Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.6% → 99.8%

Time: 7.0s

Alternatives: 11

Speedup: 1.0×

• 49.7% 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: 88.6% 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, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y\_m}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(2 \cdot \sinh y\_m\right) \cdot \mathsf{fma}\left(x, x \cdot -0.08333333333333333, 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\sinh y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y_m)) x)))
   (*
    y_s
    (if (<= t_0 (- INFINITY))
      (* (* 2.0 (sinh y_m)) (fma x (* x -0.08333333333333333) 0.5))
      (if (<= t_0 5e-107) (* (/ (sin x) x) y_m) (sinh y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double t_0 = (sin(x) * sinh(y_m)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (2.0 * sinh(y_m)) * fma(x, (x * -0.08333333333333333), 0.5);
	} else if (t_0 <= 5e-107) {
		tmp = (sin(x) / x) * y_m;
	} else {
		tmp = sinh(y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	t_0 = Float64(Float64(sin(x) * sinh(y_m)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(2.0 * sinh(y_m)) * fma(x, Float64(x * -0.08333333333333333), 0.5));
	elseif (t_0 <= 5e-107)
		tmp = Float64(Float64(sin(x) / x) * y_m);
	else
		tmp = sinh(y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(2.0 * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-107], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y$95$m), $MachinePrecision], N[Sinh[y$95$m], $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 88.6%

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

   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. Applied rewrites 63.4%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 63.4

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

   6. Applied rewrites 63.4%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.99999999999999971e-107

   1. Initial program 88.6%

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

   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 51.4

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

   4. Applied rewrites 51.4%

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

   if 4.99999999999999971e-107 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 88.6%

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

   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. metadata-eval N/A

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

      3. mult-flip N/A

         \[\leadsto \frac{e^{y} - \frac{1}{e^{y}}}{\color{blue}{2}} \]

      4. rec-exp N/A

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

      5. sinh-def N/A

         \[\leadsto \sinh y \]

      6. lift-sinh.f64 63.2

         \[\leadsto \sinh y \]

   4. Applied rewrites 63.2%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(\frac{\sinh y\_m}{x} \cdot \sin x\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m) :precision binary64 (* y_s (* (/ (sinh y_m) x) (sin x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	return y_s * ((sinh(y_m) / x) * sin(x));
}

Fortran

y\_m =     private
y\_s =     private
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(y_s, x, y_m)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = y_s * ((sinh(y_m) / x) * sin(x))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	return y_s * ((Math.sinh(y_m) / x) * Math.sin(x));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	return y_s * ((math.sinh(y_m) / x) * math.sin(x))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	return Float64(y_s * Float64(Float64(sinh(y_m) / x) * sin(x)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m)
	tmp = y_s * ((sinh(y_m) / x) * sin(x));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * N[(N[(N[Sinh[y$95$m], $MachinePrecision] / x), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sinh.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. sinh-def N/A

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

   9. rec-exp N/A

      \[\leadsto \frac{\frac{e^{y} - \color{blue}{\frac{1}{e^{y}}}}{2}}{x} \cdot \sin x \]

   10. mult-flip N/A

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

   11. metadata-eval N/A

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

   12. *-commutative N/A

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

   13. lower-/.f64 N/A

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

   14. *-commutative N/A

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

   15. metadata-eval N/A

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

   16. mult-flip N/A

       \[\leadsto \frac{\color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{2}}}{x} \cdot \sin x \]

   17. rec-exp N/A

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

   18. sinh-def N/A

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

   19. lift-sinh.f64 N/A

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

   20. lift-sin.f64 99.8

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

3. Applied rewrites 99.8%

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

Alternative 3: 87.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y\_m}{x} \leq -1 \cdot 10^{-231}:\\ \;\;\;\;\left(2 \cdot \sinh y\_m\right) \cdot \mathsf{fma}\left(x, x \cdot -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{\sinh y\_m}}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* (sin x) (sinh y_m)) x) -1e-231)
    (* (* 2.0 (sinh y_m)) (fma x (* x -0.08333333333333333) 0.5))
    (/ x (/ x (sinh y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (((sin(x) * sinh(y_m)) / x) <= -1e-231) {
		tmp = (2.0 * sinh(y_m)) * fma(x, (x * -0.08333333333333333), 0.5);
	} else {
		tmp = x / (x / sinh(y_m));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y_m)) / x) <= -1e-231)
		tmp = Float64(Float64(2.0 * sinh(y_m)) * fma(x, Float64(x * -0.08333333333333333), 0.5));
	else
		tmp = Float64(x / Float64(x / sinh(y_m)));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1e-231], N[(N[(2.0 * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(x / N[(x / N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999999e-232

   1. Initial program 88.6%

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

   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. Applied rewrites 63.4%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 63.4

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

   6. Applied rewrites 63.4%

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

   if -9.9999999999999999e-232 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 88.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-def N/A

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

      9. rec-exp N/A

         \[\leadsto \frac{\frac{e^{y} - \color{blue}{\frac{1}{e^{y}}}}{2}}{x} \cdot \sin x \]

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. mult-flip N/A

          \[\leadsto \frac{\color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{2}}}{x} \cdot \sin x \]

      17. rec-exp N/A

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

      18. sinh-def N/A

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

      19. lift-sinh.f64 N/A

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

      20. lift-sin.f64 99.8

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

   3. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. division-flip N/A

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

      4. lower-special-/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-special-/ N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sinh.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sinh.f64 N/A

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

      11. lift-/.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 74.1%

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

Alternative 4: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y\_m}{x} \leq -1 \cdot 10^{-231}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.3333333333333333, y\_m \cdot y\_m, 2\right) \cdot y\_m\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{\sinh y\_m}}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* (sin x) (sinh y_m)) x) -1e-231)
    (*
     (* (fma 0.3333333333333333 (* y_m y_m) 2.0) y_m)
     (fma (* x x) -0.08333333333333333 0.5))
    (/ x (/ x (sinh y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (((sin(x) * sinh(y_m)) / x) <= -1e-231) {
		tmp = (fma(0.3333333333333333, (y_m * y_m), 2.0) * y_m) * fma((x * x), -0.08333333333333333, 0.5);
	} else {
		tmp = x / (x / sinh(y_m));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y_m)) / x) <= -1e-231)
		tmp = Float64(Float64(fma(0.3333333333333333, Float64(y_m * y_m), 2.0) * y_m) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	else
		tmp = Float64(x / Float64(x / sinh(y_m)));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1e-231], N[(N[(N[(0.3333333333333333 * N[(y$95$m * y$95$m), $MachinePrecision] + 2.0), $MachinePrecision] * y$95$m), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(x / N[(x / N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999999e-232

   1. Initial program 88.6%

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

   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. Applied rewrites 63.4%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 55.0

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

   7. Applied rewrites 55.0%

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

   if -9.9999999999999999e-232 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 88.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-def N/A

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

      9. rec-exp N/A

         \[\leadsto \frac{\frac{e^{y} - \color{blue}{\frac{1}{e^{y}}}}{2}}{x} \cdot \sin x \]

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. mult-flip N/A

          \[\leadsto \frac{\color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{2}}}{x} \cdot \sin x \]

      17. rec-exp N/A

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

      18. sinh-def N/A

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

      19. lift-sinh.f64 N/A

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

      20. lift-sin.f64 99.8

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

   3. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. division-flip N/A

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

      4. lower-special-/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-special-/ N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sinh.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sinh.f64 N/A

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

      11. lift-/.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 74.1%

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

Alternative 5: 84.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y\_m}{x} \leq -1 \cdot 10^{-231}:\\ \;\;\;\;\frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(y\_m + y\_m\right)\right)\right) \cdot -0.08333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{\sinh y\_m}}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* (sin x) (sinh y_m)) x) -1e-231)
    (/ (* (* (* x x) (* x (+ y_m y_m))) -0.08333333333333333) x)
    (/ x (/ x (sinh y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (((sin(x) * sinh(y_m)) / x) <= -1e-231) {
		tmp = (((x * x) * (x * (y_m + y_m))) * -0.08333333333333333) / x;
	} else {
		tmp = x / (x / sinh(y_m));
	}
	return y_s * tmp;
}

Fortran

y\_m =     private
y\_s =     private
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(y_s, x, y_m)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (((sin(x) * sinh(y_m)) / x) <= (-1d-231)) then
        tmp = (((x * x) * (x * (y_m + y_m))) * (-0.08333333333333333d0)) / x
    else
        tmp = x / (x / sinh(y_m))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	double tmp;
	if (((Math.sin(x) * Math.sinh(y_m)) / x) <= -1e-231) {
		tmp = (((x * x) * (x * (y_m + y_m))) * -0.08333333333333333) / x;
	} else {
		tmp = x / (x / Math.sinh(y_m));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	tmp = 0
	if ((math.sin(x) * math.sinh(y_m)) / x) <= -1e-231:
		tmp = (((x * x) * (x * (y_m + y_m))) * -0.08333333333333333) / x
	else:
		tmp = x / (x / math.sinh(y_m))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y_m)) / x) <= -1e-231)
		tmp = Float64(Float64(Float64(Float64(x * x) * Float64(x * Float64(y_m + y_m))) * -0.08333333333333333) / x);
	else
		tmp = Float64(x / Float64(x / sinh(y_m)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m)
	tmp = 0.0;
	if (((sin(x) * sinh(y_m)) / x) <= -1e-231)
		tmp = (((x * x) * (x * (y_m + y_m))) * -0.08333333333333333) / x;
	else
		tmp = x / (x / sinh(y_m));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1e-231], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.08333333333333333), $MachinePrecision] / x), $MachinePrecision], N[(x / N[(x / N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999999e-232

   1. Initial program 88.6%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 52.2%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow3 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. rec-exp N/A

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

      10. sinh-undef-rev N/A

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

      11. lift-sinh.f64 N/A

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

      12. lift-*.f64 14.9

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

   7. Applied rewrites 14.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 13.0%

       \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(2 \cdot y\right)\right) \cdot -0.08333333333333333}{x} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. pow2 N/A

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

       4. lower-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. lower-*.f64 13.0

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

       10. lift-*.f64 N/A

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

       11. count-2-rev N/A

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

       12. lower-+.f64 13.0

           \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(y + y\right)\right)\right) \cdot -0.08333333333333333}{x} \]

   12. Applied rewrites 13.0%

       \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(y + y\right)\right)\right) \cdot -0.08333333333333333}{x} \]

   if -9.9999999999999999e-232 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 88.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. sinh-def N/A

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

      9. rec-exp N/A

         \[\leadsto \frac{\frac{e^{y} - \color{blue}{\frac{1}{e^{y}}}}{2}}{x} \cdot \sin x \]

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. mult-flip N/A

          \[\leadsto \frac{\color{blue}{\frac{e^{y} - \frac{1}{e^{y}}}{2}}}{x} \cdot \sin x \]

      17. rec-exp N/A

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

      18. sinh-def N/A

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

      19. lift-sinh.f64 N/A

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

      20. lift-sin.f64 99.8

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

   3. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sinh.f64 N/A

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

      3. division-flip N/A

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

      4. lower-special-/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-special-/ N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sinh.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sinh.f64 N/A

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

      11. lift-/.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 74.1%

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

Alternative 6: 74.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y\_m}{x} \leq -1 \cdot 10^{-231}:\\ \;\;\;\;\frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(y\_m + y\_m\right)\right)\right) \cdot -0.08333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\_m\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* (sin x) (sinh y_m)) x) -1e-231)
    (/ (* (* (* x x) (* x (+ y_m y_m))) -0.08333333333333333) x)
    (sinh y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (((sin(x) * sinh(y_m)) / x) <= -1e-231) {
		tmp = (((x * x) * (x * (y_m + y_m))) * -0.08333333333333333) / x;
	} else {
		tmp = sinh(y_m);
	}
	return y_s * tmp;
}

Fortran

y\_m =     private
y\_s =     private
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(y_s, x, y_m)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (((sin(x) * sinh(y_m)) / x) <= (-1d-231)) then
        tmp = (((x * x) * (x * (y_m + y_m))) * (-0.08333333333333333d0)) / x
    else
        tmp = sinh(y_m)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	double tmp;
	if (((Math.sin(x) * Math.sinh(y_m)) / x) <= -1e-231) {
		tmp = (((x * x) * (x * (y_m + y_m))) * -0.08333333333333333) / x;
	} else {
		tmp = Math.sinh(y_m);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	tmp = 0
	if ((math.sin(x) * math.sinh(y_m)) / x) <= -1e-231:
		tmp = (((x * x) * (x * (y_m + y_m))) * -0.08333333333333333) / x
	else:
		tmp = math.sinh(y_m)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y_m)) / x) <= -1e-231)
		tmp = Float64(Float64(Float64(Float64(x * x) * Float64(x * Float64(y_m + y_m))) * -0.08333333333333333) / x);
	else
		tmp = sinh(y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m)
	tmp = 0.0;
	if (((sin(x) * sinh(y_m)) / x) <= -1e-231)
		tmp = (((x * x) * (x * (y_m + y_m))) * -0.08333333333333333) / x;
	else
		tmp = sinh(y_m);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1e-231], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.08333333333333333), $MachinePrecision] / x), $MachinePrecision], N[Sinh[y$95$m], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999999e-232

   1. Initial program 88.6%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 52.2%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow3 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. rec-exp N/A

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

      10. sinh-undef-rev N/A

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

      11. lift-sinh.f64 N/A

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

      12. lift-*.f64 14.9

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

   7. Applied rewrites 14.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 13.0%

       \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(2 \cdot y\right)\right) \cdot -0.08333333333333333}{x} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. pow2 N/A

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

       4. lower-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. lower-*.f64 13.0

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

       10. lift-*.f64 N/A

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

       11. count-2-rev N/A

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

       12. lower-+.f64 13.0

           \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(y + y\right)\right)\right) \cdot -0.08333333333333333}{x} \]

   12. Applied rewrites 13.0%

       \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(y + y\right)\right)\right) \cdot -0.08333333333333333}{x} \]

   if -9.9999999999999999e-232 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 88.6%

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

   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. metadata-eval N/A

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

      3. mult-flip N/A

         \[\leadsto \frac{e^{y} - \frac{1}{e^{y}}}{\color{blue}{2}} \]

      4. rec-exp N/A

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

      5. sinh-def N/A

         \[\leadsto \sinh y \]

      6. lift-sinh.f64 63.2

         \[\leadsto \sinh y \]

   4. Applied rewrites 63.2%

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

Alternative 7: 74.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y\_m}{x} \leq -1 \cdot 10^{-231}:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot -0.16666666666666666\right) \cdot y\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\_m\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* (sin x) (sinh y_m)) x) -1e-231)
    (/ (* (* (* (* x x) x) -0.16666666666666666) y_m) x)
    (sinh y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (((sin(x) * sinh(y_m)) / x) <= -1e-231) {
		tmp = ((((x * x) * x) * -0.16666666666666666) * y_m) / x;
	} else {
		tmp = sinh(y_m);
	}
	return y_s * tmp;
}

Fortran

y\_m =     private
y\_s =     private
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(y_s, x, y_m)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (((sin(x) * sinh(y_m)) / x) <= (-1d-231)) then
        tmp = ((((x * x) * x) * (-0.16666666666666666d0)) * y_m) / x
    else
        tmp = sinh(y_m)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	double tmp;
	if (((Math.sin(x) * Math.sinh(y_m)) / x) <= -1e-231) {
		tmp = ((((x * x) * x) * -0.16666666666666666) * y_m) / x;
	} else {
		tmp = Math.sinh(y_m);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	tmp = 0
	if ((math.sin(x) * math.sinh(y_m)) / x) <= -1e-231:
		tmp = ((((x * x) * x) * -0.16666666666666666) * y_m) / x
	else:
		tmp = math.sinh(y_m)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y_m)) / x) <= -1e-231)
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * x) * -0.16666666666666666) * y_m) / x);
	else
		tmp = sinh(y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m)
	tmp = 0.0;
	if (((sin(x) * sinh(y_m)) / x) <= -1e-231)
		tmp = ((((x * x) * x) * -0.16666666666666666) * y_m) / x;
	else
		tmp = sinh(y_m);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1e-231], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * y$95$m), $MachinePrecision] / x), $MachinePrecision], N[Sinh[y$95$m], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999999e-232

   1. Initial program 88.6%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 52.2%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow3 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. rec-exp N/A

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

      10. sinh-undef-rev N/A

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

      11. lift-sinh.f64 N/A

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

      12. lift-*.f64 14.9

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

   7. Applied rewrites 14.9%

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

   8. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. pow3 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 13.0

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

   10. Applied rewrites 13.0%

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

   if -9.9999999999999999e-232 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 88.6%

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

   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. metadata-eval N/A

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

      3. mult-flip N/A

         \[\leadsto \frac{e^{y} - \frac{1}{e^{y}}}{\color{blue}{2}} \]

      4. rec-exp N/A

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

      5. sinh-def N/A

         \[\leadsto \sinh y \]

      6. lift-sinh.f64 63.2

         \[\leadsto \sinh y \]

   4. Applied rewrites 63.2%

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

Alternative 8: 73.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y\_m}{x} \leq -1 \cdot 10^{-231}:\\ \;\;\;\;\left(y\_m + y\_m\right) \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\sinh y\_m\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* (sin x) (sinh y_m)) x) -1e-231)
    (* (+ y_m y_m) (* (* x x) -0.08333333333333333))
    (sinh y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (((sin(x) * sinh(y_m)) / x) <= -1e-231) {
		tmp = (y_m + y_m) * ((x * x) * -0.08333333333333333);
	} else {
		tmp = sinh(y_m);
	}
	return y_s * tmp;
}

Fortran

y\_m =     private
y\_s =     private
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(y_s, x, y_m)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (((sin(x) * sinh(y_m)) / x) <= (-1d-231)) then
        tmp = (y_m + y_m) * ((x * x) * (-0.08333333333333333d0))
    else
        tmp = sinh(y_m)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	double tmp;
	if (((Math.sin(x) * Math.sinh(y_m)) / x) <= -1e-231) {
		tmp = (y_m + y_m) * ((x * x) * -0.08333333333333333);
	} else {
		tmp = Math.sinh(y_m);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	tmp = 0
	if ((math.sin(x) * math.sinh(y_m)) / x) <= -1e-231:
		tmp = (y_m + y_m) * ((x * x) * -0.08333333333333333)
	else:
		tmp = math.sinh(y_m)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y_m)) / x) <= -1e-231)
		tmp = Float64(Float64(y_m + y_m) * Float64(Float64(x * x) * -0.08333333333333333));
	else
		tmp = sinh(y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m)
	tmp = 0.0;
	if (((sin(x) * sinh(y_m)) / x) <= -1e-231)
		tmp = (y_m + y_m) * ((x * x) * -0.08333333333333333);
	else
		tmp = sinh(y_m);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1e-231], N[(N[(y$95$m + y$95$m), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision], N[Sinh[y$95$m], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999999e-232

   1. Initial program 88.6%

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

   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. Applied rewrites 63.4%

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

   5. Taylor expanded in y around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f64 36.5

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

   7. Applied rewrites 36.5%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 12.3

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

   10. Applied rewrites 12.3%

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

   if -9.9999999999999999e-232 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 88.6%

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

   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. metadata-eval N/A

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

      3. mult-flip N/A

         \[\leadsto \frac{e^{y} - \frac{1}{e^{y}}}{\color{blue}{2}} \]

      4. rec-exp N/A

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

      5. sinh-def N/A

         \[\leadsto \sinh y \]

      6. lift-sinh.f64 63.2

         \[\leadsto \sinh y \]

   4. Applied rewrites 63.2%

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

Alternative 9: 57.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 122000000000:\\ \;\;\;\;\mathsf{fma}\left(y\_m, y\_m \cdot 0.16666666666666666, 1\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y\_m \cdot y\_m\right) \cdot y\_m\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (*
  y_s
  (if (<= x 122000000000.0)
    (* (fma y_m (* y_m 0.16666666666666666) 1.0) y_m)
    (* (* (* y_m y_m) y_m) 0.16666666666666666))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (x <= 122000000000.0) {
		tmp = fma(y_m, (y_m * 0.16666666666666666), 1.0) * y_m;
	} else {
		tmp = ((y_m * y_m) * y_m) * 0.16666666666666666;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (x <= 122000000000.0)
		tmp = Float64(fma(y_m, Float64(y_m * 0.16666666666666666), 1.0) * y_m);
	else
		tmp = Float64(Float64(Float64(y_m * y_m) * y_m) * 0.16666666666666666);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[x, 122000000000.0], N[(N[(y$95$m * N[(y$95$m * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.22e11

   1. Initial program 88.6%

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

   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. metadata-eval N/A

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

      3. mult-flip N/A

         \[\leadsto \frac{e^{y} - \frac{1}{e^{y}}}{\color{blue}{2}} \]

      4. rec-exp N/A

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

      5. sinh-def N/A

         \[\leadsto \sinh y \]

      6. lift-sinh.f64 63.2

         \[\leadsto \sinh y \]

   4. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\sinh y} \]

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 52.2

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

   7. Applied rewrites 52.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 52.2

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

   9. Applied rewrites 52.2%

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

   if 1.22e11 < x

   1. Initial program 88.6%

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

   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. metadata-eval N/A

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

      3. mult-flip N/A

         \[\leadsto \frac{e^{y} - \frac{1}{e^{y}}}{\color{blue}{2}} \]

      4. rec-exp N/A

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

      5. sinh-def N/A

         \[\leadsto \sinh y \]

      6. lift-sinh.f64 63.2

         \[\leadsto \sinh y \]

   4. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\sinh y} \]

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 52.2

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

   7. Applied rewrites 52.2%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 39.3

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

   10. Applied rewrites 39.3%

       \[\leadsto \left(\left(y \cdot y\right) \cdot y\right) \cdot 0.16666666666666666 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 51.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y\_m}{x} \leq 4 \cdot 10^{-7}:\\ \;\;\;\;1 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y\_m \cdot y\_m\right) \cdot y\_m\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (* (sin x) (sinh y_m)) x) 4e-7)
    (* 1.0 y_m)
    (* (* (* y_m y_m) y_m) 0.16666666666666666))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	double tmp;
	if (((sin(x) * sinh(y_m)) / x) <= 4e-7) {
		tmp = 1.0 * y_m;
	} else {
		tmp = ((y_m * y_m) * y_m) * 0.16666666666666666;
	}
	return y_s * tmp;
}

Fortran

y\_m =     private
y\_s =     private
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(y_s, x, y_m)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (((sin(x) * sinh(y_m)) / x) <= 4d-7) then
        tmp = 1.0d0 * y_m
    else
        tmp = ((y_m * y_m) * y_m) * 0.16666666666666666d0
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	double tmp;
	if (((Math.sin(x) * Math.sinh(y_m)) / x) <= 4e-7) {
		tmp = 1.0 * y_m;
	} else {
		tmp = ((y_m * y_m) * y_m) * 0.16666666666666666;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	tmp = 0
	if ((math.sin(x) * math.sinh(y_m)) / x) <= 4e-7:
		tmp = 1.0 * y_m
	else:
		tmp = ((y_m * y_m) * y_m) * 0.16666666666666666
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	tmp = 0.0
	if (Float64(Float64(sin(x) * sinh(y_m)) / x) <= 4e-7)
		tmp = Float64(1.0 * y_m);
	else
		tmp = Float64(Float64(Float64(y_m * y_m) * y_m) * 0.16666666666666666);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m)
	tmp = 0.0;
	if (((sin(x) * sinh(y_m)) / x) <= 4e-7)
		tmp = 1.0 * y_m;
	else
		tmp = ((y_m * y_m) * y_m) * 0.16666666666666666;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], 4e-7], N[(1.0 * y$95$m), $MachinePrecision], N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 3.9999999999999998e-7

   1. Initial program 88.6%

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

   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. metadata-eval N/A

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

      3. mult-flip N/A

         \[\leadsto \frac{e^{y} - \frac{1}{e^{y}}}{\color{blue}{2}} \]

      4. rec-exp N/A

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

      5. sinh-def N/A

         \[\leadsto \sinh y \]

      6. lift-sinh.f64 63.2

         \[\leadsto \sinh y \]

   4. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\sinh y} \]

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 52.2

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

   7. Applied rewrites 52.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 27.8%

       \[\leadsto 1 \cdot y \]

   if 3.9999999999999998e-7 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 88.6%

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

   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. metadata-eval N/A

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

      3. mult-flip N/A

         \[\leadsto \frac{e^{y} - \frac{1}{e^{y}}}{\color{blue}{2}} \]

      4. rec-exp N/A

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

      5. sinh-def N/A

         \[\leadsto \sinh y \]

      6. lift-sinh.f64 63.2

         \[\leadsto \sinh y \]

   4. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\sinh y} \]

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 52.2

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

   7. Applied rewrites 52.2%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 39.3

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

   10. Applied rewrites 39.3%

       \[\leadsto \left(\left(y \cdot y\right) \cdot y\right) \cdot 0.16666666666666666 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 27.8% accurate, 13.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(1 \cdot y\_m\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m) :precision binary64 (* y_s (* 1.0 y_m)))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	return y_s * (1.0 * y_m);
}

Fortran

y\_m =     private
y\_s =     private
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(y_s, x, y_m)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = y_s * (1.0d0 * y_m)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m) {
	return y_s * (1.0 * y_m);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	return y_s * (1.0 * y_m)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	return Float64(y_s * Float64(1.0 * y_m))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m)
	tmp = y_s * (1.0 * y_m);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_] := N[(y$95$s * N[(1.0 * y$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.6%

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

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. metadata-eval N/A

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

   3. mult-flip N/A

      \[\leadsto \frac{e^{y} - \frac{1}{e^{y}}}{\color{blue}{2}} \]

   4. rec-exp N/A

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

   5. sinh-def N/A

      \[\leadsto \sinh y \]

   6. lift-sinh.f64 63.2

      \[\leadsto \sinh y \]

4. Applied rewrites 63.2%

   \[\leadsto \color{blue}{\sinh y} \]

5. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 52.2

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

7. Applied rewrites 52.2%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 27.8%

    \[\leadsto 1 \cdot y \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025132 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64
  (/ (* (sin x) (sinh y)) x))