Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.7% → 99.8%

Time: 3.9s

Alternatives: 18

Speedup: 1.0×

• 49.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

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(sin(x) * Float64(sinh(y) / x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sinh.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-sinh.f64 99.8

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

3. Applied rewrites 99.8%

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

Alternative 2: 74.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sin x \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 (- INFINITY))
     (* (* 2.0 (sinh y)) (* (* x x) -0.08333333333333333))
     (if (<= t_0 2e-104)
       (* (/ (* (sin x) (fma (* y y) 0.16666666666666666 1.0)) x) y)
       (* x (/ (sinh y) x))))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (2.0 * sinh(y)) * ((x * x) * -0.08333333333333333);
	} else if (t_0 <= 2e-104) {
		tmp = ((sin(x) * fma((y * y), 0.16666666666666666, 1.0)) / x) * y;
	} else {
		tmp = x * (sinh(y) / x);
	}
	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(2.0 * sinh(y)) * Float64(Float64(x * x) * -0.08333333333333333));
	elseif (t_0 <= 2e-104)
		tmp = Float64(Float64(Float64(sin(x) * fma(Float64(y * y), 0.16666666666666666, 1.0)) / x) * y);
	else
		tmp = Float64(x * Float64(sinh(y) / x));
	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[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-104], N[(N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[(x * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 75.0

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 24.7

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

   7. Applied rewrites 24.7%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.99999999999999985e-104

   1. Initial program 76.0%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x}{x} \cdot y} \]
   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 99.0

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

   6. Applied rewrites 99.0%

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

   if 1.99999999999999985e-104 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 75.7%

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

Alternative 3: 74.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 (- INFINITY))
     (* (* 2.0 (sinh y)) (* (* x x) -0.08333333333333333))
     (if (<= t_0 2e-104) (* (/ (sin x) x) y) (* x (/ (sinh y) x))))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (2.0 * sinh(y)) * ((x * x) * -0.08333333333333333);
	} else if (t_0 <= 2e-104) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = x * (sinh(y) / x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (Math.sin(x) * Math.sinh(y)) / x;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (2.0 * Math.sinh(y)) * ((x * x) * -0.08333333333333333);
	} else if (t_0 <= 2e-104) {
		tmp = (Math.sin(x) / x) * y;
	} else {
		tmp = x * (Math.sinh(y) / x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sin(x) * math.sinh(y)) / x
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (2.0 * math.sinh(y)) * ((x * x) * -0.08333333333333333)
	elif t_0 <= 2e-104:
		tmp = (math.sin(x) / x) * y
	else:
		tmp = x * (math.sinh(y) / x)
	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(2.0 * sinh(y)) * Float64(Float64(x * x) * -0.08333333333333333));
	elseif (t_0 <= 2e-104)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = Float64(x * Float64(sinh(y) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sin(x) * sinh(y)) / x;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (2.0 * sinh(y)) * ((x * x) * -0.08333333333333333);
	elseif (t_0 <= 2e-104)
		tmp = (sin(x) / x) * y;
	else
		tmp = x * (sinh(y) / x);
	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, (-Infinity)], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-104], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[(x * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 75.0

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 24.7

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

   7. Applied rewrites 24.7%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.99999999999999985e-104

   1. Initial program 76.0%

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

   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 98.6

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

   4. Applied rewrites 98.6%

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

   if 1.99999999999999985e-104 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 75.7%

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

Alternative 4: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -4 \cdot 10^{-211}:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(-0.08333333333333333, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -4e-211)
   (* (* 2.0 (sinh y)) (fma -0.08333333333333333 (* x x) 0.5))
   (* x (/ (sinh y) x))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -4e-211) {
		tmp = (2.0 * sinh(y)) * fma(-0.08333333333333333, (x * x), 0.5);
	} else {
		tmp = x * (sinh(y) / x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -4e-211)
		tmp = Float64(Float64(2.0 * sinh(y)) * fma(-0.08333333333333333, Float64(x * x), 0.5));
	else
		tmp = Float64(x * Float64(sinh(y) / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -4e-211], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(-0.08333333333333333 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.00000000000000034e-211

   1. Initial program 99.6%

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

   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. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 71.9

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

   4. Applied rewrites 71.9%

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

   if -4.00000000000000034e-211 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 74.7%

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

Alternative 5: 69.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -4 \cdot 10^{-211}:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -4e-211)
   (* (* 2.0 (sinh y)) (* (* x x) -0.08333333333333333))
   (* x (/ (sinh y) x))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -4e-211) {
		tmp = (2.0 * sinh(y)) * ((x * x) * -0.08333333333333333);
	} else {
		tmp = x * (sinh(y) / x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((sin(x) * sinh(y)) / x) <= (-4d-211)) then
        tmp = (2.0d0 * sinh(y)) * ((x * x) * (-0.08333333333333333d0))
    else
        tmp = x * (sinh(y) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((Math.sin(x) * Math.sinh(y)) / x) <= -4e-211) {
		tmp = (2.0 * Math.sinh(y)) * ((x * x) * -0.08333333333333333);
	} else {
		tmp = x * (Math.sinh(y) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((math.sin(x) * math.sinh(y)) / x) <= -4e-211:
		tmp = (2.0 * math.sinh(y)) * ((x * x) * -0.08333333333333333)
	else:
		tmp = x * (math.sinh(y) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -4e-211)
		tmp = Float64(Float64(2.0 * sinh(y)) * Float64(Float64(x * x) * -0.08333333333333333));
	else
		tmp = Float64(x * Float64(sinh(y) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((sin(x) * sinh(y)) / x) <= -4e-211)
		tmp = (2.0 * sinh(y)) * ((x * x) * -0.08333333333333333);
	else
		tmp = x * (sinh(y) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -4e-211], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.00000000000000034e-211

   1. Initial program 99.6%

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

   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. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 71.9

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

   4. Applied rewrites 71.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 18.9

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

   7. Applied rewrites 18.9%

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

   if -4.00000000000000034e-211 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 74.7%

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

Alternative 6: 56.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -4 \cdot 10^{-211}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \mathsf{fma}\left(-0.08333333333333333, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -4e-211)
   (*
    (* (fma 0.3333333333333333 (* y y) 2.0) y)
    (fma -0.08333333333333333 (* x x) 0.5))
   (* x (/ (sinh y) x))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -4e-211) {
		tmp = (fma(0.3333333333333333, (y * y), 2.0) * y) * fma(-0.08333333333333333, (x * x), 0.5);
	} else {
		tmp = x * (sinh(y) / x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -4e-211)
		tmp = Float64(Float64(fma(0.3333333333333333, Float64(y * y), 2.0) * y) * fma(-0.08333333333333333, Float64(x * x), 0.5));
	else
		tmp = Float64(x * Float64(sinh(y) / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -4e-211], N[(N[(N[(0.3333333333333333 * N[(y * y), $MachinePrecision] + 2.0), $MachinePrecision] * y), $MachinePrecision] * N[(-0.08333333333333333 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.00000000000000034e-211

   1. Initial program 99.6%

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

   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. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 71.9

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

   4. Applied rewrites 71.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 59.1

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

   7. Applied rewrites 59.1%

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

   if -4.00000000000000034e-211 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 74.7%

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

Alternative 7: 56.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -4 \cdot 10^{-211}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.3333333333333333, y \cdot y, 2\right) \cdot y\right) \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -4e-211)
   (*
    (* (fma 0.3333333333333333 (* y y) 2.0) y)
    (* (* x x) -0.08333333333333333))
   (* x (/ (sinh y) x))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -4e-211) {
		tmp = (fma(0.3333333333333333, (y * y), 2.0) * y) * ((x * x) * -0.08333333333333333);
	} else {
		tmp = x * (sinh(y) / x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -4e-211)
		tmp = Float64(Float64(fma(0.3333333333333333, Float64(y * y), 2.0) * y) * Float64(Float64(x * x) * -0.08333333333333333));
	else
		tmp = Float64(x * Float64(sinh(y) / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -4e-211], N[(N[(N[(0.3333333333333333 * N[(y * y), $MachinePrecision] + 2.0), $MachinePrecision] * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.00000000000000034e-211

   1. Initial program 99.6%

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

   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. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 71.9

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

   4. Applied rewrites 71.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 59.1

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

   7. Applied rewrites 59.1%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 17.7

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

   10. Applied rewrites 17.7%

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

   if -4.00000000000000034e-211 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 74.7%

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

Alternative 8: 55.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -5 \cdot 10^{-235}:\\ \;\;\;\;\left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -5e-235)
   (* (* (* (* y y) 0.3333333333333333) y) (* (* x x) -0.08333333333333333))
   (* x (/ (sinh y) x))))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -5e-235) {
		tmp = (((y * y) * 0.3333333333333333) * y) * ((x * x) * -0.08333333333333333);
	} else {
		tmp = x * (sinh(y) / x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((sin(x) * sinh(y)) / x) <= (-5d-235)) then
        tmp = (((y * y) * 0.3333333333333333d0) * y) * ((x * x) * (-0.08333333333333333d0))
    else
        tmp = x * (sinh(y) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((Math.sin(x) * Math.sinh(y)) / x) <= -5e-235) {
		tmp = (((y * y) * 0.3333333333333333) * y) * ((x * x) * -0.08333333333333333);
	} else {
		tmp = x * (Math.sinh(y) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((math.sin(x) * math.sinh(y)) / x) <= -5e-235:
		tmp = (((y * y) * 0.3333333333333333) * y) * ((x * x) * -0.08333333333333333)
	else:
		tmp = x * (math.sinh(y) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -5e-235)
		tmp = Float64(Float64(Float64(Float64(y * y) * 0.3333333333333333) * y) * Float64(Float64(x * x) * -0.08333333333333333));
	else
		tmp = Float64(x * Float64(sinh(y) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((sin(x) * sinh(y)) / x) <= -5e-235)
		tmp = (((y * y) * 0.3333333333333333) * y) * ((x * x) * -0.08333333333333333);
	else
		tmp = x * (sinh(y) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -5e-235], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.9999999999999998e-235

   1. Initial program 99.5%

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

   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. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 70.9

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

   4. Applied rewrites 70.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 58.3

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

   7. Applied rewrites 58.3%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 42.5

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

   10. Applied rewrites 42.5%

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

   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 17.5

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

   13. Applied rewrites 17.5%

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

   if -4.9999999999999998e-235 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 75.2%

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

Alternative 9: 55.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) 0.0)
   (* (* (fma -0.16666666666666666 (* x x) 1.0) x) (/ y x))
   (* (* 2.0 (sinh y)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= 0.0) {
		tmp = (fma(-0.16666666666666666, (x * x), 1.0) * x) * (y / x);
	} else {
		tmp = (2.0 * sinh(y)) * 0.5;
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0], N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 82.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 61.0

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

   6. Applied rewrites 61.0%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 48.9%

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

   if 0.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.2%

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

   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 66.2

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

   4. Applied rewrites 66.2%

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

Alternative 10: 55.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot x}{x} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) 0.0)
   (* (* (fma -0.16666666666666666 (* x x) 1.0) x) (/ y x))
   (* (/ (* (fma (* y y) 0.16666666666666666 1.0) x) x) y)))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= 0.0) {
		tmp = (fma(-0.16666666666666666, (x * x), 1.0) * x) * (y / x);
	} else {
		tmp = ((fma((y * y), 0.16666666666666666, 1.0) * x) / x) * y;
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0], N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 82.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 61.0

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

   6. Applied rewrites 61.0%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 48.9%

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

   if 0.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.2%

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

   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 75.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 52.5%

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

Alternative 11: 50.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -4 \cdot 10^{-211}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot x}{x} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -4e-211)
   (* (/ (* (fma -0.16666666666666666 (* x x) 1.0) x) x) y)
   (* (/ (* (fma (* y y) 0.16666666666666666 1.0) x) x) y)))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -4e-211) {
		tmp = ((fma(-0.16666666666666666, (x * x), 1.0) * x) / x) * y;
	} else {
		tmp = ((fma((y * y), 0.16666666666666666, 1.0) * x) / x) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -4e-211)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x) / x) * y);
	else
		tmp = Float64(Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * x) / x) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -4e-211], N[(N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.00000000000000034e-211

   1. Initial program 99.6%

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

   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 29.1

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

   4. Applied rewrites 29.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 33.0

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

   7. Applied rewrites 33.0%

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

   if -4.00000000000000034e-211 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 83.3%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 85.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 50.4%

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

Alternative 12: 44.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 8.2e+37)
   (* (fma (* y y) 0.16666666666666666 1.0) y)
   (* (* (* (* y y) 0.3333333333333333) y) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 8.2e+37) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * y;
	} else {
		tmp = (((y * y) * 0.3333333333333333) * y) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 8.2e+37)
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y);
	else
		tmp = Float64(Float64(Float64(Float64(y * y) * 0.3333333333333333) * y) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 8.2e+37], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * y), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.1999999999999996e37

   1. Initial program 85.7%

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

   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 81.9%

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

   5. Taylor expanded in 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 60.0

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

   7. Applied rewrites 60.0%

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

   if 8.1999999999999996e37 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 24.3

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

   4. Applied rewrites 24.3%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 23.2

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

   7. Applied rewrites 23.2%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 30.1

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

   10. Applied rewrites 30.1%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 44.3%

       \[\leadsto \left(\left(\left(y \cdot y\right) \cdot 0.3333333333333333\right) \cdot y\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 43.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -4 \cdot 10^{-211}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.16666666666666666 \cdot y, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) -4e-211)
   (fma (* x x) (* -0.16666666666666666 y) y)
   (* (fma (* y y) 0.16666666666666666 1.0) y)))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= -4e-211) {
		tmp = fma((x * x), (-0.16666666666666666 * y), y);
	} else {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= -4e-211)
		tmp = fma(Float64(x * x), Float64(-0.16666666666666666 * y), y);
	else
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -4e-211], N[(N[(x * x), $MachinePrecision] * N[(-0.16666666666666666 * y), $MachinePrecision] + y), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.00000000000000034e-211

   1. Initial program 99.6%

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

   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 29.1

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

   4. Applied rewrites 29.1%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 32.6

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

   7. Applied rewrites 32.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 32.6

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

   9. Applied rewrites 32.6%

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

   if -4.00000000000000034e-211 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 83.3%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 85.6%

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

   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 49.4

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

   7. Applied rewrites 49.4%

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

Alternative 14: 34.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-211}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot y\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;t\_0 \leq 10^{-19}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 -4e-211)
     (* (* (* x x) y) -0.16666666666666666)
     (if (<= t_0 1e-19) y (/ (* x y) x)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -4e-211) {
		tmp = ((x * x) * y) * -0.16666666666666666;
	} else if (t_0 <= 1e-19) {
		tmp = y;
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(x) * sinh(y)) / x
    if (t_0 <= (-4d-211)) then
        tmp = ((x * x) * y) * (-0.16666666666666666d0)
    else if (t_0 <= 1d-19) then
        tmp = y
    else
        tmp = (x * y) / x
    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 <= -4e-211) {
		tmp = ((x * x) * y) * -0.16666666666666666;
	} else if (t_0 <= 1e-19) {
		tmp = y;
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sin(x) * math.sinh(y)) / x
	tmp = 0
	if t_0 <= -4e-211:
		tmp = ((x * x) * y) * -0.16666666666666666
	elif t_0 <= 1e-19:
		tmp = y
	else:
		tmp = (x * y) / x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= -4e-211)
		tmp = Float64(Float64(Float64(x * x) * y) * -0.16666666666666666);
	elseif (t_0 <= 1e-19)
		tmp = y;
	else
		tmp = Float64(Float64(x * y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sin(x) * sinh(y)) / x;
	tmp = 0.0;
	if (t_0 <= -4e-211)
		tmp = ((x * x) * y) * -0.16666666666666666;
	elseif (t_0 <= 1e-19)
		tmp = y;
	else
		tmp = (x * y) / x;
	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, -4e-211], N[(N[(N[(x * x), $MachinePrecision] * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], If[LessEqual[t$95$0, 1e-19], y, N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -4.00000000000000034e-211

   1. Initial program 99.6%

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

   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 29.1

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

   4. Applied rewrites 29.1%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 32.6

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

   7. Applied rewrites 32.6%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 15.3

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

   10. Applied rewrites 15.3%

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

   if -4.00000000000000034e-211 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999999999998e-20

   1. Initial program 73.1%

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

   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 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 47.9%

      \[\leadsto y \]

   if 9.9999999999999998e-20 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   2. Taylor expanded in 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 6.6

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

   4. Applied rewrites 6.6%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 24.0

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

   7. Applied rewrites 24.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 24.9

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 24.9

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

   9. Applied rewrites 24.9%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 17.0%

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

Alternative 15: 34.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(y \cdot x\right), -0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) 0.0)
   (fma (* x (* y x)) -0.16666666666666666 y)
   (/ (* x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= 0.0) {
		tmp = fma((x * (y * x)), -0.16666666666666666, y);
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0], N[(N[(x * N[(y * x), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + y), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 82.0%

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

   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 61.5

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

   4. Applied rewrites 61.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 38.4

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

   7. Applied rewrites 38.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 38.4

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

   9. Applied rewrites 38.4%

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

   if 0.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.2%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 37.9

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

   4. Applied rewrites 37.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 32.3

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

   7. Applied rewrites 32.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 32.3

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 32.3

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

   9. Applied rewrites 32.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 27.3%

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

Alternative 16: 30.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.16666666666666666 \cdot y, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) 0.0)
   (fma (* x x) (* -0.16666666666666666 y) y)
   (/ (* x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= 0.0) {
		tmp = fma((x * x), (-0.16666666666666666 * y), y);
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 0.0], N[(N[(x * x), $MachinePrecision] * N[(-0.16666666666666666 * y), $MachinePrecision] + y), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 82.0%

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

   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 61.5

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

   4. Applied rewrites 61.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 38.4

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

   7. Applied rewrites 38.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 38.4

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

   9. Applied rewrites 38.4%

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

   if 0.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.2%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 37.9

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

   4. Applied rewrites 37.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 32.3

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

   7. Applied rewrites 32.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 32.3

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 32.3

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

   9. Applied rewrites 32.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 27.3%

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

Alternative 17: 29.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq 10^{-19}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (sin x) (sinh y)) x) 1e-19) y (/ (* x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (((sin(x) * sinh(y)) / x) <= 1e-19) {
		tmp = y;
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((sin(x) * sinh(y)) / x) <= 1d-19) then
        tmp = y
    else
        tmp = (x * y) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((Math.sin(x) * Math.sinh(y)) / x) <= 1e-19) {
		tmp = y;
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((math.sin(x) * math.sinh(y)) / x) <= 1e-19:
		tmp = y
	else:
		tmp = (x * y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y)) / x) <= 1e-19)
		tmp = y;
	else
		tmp = Float64(Float64(x * y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((sin(x) * sinh(y)) / x) <= 1e-19)
		tmp = y;
	else
		tmp = (x * y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 1e-19], y, N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 9.9999999999999998e-20

   1. Initial program 84.9%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 68.1

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

   4. Applied rewrites 68.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 35.4%

      \[\leadsto y \]

   if 9.9999999999999998e-20 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

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

   2. Taylor expanded in 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 6.6

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

   4. Applied rewrites 6.6%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 24.0

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

   7. Applied rewrites 24.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 24.9

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 24.9

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

   9. Applied rewrites 24.9%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 17.0%

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

Alternative 18: 27.9% accurate, 51.3× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 y)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 88.7%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lift-sin.f64 52.3

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

4. Applied rewrites 52.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 27.9%

   \[\leadsto y \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025110 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64
  (/ (* (sin x) (sinh y)) x))