N/A

         \[\leadsto \left({x}^{3} \cdot {x}^{3}\right) \cdot \frac{-1}{720} \]

      4. pow-prod-down N/A

         \[\leadsto {\left(x \cdot x\right)}^{3} \cdot \frac{-1}{720} \]

      5. pow2 N/A

         \[\leadsto {\left({x}^{2}\right)}^{3} \cdot \frac{-1}{720} \]

      6. lower-*.f64 N/A

         \[\leadsto {\left({x}^{2}\right)}^{3} \cdot \frac{-1}{720} \]

      7. pow2 N/A

         \[\leadsto {\left(x \cdot x\right)}^{3} \cdot \frac{-1}{720} \]

      8. pow-prod-down N/A

         \[\leadsto \left({x}^{3} \cdot {x}^{3}\right) \cdot \frac{-1}{720} \]

      9. metadata-eval N/A

         \[\leadsto \left({x}^{\left(\frac{6}{2}\right)} \cdot {x}^{3}\right) \cdot \frac{-1}{720} \]

      10. metadata-eval N/A

          \[\leadsto \left({x}^{\left(\frac{6}{2}\right)} \cdot {x}^{\left(\frac{6}{2}\right)}\right) \cdot \frac{-1}{720} \]

      11. lower-*.f64 N/A

          \[\leadsto \left({x}^{\left(\frac{6}{2}\right)} \cdot {x}^{\left(\frac{6}{2}\right)}\right) \cdot \frac{-1}{720} \]

      12. metadata-eval N/A

          \[\leadsto \left({x}^{3} \cdot {x}^{\left(\frac{6}{2}\right)}\right) \cdot \frac{-1}{720} \]

      13. unpow3 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot {x}^{\left(\frac{6}{2}\right)}\right) \cdot \frac{-1}{720} \]

      14. pow2 N/A

          \[\leadsto \left(\left({x}^{2} \cdot x\right) \cdot {x}^{\left(\frac{6}{2}\right)}\right) \cdot \frac{-1}{720} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\left({x}^{2} \cdot x\right) \cdot {x}^{\left(\frac{6}{2}\right)}\right) \cdot \frac{-1}{720} \]

      16. pow2 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot {x}^{\left(\frac{6}{2}\right)}\right) \cdot \frac{-1}{720} \]

      17. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot {x}^{\left(\frac{6}{2}\right)}\right) \cdot \frac{-1}{720} \]

      18. metadata-eval N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot {x}^{3}\right) \cdot \frac{-1}{720} \]

      19. unpow3 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{-1}{720} \]

      20. pow2 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left({x}^{2} \cdot x\right)\right) \cdot \frac{-1}{720} \]

      21. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left({x}^{2} \cdot x\right)\right) \cdot \frac{-1}{720} \]

      22. pow2 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{-1}{720} \]

      23. lift-*.f64 44.3

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot -0.001388888888888889 \]

   10. Applied rewrites 44.3%

       \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot -0.001388888888888889 \]

   if -0.0100000000000000002 < (*.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 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 73.9

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

   7. Applied rewrites 73.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 73.9

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

   9. Applied rewrites 73.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y, y, 1\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. lift-fma.f64 N/A

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

       6. lift-*.f64 73.9

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

   11. Applied rewrites 73.9%

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

Alternative 10: 63.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot y, y, 0.16666666666666666\right) \cdot y, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.01)
   (fma -0.5 (* x x) 1.0)
   (fma (* (fma (* 0.008333333333333333 y) y 0.16666666666666666) y) y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.01) {
		tmp = fma(-0.5, (x * x), 1.0);
	} else {
		tmp = fma((fma((0.008333333333333333 * y), y, 0.16666666666666666) * y), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.01)
		tmp = fma(-0.5, Float64(x * x), 1.0);
	else
		tmp = fma(Float64(fma(Float64(0.008333333333333333 * y), y, 0.16666666666666666) * y), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.01], N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * y), $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lift-cos.f64 50.5

         \[\leadsto \cos x \]

   4. Applied rewrites 50.5%

      \[\leadsto \color{blue}{\cos x} \]

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{\color{blue}{2}}, 1\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {x}^{2}, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) \]

      14. lift-*.f64 44.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) \]

   7. Applied rewrites 44.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 30.3%

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

   if -0.0100000000000000002 < (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 73.9

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

   7. Applied rewrites 73.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 73.9

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

   9. Applied rewrites 73.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right) \cdot y, y, 1\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. lift-fma.f64 N/A

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

       6. lift-*.f64 73.9

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

   11. Applied rewrites 73.9%

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

Alternative 11: 63.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -0.01)
     (fma -0.5 (* x x) 1.0)
     (if (<= t_0 2.0)
       (fma (* 0.16666666666666666 y) y 1.0)
       (* (* y y) (* (* y y) 0.008333333333333333))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = fma(-0.5, (x * x), 1.0);
	} else if (t_0 <= 2.0) {
		tmp = fma((0.16666666666666666 * y), y, 1.0);
	} else {
		tmp = (y * y) * ((y * y) * 0.008333333333333333);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = fma(-0.5, Float64(x * x), 1.0);
	elseif (t_0 <= 2.0)
		tmp = fma(Float64(0.16666666666666666 * y), y, 1.0);
	else
		tmp = Float64(Float64(y * y) * Float64(Float64(y * y) * 0.008333333333333333));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -0.0100000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lift-cos.f64 50.4

         \[\leadsto \cos x \]

   4. Applied rewrites 50.4%

      \[\leadsto \color{blue}{\cos x} \]

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{\color{blue}{2}}, 1\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {x}^{2}, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) \]

      14. lift-*.f64 44.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) \]

   7. Applied rewrites 44.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 30.4%

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

   if -0.0100000000000000002 < (*.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 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 99.7

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

   4. Applied rewrites 99.7%

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

   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 72.3

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

   7. Applied rewrites 72.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 72.3

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

   9. Applied rewrites 72.3%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 72.2%

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

   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 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 75.6

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

   4. Applied rewrites 75.6%

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

   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 75.6

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

   7. Applied rewrites 75.6%

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

   8. Taylor expanded in y around inf

      \[\leadsto \frac{1}{120} \cdot {y}^{\color{blue}{4}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {y}^{4} \cdot \frac{1}{120} \]

      2. lower-*.f64 N/A

         \[\leadsto {y}^{4} \cdot \frac{1}{120} \]

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

         \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120} \]

      9. lift-*.f64 75.6

         \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot 0.008333333333333333 \]

   10. Applied rewrites 75.6%

       \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot 0.008333333333333333 \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120} \]

       3. lift-*.f64 N/A

          \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120} \]

       4. lift-*.f64 N/A

          \[\leadsto \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \frac{1}{120} \]

       5. associate-*l* N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120}\right) \]

       6. *-commutative N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot \left(y \cdot y\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot \left(y \cdot \color{blue}{y}\right)\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\frac{1}{120} \cdot \left(y \cdot y\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \frac{1}{120}\right) \]

       11. lift-*.f64 75.5

           \[\leadsto \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right) \]

   12. Applied rewrites 75.5%

       \[\leadsto \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot 0.008333333333333333\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 62.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.01)
   (fma -0.5 (* x x) 1.0)
   (fma (* (* y y) 0.008333333333333333) (* y y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.01) {
		tmp = fma(-0.5, (x * x), 1.0);
	} else {
		tmp = fma(((y * y) * 0.008333333333333333), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.01)
		tmp = fma(-0.5, Float64(x * x), 1.0);
	else
		tmp = fma(Float64(Float64(y * y) * 0.008333333333333333), Float64(y * y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.01], N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lift-cos.f64 50.5

         \[\leadsto \cos x \]

   4. Applied rewrites 50.5%

      \[\leadsto \color{blue}{\cos x} \]

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{\color{blue}{2}}, 1\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {x}^{2}, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) \]

      14. lift-*.f64 44.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) \]

   7. Applied rewrites 44.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 30.3%

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

   if -0.0100000000000000002 < (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 73.9

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

   7. Applied rewrites 73.9%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 73.6

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

   10. Applied rewrites 73.6%

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

Alternative 13: 54.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.01)
   (fma -0.5 (* x x) 1.0)
   (fma (* 0.16666666666666666 y) y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.01) {
		tmp = fma(-0.5, (x * x), 1.0);
	} else {
		tmp = fma((0.16666666666666666 * y), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.01)
		tmp = fma(-0.5, Float64(x * x), 1.0);
	else
		tmp = fma(Float64(0.16666666666666666 * y), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.01], N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lift-cos.f64 50.5

         \[\leadsto \cos x \]

   4. Applied rewrites 50.5%

      \[\leadsto \color{blue}{\cos x} \]

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{\color{blue}{2}}, 1\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {x}^{2}, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) \]

      14. lift-*.f64 44.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) \]

   7. Applied rewrites 44.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 30.3%

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

   if -0.0100000000000000002 < (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 73.9

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

   7. Applied rewrites 73.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 73.9

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

   9. Applied rewrites 73.9%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 62.0%

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

Alternative 14: 35.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.01:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.01) (fma -0.5 (* x x) 1.0) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.01) {
		tmp = fma(-0.5, (x * x), 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.01)
		tmp = fma(-0.5, Float64(x * x), 1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.01], N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -0.0100000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lift-cos.f64 50.5

         \[\leadsto \cos x \]

   4. Applied rewrites 50.5%

      \[\leadsto \color{blue}{\cos x} \]

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1 \]

      2. *-commutative N/A

         \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{\color{blue}{2}}, 1\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {x}^{2}, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right) \]

      14. lift-*.f64 44.2

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) \]

   7. Applied rewrites 44.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, \color{blue}{x \cdot x}, 1\right) \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 30.3%

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

   if -0.0100000000000000002 < (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 73.9

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

   7. Applied rewrites 73.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 37.6%

       \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 28.5% accurate, 51.4× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 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
end function

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

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 55.8

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

7. Applied rewrites 55.8%

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

8. Taylor expanded in y around 0

   \[\leadsto 1 \]
9. Step-by-step derivation

10. Applied rewrites 28.5%

    \[\leadsto 1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025113 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))