Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 3.0s

Alternatives: 13

Speedup: 0.3×

• 49.8% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\left(\left(x \cdot x\right) \cdot -0.5\right) \cdot \sinh y}{y}\\ \mathbf{elif}\;t\_0 \leq 0.9999999855652147:\\ \;\;\;\;\cos x \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y \cdot 1}{y}\\ \end{array} \end{array} \]

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 0.9999999855652147)
       (* (cos x) (fma (* y y) 0.16666666666666666 1.0))
       (/ (* (sinh y) 1.0) 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 <= 0.9999999855652147) {
		tmp = cos(x) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = (sinh(y) * 1.0) / 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 <= 0.9999999855652147)
		tmp = Float64(cos(x) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(Float64(sinh(y) * 1.0) / 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, 0.9999999855652147], N[(N[Cos[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sinh[y], $MachinePrecision] * 1.0), $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)) < 0.999999985565214744

   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 0.999999985565214744 < (*.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 99.8%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sinh.f64 99.8

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

   6. Applied rewrites 99.8%

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

Alternative 2: 99.7% 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 3: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\left(\left(x \cdot x\right) \cdot -0.5\right) \cdot \sinh y}{y}\\ \mathbf{elif}\;t\_0 \leq 0.9999999855652147:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y \cdot 1}{y}\\ \end{array} \end{array} \]

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 0.9999999855652147) (cos x) (/ (* (sinh y) 1.0) 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 <= 0.9999999855652147) {
		tmp = cos(x);
	} else {
		tmp = (sinh(y) * 1.0) / 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 <= 0.9999999855652147) {
		tmp = Math.cos(x);
	} else {
		tmp = (Math.sinh(y) * 1.0) / 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 <= 0.9999999855652147:
		tmp = math.cos(x)
	else:
		tmp = (math.sinh(y) * 1.0) / 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 <= 0.9999999855652147)
		tmp = cos(x);
	else
		tmp = Float64(Float64(sinh(y) * 1.0) / 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 <= 0.9999999855652147)
		tmp = cos(x);
	else
		tmp = (sinh(y) * 1.0) / 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, 0.9999999855652147], N[Cos[x], $MachinePrecision], N[(N[(N[Sinh[y], $MachinePrecision] * 1.0), $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)) < 0.999999985565214744

   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 98.4

         \[\leadsto \cos x \]

   4. Applied rewrites 98.4%

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

   if 0.999999985565214744 < (*.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 99.8%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sinh.f64 99.8

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

   6. Applied rewrites 99.8%

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

Alternative 4: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \cdot \frac{\sinh y}{y} \leq -0.0002:\\ \;\;\;\;\frac{\left(\left(x \cdot x\right) \cdot -0.5\right) \cdot \sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y \cdot 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (cos x) (/ (sinh y) y)) -0.0002)
   (/ (* (* (* x x) -0.5) (sinh y)) y)
   (/ (* (sinh y) 1.0) y)))

C

double code(double x, double y) {
	double tmp;
	if ((cos(x) * (sinh(y) / y)) <= -0.0002) {
		tmp = (((x * x) * -0.5) * sinh(y)) / y;
	} else {
		tmp = (sinh(y) * 1.0) / 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)) <= (-0.0002d0)) then
        tmp = (((x * x) * (-0.5d0)) * sinh(y)) / y
    else
        tmp = (sinh(y) * 1.0d0) / 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)) <= -0.0002) {
		tmp = (((x * x) * -0.5) * Math.sinh(y)) / y;
	} else {
		tmp = (Math.sinh(y) * 1.0) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.cos(x) * (math.sinh(y) / y)) <= -0.0002:
		tmp = (((x * x) * -0.5) * math.sinh(y)) / y
	else:
		tmp = (math.sinh(y) * 1.0) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(cos(x) * Float64(sinh(y) / y)) <= -0.0002)
		tmp = Float64(Float64(Float64(Float64(x * x) * -0.5) * sinh(y)) / y);
	else
		tmp = Float64(Float64(sinh(y) * 1.0) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((cos(x) * (sinh(y) / y)) <= -0.0002)
		tmp = (((x * x) * -0.5) * sinh(y)) / y;
	else
		tmp = (sinh(y) * 1.0) / 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], -0.0002], N[(N[(N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[Sinh[y], $MachinePrecision] * 1.0), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -2.0000000000000001e-4

   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 51.6

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

   4. Applied rewrites 51.6%

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

   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 51.6

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

   7. Applied rewrites 51.6%

      \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{-0.5}\right) \cdot \frac{\sinh y}{y} \]
   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 51.6

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

   9. Applied rewrites 51.6%

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

   if -2.0000000000000001e-4 < (*.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 86.8%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sinh.f64 86.8

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

   6. Applied rewrites 86.8%

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

Alternative 5: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.0002:\\ \;\;\;\;\mathsf{fma}\left(-0.5, x \cdot x, 1\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y \cdot 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.0002)
   (* (fma -0.5 (* x x) 1.0) (fma (* y y) 0.16666666666666666 1.0))
   (/ (* (sinh y) 1.0) y)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.0002) {
		tmp = fma(-0.5, (x * x), 1.0) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = (sinh(y) * 1.0) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.0002)
		tmp = Float64(fma(-0.5, Float64(x * x), 1.0) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(Float64(sinh(y) * 1.0) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.0002], N[(N[(-0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sinh[y], $MachinePrecision] * 1.0), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -2.0000000000000001e-4

   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 0.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 1.1%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 0.9

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

   10. Applied rewrites 0.9%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 45.6

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

   13. Applied rewrites 45.6%

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

   if -2.0000000000000001e-4 < (cos.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 86.8%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sinh.f64 86.8

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

   6. Applied rewrites 86.8%

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

Alternative 6: 72.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.0002:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot x, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y \cdot 1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.0002) (fma (* -0.5 x) x 1.0) (/ (* (sinh y) 1.0) y)))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.0002) {
		tmp = fma((-0.5 * x), x, 1.0);
	} else {
		tmp = (sinh(y) * 1.0) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.0002)
		tmp = fma(Float64(-0.5 * x), x, 1.0);
	else
		tmp = Float64(Float64(sinh(y) * 1.0) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.0002], N[(N[(-0.5 * x), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(N[(N[Sinh[y], $MachinePrecision] * 1.0), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -2.0000000000000001e-4

   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 51.2

         \[\leadsto \cos x \]

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 0.5

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

   7. Applied rewrites 0.5%

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

   8. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 28.5

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

   10. Applied rewrites 28.5%

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

   if -2.0000000000000001e-4 < (cos.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 86.8%

      \[\leadsto \color{blue}{1} \cdot \frac{\sinh y}{y} \]
   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. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sinh.f64 86.8

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

   6. Applied rewrites 86.8%

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

Alternative 7: 60.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.0002:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot x, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\mathsf{fma}\left(y \cdot y, y \cdot 0.16666666666666666, y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.0002)
   (fma (* -0.5 x) x 1.0)
   (* 1.0 (/ (fma (* y y) (* y 0.16666666666666666) y) y))))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.0002) {
		tmp = fma((-0.5 * x), x, 1.0);
	} else {
		tmp = 1.0 * (fma((y * y), (y * 0.16666666666666666), y) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.0002)
		tmp = fma(Float64(-0.5 * x), x, 1.0);
	else
		tmp = Float64(1.0 * Float64(fma(Float64(y * y), Float64(y * 0.16666666666666666), y) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.0002], N[(N[(-0.5 * x), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(1.0 * N[(N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -2.0000000000000001e-4

   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 51.2

         \[\leadsto \cos x \]

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 0.5

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

   7. Applied rewrites 0.5%

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

   8. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 28.5

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

   10. Applied rewrites 28.5%

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

   if -2.0000000000000001e-4 < (cos.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 86.8%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 70.9

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

   7. Applied rewrites 70.9%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 70.9

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

   9. Applied rewrites 70.9%

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

Alternative 8: 60.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.0002:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot x, x, 1\right)\\ \mathbf{elif}\;t\_0 \leq 20:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\left(\left(y \cdot y\right) \cdot y\right) \cdot 0.16666666666666666}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -0.0002)
     (fma (* -0.5 x) x 1.0)
     (if (<= t_0 20.0)
       (* 1.0 (fma (* 0.16666666666666666 y) y 1.0))
       (* 1.0 (/ (* (* (* y y) y) 0.16666666666666666) y))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.0002) {
		tmp = fma((-0.5 * x), x, 1.0);
	} else if (t_0 <= 20.0) {
		tmp = 1.0 * fma((0.16666666666666666 * y), y, 1.0);
	} else {
		tmp = 1.0 * ((((y * y) * y) * 0.16666666666666666) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.0002)
		tmp = fma(Float64(-0.5 * x), x, 1.0);
	elseif (t_0 <= 20.0)
		tmp = Float64(1.0 * fma(Float64(0.16666666666666666 * y), y, 1.0));
	else
		tmp = Float64(1.0 * Float64(Float64(Float64(Float64(y * y) * y) * 0.16666666666666666) / 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, -0.0002], N[(N[(-0.5 * x), $MachinePrecision] * x + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 20.0], N[(1.0 * N[(N[(0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -2.0000000000000001e-4

   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 51.2

         \[\leadsto \cos x \]

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 0.5

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

   7. Applied rewrites 0.5%

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

   8. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 28.5

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

   10. Applied rewrites 28.5%

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

   if -2.0000000000000001e-4 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 20

   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 73.7%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 73.0%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 73.4

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

   10. Applied rewrites 73.4%

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

       1. lift-fma.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. pow2 N/A

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

       4. *-commutative N/A

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

       5. pow2 N/A

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

       6. associate-*r* N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-*.f64 73.4

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

   12. Applied rewrites 73.4%

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

   if 20 < (*.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 99.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 68.4

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

   7. Applied rewrites 68.4%

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

   8. Taylor expanded in y around inf

      \[\leadsto 1 \cdot \frac{\frac{1}{6} \cdot \color{blue}{{y}^{3}}}{y} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 \cdot \frac{{y}^{3} \cdot \frac{1}{6}}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 \cdot \frac{{y}^{3} \cdot \frac{1}{6}}{y} \]

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 68.5

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

   10. Applied rewrites 68.5%

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

Alternative 9: 55.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.0002:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot x, x, 1\right)\\ \mathbf{elif}\;\cos x \leq 0.97:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.0002)
   (fma (* -0.5 x) x 1.0)
   (if (<= (cos x) 0.97)
     (* (* (* (* x x) x) x) 0.041666666666666664)
     (* 1.0 (fma (* 0.16666666666666666 y) y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.0002) {
		tmp = fma((-0.5 * x), x, 1.0);
	} else if (cos(x) <= 0.97) {
		tmp = (((x * x) * x) * x) * 0.041666666666666664;
	} else {
		tmp = 1.0 * fma((0.16666666666666666 * y), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.0002)
		tmp = fma(Float64(-0.5 * x), x, 1.0);
	elseif (cos(x) <= 0.97)
		tmp = Float64(Float64(Float64(Float64(x * x) * x) * x) * 0.041666666666666664);
	else
		tmp = Float64(1.0 * fma(Float64(0.16666666666666666 * y), y, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.0002], N[(N[(-0.5 * x), $MachinePrecision] * x + 1.0), $MachinePrecision], If[LessEqual[N[Cos[x], $MachinePrecision], 0.97], N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision], N[(1.0 * N[(N[(0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 x) < -2.0000000000000001e-4

   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 51.2

         \[\leadsto \cos x \]

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 0.5

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

   7. Applied rewrites 0.5%

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

   8. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 28.5

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

   10. Applied rewrites 28.5%

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

   if -2.0000000000000001e-4 < (cos.f64 x) < 0.96999999999999997

   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 49.9

         \[\leadsto \cos x \]

   4. Applied rewrites 49.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 40.3

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

   7. Applied rewrites 40.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{24} \cdot {x}^{\color{blue}{4}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {x}^{4} \cdot \frac{1}{24} \]

      2. lower-*.f64 N/A

         \[\leadsto {x}^{4} \cdot \frac{1}{24} \]

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{24} \]

      10. lift-*.f64 40.2

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.041666666666666664 \]

   10. Applied rewrites 40.2%

       \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.041666666666666664 \]

   if 0.96999999999999997 < (cos.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 97.4%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 48.7%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 73.3

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

   10. Applied rewrites 73.3%

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

       1. lift-fma.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. pow2 N/A

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

       4. *-commutative N/A

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

       5. pow2 N/A

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

       6. associate-*r* N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-*.f64 73.2

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

   12. Applied rewrites 73.2%

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

Alternative 10: 54.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.0002:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot x, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(0.16666666666666666 \cdot y, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.0002)
   (fma (* -0.5 x) x 1.0)
   (* 1.0 (fma (* 0.16666666666666666 y) y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.0002) {
		tmp = fma((-0.5 * x), x, 1.0);
	} else {
		tmp = 1.0 * fma((0.16666666666666666 * y), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (cos(x) <= -0.0002)
		tmp = fma(Float64(-0.5 * x), x, 1.0);
	else
		tmp = Float64(1.0 * fma(Float64(0.16666666666666666 * y), y, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.0002], N[(N[(-0.5 * x), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(1.0 * N[(N[(0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -2.0000000000000001e-4

   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 51.2

         \[\leadsto \cos x \]

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 0.5

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

   7. Applied rewrites 0.5%

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

   8. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 28.5

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

   10. Applied rewrites 28.5%

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

   if -2.0000000000000001e-4 < (cos.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 86.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 38.0%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 62.7

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

   10. Applied rewrites 62.7%

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

       1. lift-fma.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. pow2 N/A

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

       4. *-commutative N/A

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

       5. pow2 N/A

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

       6. associate-*r* N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-*.f64 62.7

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

   12. Applied rewrites 62.7%

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

Alternative 11: 54.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.0002:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot x, x, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (/ (sinh y) y))))
   (if (<= t_0 -0.0002)
     (fma (* -0.5 x) x 1.0)
     (if (<= t_0 2.0) 1.0 (* 1.0 (* (* y y) 0.16666666666666666))))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.0002) {
		tmp = fma((-0.5 * x), x, 1.0);
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 * ((y * y) * 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.0002)
		tmp = fma(Float64(-0.5 * x), x, 1.0);
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(1.0 * Float64(Float64(y * y) * 0.16666666666666666));
	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, -0.0002], N[(N[(-0.5 * x), $MachinePrecision] * x + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, N[(1.0 * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < -2.0000000000000001e-4

   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 51.2

         \[\leadsto \cos x \]

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 0.5

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

   7. Applied rewrites 0.5%

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

   8. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 28.5

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

   10. Applied rewrites 28.5%

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

   if -2.0000000000000001e-4 < (*.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

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

      1. lift-cos.f64 98.9

         \[\leadsto \cos x \]

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 68.0

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

   7. Applied rewrites 68.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 73.1%

       \[\leadsto 1 \]

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 3.2%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 52.1

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

   10. Applied rewrites 52.1%

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

   11. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 52.1

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

   13. Applied rewrites 52.1%

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

Alternative 12: 35.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos x \leq -0.0002:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot x, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cos x) -0.0002) (fma (* -0.5 x) x 1.0) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (cos(x) <= -0.0002) {
		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) <= -0.0002)
		tmp = fma(Float64(-0.5 * x), x, 1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cos[x], $MachinePrecision], -0.0002], N[(N[(-0.5 * x), $MachinePrecision] * x + 1.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 x) < -2.0000000000000001e-4

   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 51.2

         \[\leadsto \cos x \]

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 0.5

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

   7. Applied rewrites 0.5%

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

   8. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 28.5

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

   10. Applied rewrites 28.5%

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

   if -2.0000000000000001e-4 < (cos.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lift-cos.f64 50.9

         \[\leadsto \cos x \]

   4. Applied rewrites 50.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 47.9

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

   7. Applied rewrites 47.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 38.0%

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

Alternative 13: 28.8% 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

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

   1. lift-cos.f64 51.0

      \[\leadsto \cos x \]

4. Applied rewrites 51.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 36.0

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

7. Applied rewrites 36.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 28.8%

    \[\leadsto 1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025114 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))