Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.1% → 99.9%

Time: 3.3s

Alternatives: 12

Speedup: 1.0×

• 49.4% 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: 89.1% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sinh.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-sin.f64 99.9

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

3. Applied rewrites 99.9%

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

Alternative 2: 83.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)}{x} \cdot y\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sin x \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 (- INFINITY))
     (*
      (/
       (*
        (* (* y y) 0.16666666666666666)
        (* (fma -0.16666666666666666 (* x x) 1.0) x))
       x)
      y)
     (if (<= t_0 4e-51)
       (* (/ (* (sin x) (fma (* y y) 0.16666666666666666 1.0)) x) y)
       (* (sinh y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((((y * y) * 0.16666666666666666) * (fma(-0.16666666666666666, (x * x), 1.0) * x)) / x) * y;
	} else if (t_0 <= 4e-51) {
		tmp = ((sin(x) * fma((y * y), 0.16666666666666666, 1.0)) / x) * y;
	} else {
		tmp = sinh(y) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(Float64(y * y) * 0.16666666666666666) * Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x)) / x) * y);
	elseif (t_0 <= 4e-51)
		tmp = Float64(Float64(Float64(sin(x) * fma(Float64(y * y), 0.16666666666666666, 1.0)) / x) * y);
	else
		tmp = Float64(sinh(y) * 1.0);
	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]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 4e-51], N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 89.1%

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

   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 80.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 53.8

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

   7. Applied rewrites 53.8%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 32.0

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

   10. Applied rewrites 32.0%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4e-51

   1. Initial program 89.1%

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

   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 80.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x}{x} \cdot y} \]
   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 80.7

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

   6. Applied rewrites 80.7%

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

   if 4e-51 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 89.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sinh.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sin.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 63.1%

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

Alternative 3: 82.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)}{x} \cdot y\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 (- INFINITY))
     (*
      (/
       (*
        (* (* y y) 0.16666666666666666)
        (* (fma -0.16666666666666666 (* x x) 1.0) x))
       x)
      y)
     (if (<= t_0 4e-51) (* (/ (sin x) x) y) (* (sinh y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((((y * y) * 0.16666666666666666) * (fma(-0.16666666666666666, (x * x), 1.0) * x)) / x) * y;
	} else if (t_0 <= 4e-51) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = sinh(y) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(Float64(y * y) * 0.16666666666666666) * Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x)) / x) * y);
	elseif (t_0 <= 4e-51)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = Float64(sinh(y) * 1.0);
	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]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 4e-51], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 89.1%

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

   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 80.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 53.8

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

   7. Applied rewrites 53.8%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 32.0

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

   10. Applied rewrites 32.0%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4e-51

   1. Initial program 89.1%

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

   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 51.8

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

   4. Applied rewrites 51.8%

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

   if 4e-51 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 89.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sinh.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sin.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 63.1%

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

Alternative 4: 63.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)}{x} \cdot y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-263}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.008333333333333333 - 0.16666666666666666, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -2e-310)
     (*
      (/
       (*
        (* (* y y) 0.16666666666666666)
        (* (fma -0.16666666666666666 (* x x) 1.0) x))
       x)
      y)
     (if (<= t_0 2e-263)
       (* x (/ y x))
       (*
        (sinh y)
        (fma
         (- (* (* x x) 0.008333333333333333) 0.16666666666666666)
         (* x x)
         1.0))))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -2e-310) {
		tmp = ((((y * y) * 0.16666666666666666) * (fma(-0.16666666666666666, (x * x), 1.0) * x)) / x) * y;
	} else if (t_0 <= 2e-263) {
		tmp = x * (y / x);
	} else {
		tmp = sinh(y) * fma((((x * x) * 0.008333333333333333) - 0.16666666666666666), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -2e-310)
		tmp = Float64(Float64(Float64(Float64(Float64(y * y) * 0.16666666666666666) * Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x)) / x) * y);
	elseif (t_0 <= 2e-263)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(sinh(y) * fma(Float64(Float64(Float64(x * x) * 0.008333333333333333) - 0.16666666666666666), Float64(x * x), 1.0));
	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]}, If[LessEqual[t$95$0, -2e-310], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 2e-263], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] * N[(N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.999999999999994e-310

   1. Initial program 89.1%

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

   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 80.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 53.8

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

   7. Applied rewrites 53.8%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 32.0

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

   10. Applied rewrites 32.0%

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

   if -1.999999999999994e-310 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2e-263

   1. Initial program 89.1%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 41.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 22.2%

      \[\leadsto \frac{\color{blue}{x} \cdot y}{x} \]
   8. 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 50.0

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

   9. Applied rewrites 50.0%

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

   if 2e-263 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 89.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sinh.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sin.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 62.4

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 62.4

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

   6. Applied rewrites 62.4%

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

Alternative 5: 63.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)}{x} \cdot y\\ \mathbf{elif}\;t\_0 \leq 10^{-314}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -2e-310)
     (*
      (/
       (*
        (* (* y y) 0.16666666666666666)
        (* (fma -0.16666666666666666 (* x x) 1.0) x))
       x)
      y)
     (if (<= t_0 1e-314) (* x (/ y x)) (* (sinh y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -2e-310) {
		tmp = ((((y * y) * 0.16666666666666666) * (fma(-0.16666666666666666, (x * x), 1.0) * x)) / x) * y;
	} else if (t_0 <= 1e-314) {
		tmp = x * (y / x);
	} else {
		tmp = sinh(y) * 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -2e-310)
		tmp = Float64(Float64(Float64(Float64(Float64(y * y) * 0.16666666666666666) * Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x)) / x) * y);
	elseif (t_0 <= 1e-314)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(sinh(y) * 1.0);
	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]}, If[LessEqual[t$95$0, -2e-310], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 1e-314], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.999999999999994e-310

   1. Initial program 89.1%

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

   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 80.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 53.8

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

   7. Applied rewrites 53.8%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 32.0

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

   10. Applied rewrites 32.0%

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

   if -1.999999999999994e-310 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999996e-315

   1. Initial program 89.1%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 41.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 22.2%

      \[\leadsto \frac{\color{blue}{x} \cdot y}{x} \]
   8. 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 50.0

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

   9. Applied rewrites 50.0%

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

   if 9.9999999996e-315 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 89.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sinh.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sin.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 63.1%

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

Alternative 6: 55.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-314}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -2e-310)
     (* (* (* (* y y) 0.3333333333333333) y) (* (* x x) -0.08333333333333333))
     (if (<= t_0 1e-314) (* x (/ y x)) (* (sinh y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -2e-310) {
		tmp = (((y * y) * 0.3333333333333333) * y) * ((x * x) * -0.08333333333333333);
	} else if (t_0 <= 1e-314) {
		tmp = x * (y / x);
	} else {
		tmp = sinh(y) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(x) * sinh(y)) / x
    if (t_0 <= (-2d-310)) then
        tmp = (((y * y) * 0.3333333333333333d0) * y) * ((x * x) * (-0.08333333333333333d0))
    else if (t_0 <= 1d-314) then
        tmp = x * (y / x)
    else
        tmp = sinh(y) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (Math.sin(x) * Math.sinh(y)) / x;
	double tmp;
	if (t_0 <= -2e-310) {
		tmp = (((y * y) * 0.3333333333333333) * y) * ((x * x) * -0.08333333333333333);
	} else if (t_0 <= 1e-314) {
		tmp = x * (y / x);
	} else {
		tmp = Math.sinh(y) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sin(x) * math.sinh(y)) / x
	tmp = 0
	if t_0 <= -2e-310:
		tmp = (((y * y) * 0.3333333333333333) * y) * ((x * x) * -0.08333333333333333)
	elif t_0 <= 1e-314:
		tmp = x * (y / x)
	else:
		tmp = math.sinh(y) * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -2e-310)
		tmp = Float64(Float64(Float64(Float64(y * y) * 0.3333333333333333) * y) * Float64(Float64(x * x) * -0.08333333333333333));
	elseif (t_0 <= 1e-314)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(sinh(y) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sin(x) * sinh(y)) / x;
	tmp = 0.0;
	if (t_0 <= -2e-310)
		tmp = (((y * y) * 0.3333333333333333) * y) * ((x * x) * -0.08333333333333333);
	elseif (t_0 <= 1e-314)
		tmp = x * (y / x);
	else
		tmp = sinh(y) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-310], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-314], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.999999999999994e-310

   1. Initial program 89.1%

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

   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 62.4

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

   4. Applied rewrites 62.4%

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

   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 53.6

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

   7. Applied rewrites 53.6%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 33.2

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

   10. Applied rewrites 33.2%

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

   11. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 15.9

          \[\leadsto \left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333\right) \]

   13. Applied rewrites 15.9%

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

   if -1.999999999999994e-310 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999996e-315

   1. Initial program 89.1%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 41.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 22.2%

      \[\leadsto \frac{\color{blue}{x} \cdot y}{x} \]
   8. 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 50.0

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

   9. Applied rewrites 50.0%

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

   if 9.9999999996e-315 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 89.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sinh.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sin.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 63.1%

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

Alternative 7: 54.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-210}:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot x\right) \cdot y}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{-314}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -5e-210)
     (/ (* (* (* (* x x) -0.16666666666666666) x) y) x)
     (if (<= t_0 1e-314) (* x (/ y x)) (* (sinh y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -5e-210) {
		tmp = ((((x * x) * -0.16666666666666666) * x) * y) / x;
	} else if (t_0 <= 1e-314) {
		tmp = x * (y / x);
	} else {
		tmp = sinh(y) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(x) * sinh(y)) / x
    if (t_0 <= (-5d-210)) then
        tmp = ((((x * x) * (-0.16666666666666666d0)) * x) * y) / x
    else if (t_0 <= 1d-314) then
        tmp = x * (y / x)
    else
        tmp = sinh(y) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (Math.sin(x) * Math.sinh(y)) / x;
	double tmp;
	if (t_0 <= -5e-210) {
		tmp = ((((x * x) * -0.16666666666666666) * x) * y) / x;
	} else if (t_0 <= 1e-314) {
		tmp = x * (y / x);
	} else {
		tmp = Math.sinh(y) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sin(x) * math.sinh(y)) / x
	tmp = 0
	if t_0 <= -5e-210:
		tmp = ((((x * x) * -0.16666666666666666) * x) * y) / x
	elif t_0 <= 1e-314:
		tmp = x * (y / x)
	else:
		tmp = math.sinh(y) * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -5e-210)
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * -0.16666666666666666) * x) * y) / x);
	elseif (t_0 <= 1e-314)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(sinh(y) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sin(x) * sinh(y)) / x;
	tmp = 0.0;
	if (t_0 <= -5e-210)
		tmp = ((((x * x) * -0.16666666666666666) * x) * y) / x;
	elseif (t_0 <= 1e-314)
		tmp = x * (y / x);
	else
		tmp = sinh(y) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-210], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1e-314], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -5.0000000000000002e-210

   1. Initial program 89.1%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 41.0%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 25.8

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

   7. Applied rewrites 25.8%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 12.5

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

   10. Applied rewrites 12.5%

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

   if -5.0000000000000002e-210 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999996e-315

   1. Initial program 89.1%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 41.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 22.2%

      \[\leadsto \frac{\color{blue}{x} \cdot y}{x} \]
   8. 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 50.0

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

   9. Applied rewrites 50.0%

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

   if 9.9999999996e-315 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 89.1%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sinh.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sin.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 63.1%

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

Alternative 8: 54.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-210}:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot x\right) \cdot y}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-263}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -5e-210)
     (/ (* (* (* (* x x) -0.16666666666666666) x) y) x)
     (if (<= t_0 2e-263)
       (* x (/ y x))
       (* (fma (* y y) 0.16666666666666666 1.0) y)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -5e-210) {
		tmp = ((((x * x) * -0.16666666666666666) * x) * y) / x;
	} else if (t_0 <= 2e-263) {
		tmp = x * (y / x);
	} else {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -5e-210)
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * -0.16666666666666666) * x) * y) / x);
	elseif (t_0 <= 2e-263)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y);
	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]}, If[LessEqual[t$95$0, -5e-210], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 2e-263], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -5.0000000000000002e-210

   1. Initial program 89.1%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 41.0%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 25.8

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

   7. Applied rewrites 25.8%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 12.5

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

   10. Applied rewrites 12.5%

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

   if -5.0000000000000002e-210 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2e-263

   1. Initial program 89.1%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 41.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 22.2%

      \[\leadsto \frac{\color{blue}{x} \cdot y}{x} \]
   8. 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 50.0

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

   9. Applied rewrites 50.0%

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

   if 2e-263 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 89.1%

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

   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 80.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 51.3

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

   7. Applied rewrites 51.3%

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

Alternative 9: 53.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-210}:\\ \;\;\;\;\left(y + y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-263}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -5e-210)
     (* (+ y y) (fma (* x x) -0.08333333333333333 0.5))
     (if (<= t_0 2e-263)
       (* x (/ y x))
       (* (fma (* y y) 0.16666666666666666 1.0) y)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -5e-210) {
		tmp = (y + y) * fma((x * x), -0.08333333333333333, 0.5);
	} else if (t_0 <= 2e-263) {
		tmp = x * (y / x);
	} else {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -5e-210)
		tmp = Float64(Float64(y + y) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	elseif (t_0 <= 2e-263)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y);
	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]}, If[LessEqual[t$95$0, -5e-210], N[(N[(y + y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-263], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -5.0000000000000002e-210

   1. Initial program 89.1%

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

   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 62.4

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

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in y around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 35.6

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

   7. Applied rewrites 35.6%

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

   if -5.0000000000000002e-210 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2e-263

   1. Initial program 89.1%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 41.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 22.2%

      \[\leadsto \frac{\color{blue}{x} \cdot y}{x} \]
   8. 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 50.0

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

   9. Applied rewrites 50.0%

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

   if 2e-263 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 89.1%

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

   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 80.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 51.3

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

   7. Applied rewrites 51.3%

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

Alternative 10: 50.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e+97) (* x (/ y x)) (* (fma (* y y) 0.16666666666666666 1.0) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.1e+97) {
		tmp = x * (y / x);
	} else {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.1e+97)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.1e+97], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.09999999999999981e97

   1. Initial program 89.1%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 41.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 22.2%

      \[\leadsto \frac{\color{blue}{x} \cdot y}{x} \]
   8. 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 50.0

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

   9. Applied rewrites 50.0%

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

   if 3.09999999999999981e97 < y

   1. Initial program 89.1%

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

   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 80.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 51.3

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

   7. Applied rewrites 51.3%

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

Alternative 11: 47.9% 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 89.1%

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

2. Taylor expanded in y around 0

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

4. Applied rewrites 41.0%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 22.2%

   \[\leadsto \frac{\color{blue}{x} \cdot y}{x} \]
8. 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 50.0

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

9. Applied rewrites 50.0%

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

Alternative 12: 27.6% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \left(y + y\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (+ y y) 0.5))

C

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

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 + y) * 0.5d0
end function

Java

public static double code(double x, double y) {
	return (y + y) * 0.5;
}

Python

def code(x, y):
	return (y + y) * 0.5

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (y + y) * 0.5;
end

Wolfram

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

Derivation

1. Initial program 89.1%

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

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 63.1

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

4. Applied rewrites 63.1%

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

5. Taylor expanded in y around 0

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

   1. count-2-rev N/A

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

   2. lower-+.f64 27.6

      \[\leadsto \left(y + y\right) \cdot 0.5 \]

7. Applied rewrites 27.6%

   \[\leadsto \left(y + y\right) \cdot 0.5 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025131 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64
  (/ (* (sin x) (sinh y)) x))