Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.7% → 99.9%

Time: 6.1s

Alternatives: 18

Speedup: 1.0×

• 48.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sinh.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-sinh.f64 N/A

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

   10. lift-sin.f64 99.9

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

3. Applied rewrites 99.9%

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

Alternative 2: 92.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ t_1 := \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+132}:\\ \;\;\;\;\frac{\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq -27:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-13}:\\ \;\;\;\;\left(t\_1 \cdot \frac{\sin x}{x}\right) \cdot y\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+106}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot t\_1}{x} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 2.0 (sinh y)) (fma (* x x) -0.08333333333333333 0.5)))
        (t_1 (fma (* y y) 0.16666666666666666 1.0)))
   (if (<= y -2.4e+132)
     (* (/ (* (* (* y y) 0.16666666666666666) (sin x)) x) y)
     (if (<= y -27.0)
       t_0
       (if (<= y 2.1e-13)
         (* (* t_1 (/ (sin x) x)) y)
         (if (<= y 1.55e+106) t_0 (* (/ (* (sin x) t_1) x) y)))))))

C

double code(double x, double y) {
	double t_0 = (2.0 * sinh(y)) * fma((x * x), -0.08333333333333333, 0.5);
	double t_1 = fma((y * y), 0.16666666666666666, 1.0);
	double tmp;
	if (y <= -2.4e+132) {
		tmp = ((((y * y) * 0.16666666666666666) * sin(x)) / x) * y;
	} else if (y <= -27.0) {
		tmp = t_0;
	} else if (y <= 2.1e-13) {
		tmp = (t_1 * (sin(x) / x)) * y;
	} else if (y <= 1.55e+106) {
		tmp = t_0;
	} else {
		tmp = ((sin(x) * t_1) / x) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 * sinh(y)) * fma(Float64(x * x), -0.08333333333333333, 0.5))
	t_1 = fma(Float64(y * y), 0.16666666666666666, 1.0)
	tmp = 0.0
	if (y <= -2.4e+132)
		tmp = Float64(Float64(Float64(Float64(Float64(y * y) * 0.16666666666666666) * sin(x)) / x) * y);
	elseif (y <= -27.0)
		tmp = t_0;
	elseif (y <= 2.1e-13)
		tmp = Float64(Float64(t_1 * Float64(sin(x) / x)) * y);
	elseif (y <= 1.55e+106)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(sin(x) * t_1) / x) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]}, If[LessEqual[y, -2.4e+132], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, -27.0], t$95$0, If[LessEqual[y, 2.1e-13], N[(N[(t$95$1 * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.55e+106], t$95$0, N[(N[(N[(N[Sin[x], $MachinePrecision] * t$95$1), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.4000000000000001e132

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 96.7

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

   7. Applied rewrites 96.7%

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

   if -2.4000000000000001e132 < y < -27 or 2.09999999999999989e-13 < y < 1.55e106

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {x}^{2}\right) \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + \color{blue}{\frac{1}{2}} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      4. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      5. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      7. lift-sinh.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 71.4

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   4. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)} \]

   if -27 < y < 2.09999999999999989e-13

   1. Initial program 77.3%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-/l* N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-sin.f64 99.4

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

   6. Applied rewrites 99.4%

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

   if 1.55e106 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 93.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 93.4

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

   6. Applied rewrites 93.4%

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

Alternative 3: 92.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ t_1 := \frac{\sin x \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+132}:\\ \;\;\;\;\frac{\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq -27:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+106}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 2.0 (sinh y)) (fma (* x x) -0.08333333333333333 0.5)))
        (t_1 (* (/ (* (sin x) (fma (* y y) 0.16666666666666666 1.0)) x) y)))
   (if (<= y -2.4e+132)
     (* (/ (* (* (* y y) 0.16666666666666666) (sin x)) x) y)
     (if (<= y -27.0)
       t_0
       (if (<= y 2.1e-13) t_1 (if (<= y 1.55e+106) t_0 t_1))))))

C

double code(double x, double y) {
	double t_0 = (2.0 * sinh(y)) * fma((x * x), -0.08333333333333333, 0.5);
	double t_1 = ((sin(x) * fma((y * y), 0.16666666666666666, 1.0)) / x) * y;
	double tmp;
	if (y <= -2.4e+132) {
		tmp = ((((y * y) * 0.16666666666666666) * sin(x)) / x) * y;
	} else if (y <= -27.0) {
		tmp = t_0;
	} else if (y <= 2.1e-13) {
		tmp = t_1;
	} else if (y <= 1.55e+106) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 * sinh(y)) * fma(Float64(x * x), -0.08333333333333333, 0.5))
	t_1 = Float64(Float64(Float64(sin(x) * fma(Float64(y * y), 0.16666666666666666, 1.0)) / x) * y)
	tmp = 0.0
	if (y <= -2.4e+132)
		tmp = Float64(Float64(Float64(Float64(Float64(y * y) * 0.16666666666666666) * sin(x)) / x) * y);
	elseif (y <= -27.0)
		tmp = t_0;
	elseif (y <= 2.1e-13)
		tmp = t_1;
	elseif (y <= 1.55e+106)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -2.4e+132], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, -27.0], t$95$0, If[LessEqual[y, 2.1e-13], t$95$1, If[LessEqual[y, 1.55e+106], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4000000000000001e132

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 96.7

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

   7. Applied rewrites 96.7%

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

   if -2.4000000000000001e132 < y < -27 or 2.09999999999999989e-13 < y < 1.55e106

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {x}^{2}\right) \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + \color{blue}{\frac{1}{2}} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      4. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      5. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      7. lift-sinh.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 71.4

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   4. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)} \]

   if -27 < y < 2.09999999999999989e-13 or 1.55e106 < y

   1. Initial program 82.9%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 97.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 97.9

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

   6. Applied rewrites 97.9%

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

Alternative 4: 92.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ t_1 := \frac{\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \sin x}{x} \cdot y\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -27:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-13}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+106}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 2.0 (sinh y)) (fma (* x x) -0.08333333333333333 0.5)))
        (t_1 (* (/ (* (* (* y y) 0.16666666666666666) (sin x)) x) y)))
   (if (<= y -2.4e+132)
     t_1
     (if (<= y -27.0)
       t_0
       (if (<= y 2.1e-13)
         (* (/ (sin x) x) y)
         (if (<= y 1.55e+106) t_0 t_1))))))

C

double code(double x, double y) {
	double t_0 = (2.0 * sinh(y)) * fma((x * x), -0.08333333333333333, 0.5);
	double t_1 = ((((y * y) * 0.16666666666666666) * sin(x)) / x) * y;
	double tmp;
	if (y <= -2.4e+132) {
		tmp = t_1;
	} else if (y <= -27.0) {
		tmp = t_0;
	} else if (y <= 2.1e-13) {
		tmp = (sin(x) / x) * y;
	} else if (y <= 1.55e+106) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 * sinh(y)) * fma(Float64(x * x), -0.08333333333333333, 0.5))
	t_1 = Float64(Float64(Float64(Float64(Float64(y * y) * 0.16666666666666666) * sin(x)) / x) * y)
	tmp = 0.0
	if (y <= -2.4e+132)
		tmp = t_1;
	elseif (y <= -27.0)
		tmp = t_0;
	elseif (y <= 2.1e-13)
		tmp = Float64(Float64(sin(x) / x) * y);
	elseif (y <= 1.55e+106)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -2.4e+132], t$95$1, If[LessEqual[y, -27.0], t$95$0, If[LessEqual[y, 2.1e-13], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.55e+106], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4000000000000001e132 or 1.55e106 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 94.9

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

   7. Applied rewrites 94.9%

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

   if -2.4000000000000001e132 < y < -27 or 2.09999999999999989e-13 < y < 1.55e106

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {x}^{2}\right) \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + \color{blue}{\frac{1}{2}} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      4. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      5. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      7. lift-sinh.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 71.4

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   4. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)} \]

   if -27 < y < 2.09999999999999989e-13

   1. Initial program 77.3%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 99.1

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

   4. Applied rewrites 99.1%

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

Alternative 5: 86.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ t_1 := \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x))
        (t_1 (* (* 2.0 (sinh y)) (fma (* x x) -0.08333333333333333 0.5))))
   (if (<= t_0 -2e-6) t_1 (if (<= t_0 5e-48) (* (/ (sin x) x) y) t_1))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double t_1 = (2.0 * sinh(y)) * fma((x * x), -0.08333333333333333, 0.5);
	double tmp;
	if (t_0 <= -2e-6) {
		tmp = t_1;
	} else if (t_0 <= 5e-48) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	t_1 = Float64(Float64(2.0 * sinh(y)) * fma(Float64(x * x), -0.08333333333333333, 0.5))
	tmp = 0.0
	if (t_0 <= -2e-6)
		tmp = t_1;
	elseif (t_0 <= 5e-48)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-6], t$95$1, If[LessEqual[t$95$0, 5e-48], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.99999999999999991e-6 or 4.9999999999999999e-48 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {x}^{2}\right) \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + \color{blue}{\frac{1}{2}} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      4. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      5. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      7. lift-sinh.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 74.8

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   4. Applied rewrites 74.8%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)} \]

   if -1.99999999999999991e-6 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.9999999999999999e-48

   1. Initial program 76.9%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 99.2

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

   4. Applied rewrites 99.2%

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

Alternative 6: 73.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+115}:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{y} - e^{-y}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.5e+115)
   (* (* 2.0 (sinh y)) (fma (* x x) -0.08333333333333333 0.5))
   (* (- (exp y) (exp (- y))) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.5e+115) {
		tmp = (2.0 * sinh(y)) * fma((x * x), -0.08333333333333333, 0.5);
	} else {
		tmp = (exp(y) - exp(-y)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.5e+115)
		tmp = Float64(Float64(2.0 * sinh(y)) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	else
		tmp = Float64(Float64(exp(y) - exp(Float64(-y))) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.5e+115], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[y], $MachinePrecision] - N[Exp[(-y)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.49999999999999963e115

   1. Initial program 86.7%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {x}^{2}\right) \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + \color{blue}{\frac{1}{2}} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      4. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      5. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      7. lift-sinh.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 68.5

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   4. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)} \]

   if 4.49999999999999963e115 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 26.7

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   4. Applied rewrites 26.7%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. sinh-undef-rev N/A

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

      4. rec-exp N/A

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

      5. lower--.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. rec-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 57.6

         \[\leadsto \left(e^{y} - e^{-y}\right) \cdot 0.5 \]

   6. Applied rewrites 57.6%

      \[\leadsto \left(e^{y} - e^{-y}\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 73.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+115}:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(\left(y \cdot y\right) \cdot y\right) \cdot 0.16666666666666666\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.5e+115)
   (* (* 2.0 (sinh y)) (fma (* x x) -0.08333333333333333 0.5))
   (/ (* x (* (* (* y y) y) 0.16666666666666666)) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.5e+115) {
		tmp = (2.0 * sinh(y)) * fma((x * x), -0.08333333333333333, 0.5);
	} else {
		tmp = (x * (((y * y) * y) * 0.16666666666666666)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.5e+115)
		tmp = Float64(Float64(2.0 * sinh(y)) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	else
		tmp = Float64(Float64(x * Float64(Float64(Float64(y * y) * y) * 0.16666666666666666)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.5e+115], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.49999999999999963e115

   1. Initial program 86.7%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {x}^{2}\right) \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + \color{blue}{\frac{1}{2}} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      4. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      5. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      7. lift-sinh.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 68.5

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   4. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)} \]

   if 4.49999999999999963e115 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 26.7%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 24.5

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

   7. Applied rewrites 24.5%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 51.4

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

   10. Applied rewrites 51.4%

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

Alternative 8: 67.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \sinh y\\ \mathbf{if}\;x \leq 16:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+115}:\\ \;\;\;\;t\_0 \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(\left(y \cdot y\right) \cdot y\right) \cdot 0.16666666666666666\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 2.0 (sinh y))))
   (if (<= x 16.0)
     (* t_0 0.5)
     (if (<= x 4.5e+115)
       (* t_0 (* (* x x) -0.08333333333333333))
       (/ (* x (* (* (* y y) y) 0.16666666666666666)) x)))))

C

double code(double x, double y) {
	double t_0 = 2.0 * sinh(y);
	double tmp;
	if (x <= 16.0) {
		tmp = t_0 * 0.5;
	} else if (x <= 4.5e+115) {
		tmp = t_0 * ((x * x) * -0.08333333333333333);
	} else {
		tmp = (x * (((y * y) * y) * 0.16666666666666666)) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * sinh(y)
    if (x <= 16.0d0) then
        tmp = t_0 * 0.5d0
    else if (x <= 4.5d+115) then
        tmp = t_0 * ((x * x) * (-0.08333333333333333d0))
    else
        tmp = (x * (((y * y) * y) * 0.16666666666666666d0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 2.0 * Math.sinh(y);
	double tmp;
	if (x <= 16.0) {
		tmp = t_0 * 0.5;
	} else if (x <= 4.5e+115) {
		tmp = t_0 * ((x * x) * -0.08333333333333333);
	} else {
		tmp = (x * (((y * y) * y) * 0.16666666666666666)) / x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 2.0 * math.sinh(y)
	tmp = 0
	if x <= 16.0:
		tmp = t_0 * 0.5
	elif x <= 4.5e+115:
		tmp = t_0 * ((x * x) * -0.08333333333333333)
	else:
		tmp = (x * (((y * y) * y) * 0.16666666666666666)) / x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(2.0 * sinh(y))
	tmp = 0.0
	if (x <= 16.0)
		tmp = Float64(t_0 * 0.5);
	elseif (x <= 4.5e+115)
		tmp = Float64(t_0 * Float64(Float64(x * x) * -0.08333333333333333));
	else
		tmp = Float64(Float64(x * Float64(Float64(Float64(y * y) * y) * 0.16666666666666666)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 2.0 * sinh(y);
	tmp = 0.0;
	if (x <= 16.0)
		tmp = t_0 * 0.5;
	elseif (x <= 4.5e+115)
		tmp = t_0 * ((x * x) * -0.08333333333333333);
	else
		tmp = (x * (((y * y) * y) * 0.16666666666666666)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 16.0], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[x, 4.5e+115], N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 16

   1. Initial program 85.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 74.6

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   4. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   if 16 < x < 4.49999999999999963e115

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {x}^{2}\right) \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + \color{blue}{\frac{1}{2}} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      4. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      5. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      7. lift-sinh.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 24.2

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   4. Applied rewrites 24.2%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)} \]

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12}\right) \]

      3. pow2 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{12}\right) \]

      4. lift-*.f64 24.2

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333\right) \]

   7. Applied rewrites 24.2%

      \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{-0.08333333333333333}\right) \]

   if 4.49999999999999963e115 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 26.7%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 24.5

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

   7. Applied rewrites 24.5%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 51.4

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

   10. Applied rewrites 51.4%

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

Alternative 9: 66.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+46}:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+115}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(\left(y \cdot y\right) \cdot y\right) \cdot 0.16666666666666666\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4e+46)
   (* (* 2.0 (sinh y)) 0.5)
   (if (<= x 4.5e+115)
     (*
      (* (fma 0.3333333333333333 (* y y) 2.0) y)
      (* (* x x) -0.08333333333333333))
     (/ (* x (* (* (* y y) y) 0.16666666666666666)) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4e+46) {
		tmp = (2.0 * sinh(y)) * 0.5;
	} else if (x <= 4.5e+115) {
		tmp = (fma(0.3333333333333333, (y * y), 2.0) * y) * ((x * x) * -0.08333333333333333);
	} else {
		tmp = (x * (((y * y) * y) * 0.16666666666666666)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4e+46)
		tmp = Float64(Float64(2.0 * sinh(y)) * 0.5);
	elseif (x <= 4.5e+115)
		tmp = Float64(Float64(fma(0.3333333333333333, Float64(y * y), 2.0) * y) * Float64(Float64(x * x) * -0.08333333333333333));
	else
		tmp = Float64(Float64(x * Float64(Float64(Float64(y * y) * y) * 0.16666666666666666)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4e+46], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 4.5e+115], N[(N[(N[(0.3333333333333333 * N[(y * y), $MachinePrecision] + 2.0), $MachinePrecision] * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4e46

   1. Initial program 85.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 72.4

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   4. Applied rewrites 72.4%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   if 4e46 < x < 4.49999999999999963e115

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {x}^{2}\right) \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + \color{blue}{\frac{1}{2}} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      4. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      5. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      7. lift-sinh.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 23.8

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   4. Applied rewrites 23.8%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)} \]

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12}\right) \]

      3. pow2 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{12}\right) \]

      4. lift-*.f64 23.8

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333\right) \]

   7. Applied rewrites 23.8%

      \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{-0.08333333333333333}\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + \frac{1}{3} \cdot {y}^{2}\right)\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{12}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(2 + \frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \left(\left(x \cdot \color{blue}{x}\right) \cdot \frac{-1}{12}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(2 + \frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \left(\left(x \cdot \color{blue}{x}\right) \cdot \frac{-1}{12}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{1}{3} \cdot {y}^{2} + 2\right) \cdot y\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{12}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3}, {y}^{2}, 2\right) \cdot y\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{12}\right) \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3}, y \cdot y, 2\right) \cdot y\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{12}\right) \]

      6. lift-*.f64 21.4

         \[\leadsto \left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333\right) \]

   10. Applied rewrites 21.4%

       \[\leadsto \left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot -0.08333333333333333\right) \]

   if 4.49999999999999963e115 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 26.7%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 24.5

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

   7. Applied rewrites 24.5%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 51.4

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

   10. Applied rewrites 51.4%

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

Alternative 10: 66.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+162}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.1e+162)
   (*
    (* (fma 0.3333333333333333 (* y y) 2.0) y)
    (fma (* x x) -0.08333333333333333 0.5))
   (* x (/ (sinh y) x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.1e+162) {
		tmp = (fma(0.3333333333333333, (y * y), 2.0) * y) * fma((x * x), -0.08333333333333333, 0.5);
	} else {
		tmp = x * (sinh(y) / x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.1e+162)
		tmp = Float64(Float64(fma(0.3333333333333333, Float64(y * y), 2.0) * y) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	else
		tmp = Float64(x * Float64(sinh(y) / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.1e+162], N[(N[(N[(0.3333333333333333 * N[(y * y), $MachinePrecision] + 2.0), $MachinePrecision] * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1000000000000001e162

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {x}^{2}\right) \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + \color{blue}{\frac{1}{2}} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      4. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      5. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      7. lift-sinh.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 75.2

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   4. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)} \]

   5. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(2 + \frac{1}{3} \cdot {y}^{2}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{12}, \frac{1}{2}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(2 + \frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot \color{blue}{x}, \frac{-1}{12}, \frac{1}{2}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(2 + \frac{1}{3} \cdot {y}^{2}\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot \color{blue}{x}, \frac{-1}{12}, \frac{1}{2}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(\frac{1}{3} \cdot {y}^{2} + 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3}, {y}^{2}, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{1}{3}, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      6. lower-*.f64 75.2

         \[\leadsto \left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   7. Applied rewrites 75.2%

      \[\leadsto \left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.08333333333333333, 0.5\right) \]

   if -1.1000000000000001e162 < y

   1. Initial program 87.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 48.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 N/A

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

      7. lift-/.f64 73.6

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

   6. Applied rewrites 73.6%

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

Alternative 11: 66.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 550000:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(\left(y \cdot y\right) \cdot y\right) \cdot 0.16666666666666666\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 550000.0)
   (* (* 2.0 (sinh y)) 0.5)
   (/ (* x (* (* (* y y) y) 0.16666666666666666)) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 550000.0) {
		tmp = (2.0 * sinh(y)) * 0.5;
	} else {
		tmp = (x * (((y * y) * y) * 0.16666666666666666)) / x;
	}
	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 (x <= 550000.0d0) then
        tmp = (2.0d0 * sinh(y)) * 0.5d0
    else
        tmp = (x * (((y * y) * y) * 0.16666666666666666d0)) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 550000.0) {
		tmp = (2.0 * Math.sinh(y)) * 0.5;
	} else {
		tmp = (x * (((y * y) * y) * 0.16666666666666666)) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 550000.0:
		tmp = (2.0 * math.sinh(y)) * 0.5
	else:
		tmp = (x * (((y * y) * y) * 0.16666666666666666)) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 550000.0)
		tmp = Float64(Float64(2.0 * sinh(y)) * 0.5);
	else
		tmp = Float64(Float64(x * Float64(Float64(Float64(y * y) * y) * 0.16666666666666666)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 550000.0)
		tmp = (2.0 * sinh(y)) * 0.5;
	else
		tmp = (x * (((y * y) * y) * 0.16666666666666666)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 550000.0], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(x * N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5e5

   1. Initial program 85.1%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 74.4

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   4. Applied rewrites 74.4%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   if 5.5e5 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 28.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 24.1

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

   7. Applied rewrites 24.1%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 44.2

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

   10. Applied rewrites 44.2%

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

Alternative 12: 66.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 52.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sinh.f64 N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-sinh.f64 N/A

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

   7. lift-/.f64 73.8

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

6. Applied rewrites 73.8%

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

Alternative 13: 65.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 550000:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(\left(y \cdot y\right) \cdot y\right) \cdot 0.16666666666666666\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 550000.0)
   (* x (/ (* (fma (* y y) 0.16666666666666666 1.0) y) x))
   (/ (* x (* (* (* y y) y) 0.16666666666666666)) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 550000.0) {
		tmp = x * ((fma((y * y), 0.16666666666666666, 1.0) * y) / x);
	} else {
		tmp = (x * (((y * y) * y) * 0.16666666666666666)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 550000.0)
		tmp = Float64(x * Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y) / x));
	else
		tmp = Float64(Float64(x * Float64(Float64(Float64(y * y) * y) * 0.16666666666666666)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 550000.0], N[(x * N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5e5

   1. Initial program 85.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 59.6%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 48.3

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

   7. Applied rewrites 48.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 74.1

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

   9. Applied rewrites 74.1%

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

   if 5.5e5 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 28.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 24.1

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

   7. Applied rewrites 24.1%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 44.2

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

   10. Applied rewrites 44.2%

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

Alternative 14: 61.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 550000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(\left(y \cdot y\right) \cdot y\right) \cdot 0.16666666666666666\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 550000.0)
   (* (fma (* y y) 0.16666666666666666 1.0) y)
   (/ (* x (* (* (* y y) y) 0.16666666666666666)) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 550000.0) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * y;
	} else {
		tmp = (x * (((y * y) * y) * 0.16666666666666666)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 550000.0)
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y);
	else
		tmp = Float64(Float64(x * Float64(Float64(Float64(y * y) * y) * 0.16666666666666666)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 550000.0], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(x * N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.5e5

   1. Initial program 85.1%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 74.4

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   4. Applied rewrites 74.4%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 61.7

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

   7. Applied rewrites 61.7%

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

   if 5.5e5 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 28.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 24.1

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

   7. Applied rewrites 24.1%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 44.2

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

   10. Applied rewrites 44.2%

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

Alternative 15: 60.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{-143}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-130}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.16666666666666666 \cdot y, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (* y y) 0.16666666666666666 1.0) y)))
   (if (<= y -6.2e-143)
     t_0
     (if (<= y 2e-130)
       (* x (/ y x))
       (if (<= y 1.35e+82) (fma (* x x) (* -0.16666666666666666 y) y) t_0)))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), 0.16666666666666666, 1.0) * y;
	double tmp;
	if (y <= -6.2e-143) {
		tmp = t_0;
	} else if (y <= 2e-130) {
		tmp = x * (y / x);
	} else if (y <= 1.35e+82) {
		tmp = fma((x * x), (-0.16666666666666666 * y), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y)
	tmp = 0.0
	if (y <= -6.2e-143)
		tmp = t_0;
	elseif (y <= 2e-130)
		tmp = Float64(x * Float64(y / x));
	elseif (y <= 1.35e+82)
		tmp = fma(Float64(x * x), Float64(-0.16666666666666666 * y), y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -6.2e-143], t$95$0, If[LessEqual[y, 2e-130], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+82], N[(N[(x * x), $MachinePrecision] * N[(-0.16666666666666666 * y), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.20000000000000015e-143 or 1.35e82 < y

   1. Initial program 98.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 69.3

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   4. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 56.8

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

   7. Applied rewrites 56.8%

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

   if -6.20000000000000015e-143 < y < 2.0000000000000002e-130

   1. Initial program 67.7%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 21.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 21.0

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

   7. Applied rewrites 21.0%

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

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot y}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 21.0%

       \[\leadsto \frac{x \cdot y}{x} \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 83.4

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

   12. Applied rewrites 83.4%

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

   if 2.0000000000000002e-130 < y < 1.35e82

   1. Initial program 94.2%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 62.4

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

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 35.5

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

   7. Applied rewrites 35.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 35.5

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

   9. Applied rewrites 35.5%

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

Alternative 16: 57.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{-143}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+86}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (fma (* y y) 0.16666666666666666 1.0) y)))
   (if (<= y -6.2e-143) t_0 (if (<= y 8.5e+86) (* x (/ y x)) t_0))))

C

double code(double x, double y) {
	double t_0 = fma((y * y), 0.16666666666666666, 1.0) * y;
	double tmp;
	if (y <= -6.2e-143) {
		tmp = t_0;
	} else if (y <= 8.5e+86) {
		tmp = x * (y / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y)
	tmp = 0.0
	if (y <= -6.2e-143)
		tmp = t_0;
	elseif (y <= 8.5e+86)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -6.2e-143], t$95$0, If[LessEqual[y, 8.5e+86], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.20000000000000015e-143 or 8.5000000000000005e86 < y

   1. Initial program 98.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 69.4

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   4. Applied rewrites 69.4%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 57.1

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

   7. Applied rewrites 57.1%

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

   if -6.20000000000000015e-143 < y < 8.5000000000000005e86

   1. Initial program 77.7%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 33.7%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 24.8

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

   7. Applied rewrites 24.8%

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

   8. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot y}{x} \]
   9. Step-by-step derivation

   10. Applied rewrites 23.4%

       \[\leadsto \frac{x \cdot y}{x} \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 66.9

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

   12. Applied rewrites 66.9%

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

Alternative 17: 50.1% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 52.0%

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

5. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 42.5

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

7. Applied rewrites 42.5%

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

8. Taylor expanded in y around 0

   \[\leadsto \frac{x \cdot y}{x} \]
9. Step-by-step derivation

10. Applied rewrites 22.6%

    \[\leadsto \frac{x \cdot y}{x} \]
11. Step-by-step derivation

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-/l* N/A

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

    4. lower-*.f64 N/A

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

    5. lower-/.f64 50.1

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

12. Applied rewrites 50.1%

    \[\leadsto \color{blue}{x \cdot \frac{y}{x}} \]
13. Add Preprocessing

Alternative 18: 28.0% accurate, 51.3× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 y)

C

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

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 88.7%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lift-sin.f64 52.3

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

4. Applied rewrites 52.3%

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

5. Taylor expanded in x around 0

   \[\leadsto y \]
6. Step-by-step derivation

7. Applied rewrites 28.0%

   \[\leadsto y \]
8. Add Preprocessing

Reproduce

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