Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.6% → 99.8%

Time: 3.9s

Alternatives: 14

Speedup: 1.0×

• 49.5% 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

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y\_m}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(\left(y\_m \cdot y\_m\right) \cdot 0.3333333333333333\right) \cdot y\_m\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-87}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \sinh y\_m}{x}\\ \end{array} \end{array} \end{array} \]

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))
      (*
       (* (* (* y_m y_m) 0.3333333333333333) y_m)
       (fma (* x x) -0.08333333333333333 0.5))
      (if (<= t_0 2e-87)
        (* (/ (sin x) x) y_m)
        (/
         (*
          (*
           (fma
            (- (* 0.008333333333333333 (* x x)) 0.16666666666666666)
            (* x x)
            1.0)
           x)
          (sinh y_m))
         x))))))

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 = (((y_m * y_m) * 0.3333333333333333) * y_m) * fma((x * x), -0.08333333333333333, 0.5);
	} else if (t_0 <= 2e-87) {
		tmp = (sin(x) / x) * y_m;
	} else {
		tmp = ((fma(((0.008333333333333333 * (x * x)) - 0.16666666666666666), (x * x), 1.0) * x) * sinh(y_m)) / x;
	}
	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(Float64(Float64(y_m * y_m) * 0.3333333333333333) * y_m) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	elseif (t_0 <= 2e-87)
		tmp = Float64(Float64(sin(x) / x) * y_m);
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(0.008333333333333333 * Float64(x * x)) - 0.16666666666666666), Float64(x * x), 1.0) * x) * sinh(y_m)) / x);
	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[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * y$95$m), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-87], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   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 93.0

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

   7. Applied rewrites 93.0%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 93.0

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

   10. Applied rewrites 93.0%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2.00000000000000004e-87

   1. Initial program 74.0%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 98.9

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

   4. Applied rewrites 98.9%

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

   if 2.00000000000000004e-87 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

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

Alternative 2: 98.1% 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. lower-/.f64 N/A

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

   9. lift-sinh.f64 N/A

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

   10. lift-sin.f64 99.8

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

3. Applied rewrites 99.8%

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

Alternative 3: 86.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y\_m}{x} \leq -1 \cdot 10^{-300}:\\ \;\;\;\;\left(\left(\left(y\_m \cdot y\_m\right) \cdot 0.3333333333333333\right) \cdot y\_m\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2 \cdot \sinh y\_m}{x} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

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-300)
    (*
     (* (* (* y_m y_m) 0.3333333333333333) y_m)
     (fma (* x x) -0.08333333333333333 0.5))
    (* (* (/ (* 2.0 (sinh y_m)) x) 0.5) x))))

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-300) {
		tmp = (((y_m * y_m) * 0.3333333333333333) * y_m) * fma((x * x), -0.08333333333333333, 0.5);
	} else {
		tmp = (((2.0 * sinh(y_m)) / x) * 0.5) * x;
	}
	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-300)
		tmp = Float64(Float64(Float64(Float64(y_m * y_m) * 0.3333333333333333) * y_m) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	else
		tmp = Float64(Float64(Float64(Float64(2.0 * sinh(y_m)) / x) * 0.5) * x);
	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-300], N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * y$95$m), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.00000000000000003e-300

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 68.0

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

   4. Applied rewrites 68.0%

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

   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 63.4

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

   7. Applied rewrites 63.4%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 63.8

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

   10. Applied rewrites 63.8%

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

   if -1.00000000000000003e-300 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 86.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 91.1%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 91.1

         \[\leadsto \left(\frac{2 \cdot \sinh y}{x} \cdot 0.5\right) \cdot x \]

   9. Applied rewrites 91.1%

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

Alternative 4: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y\_m}{x} \leq -1 \cdot 10^{-300}:\\ \;\;\;\;\left(\left(\left(y\_m \cdot y\_m\right) \cdot 0.3333333333333333\right) \cdot y\_m\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y\_m}{x} \cdot x\\ \end{array} \end{array} \]

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-300)
    (*
     (* (* (* y_m y_m) 0.3333333333333333) y_m)
     (fma (* x x) -0.08333333333333333 0.5))
    (* (/ (sinh y_m) x) x))))

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-300) {
		tmp = (((y_m * y_m) * 0.3333333333333333) * y_m) * fma((x * x), -0.08333333333333333, 0.5);
	} else {
		tmp = (sinh(y_m) / x) * x;
	}
	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-300)
		tmp = Float64(Float64(Float64(Float64(y_m * y_m) * 0.3333333333333333) * y_m) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	else
		tmp = Float64(Float64(sinh(y_m) / x) * x);
	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-300], N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * y$95$m), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sinh[y$95$m], $MachinePrecision] / x), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.00000000000000003e-300

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 68.0

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

   4. Applied rewrites 68.0%

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

   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 63.4

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

   7. Applied rewrites 63.4%

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

   8. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 63.8

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

   10. Applied rewrites 63.8%

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

   if -1.00000000000000003e-300 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 86.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 91.1%

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

Alternative 5: 84.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y\_m}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot x\right) \cdot y\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\frac{y\_m + y\_m}{x} \cdot 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sinh y\_m\right) \cdot 0.5\\ \end{array} \end{array} \end{array} \]

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 -2e-213)
      (/ (* (* (* (* x x) -0.16666666666666666) x) y_m) x)
      (if (<= t_0 0.0)
        (* (* (/ (+ y_m y_m) x) 0.5) x)
        (* (* 2.0 (sinh y_m)) 0.5))))))

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 <= -2e-213) {
		tmp = ((((x * x) * -0.16666666666666666) * x) * y_m) / x;
	} else if (t_0 <= 0.0) {
		tmp = (((y_m + y_m) / x) * 0.5) * x;
	} else {
		tmp = (2.0 * sinh(y_m)) * 0.5;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (sin(x) * sinh(y_m)) / x
    if (t_0 <= (-2d-213)) then
        tmp = ((((x * x) * (-0.16666666666666666d0)) * x) * y_m) / x
    else if (t_0 <= 0.0d0) then
        tmp = (((y_m + y_m) / x) * 0.5d0) * x
    else
        tmp = (2.0d0 * sinh(y_m)) * 0.5d0
    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 t_0 = (Math.sin(x) * Math.sinh(y_m)) / x;
	double tmp;
	if (t_0 <= -2e-213) {
		tmp = ((((x * x) * -0.16666666666666666) * x) * y_m) / x;
	} else if (t_0 <= 0.0) {
		tmp = (((y_m + y_m) / x) * 0.5) * x;
	} else {
		tmp = (2.0 * Math.sinh(y_m)) * 0.5;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	t_0 = (math.sin(x) * math.sinh(y_m)) / x
	tmp = 0
	if t_0 <= -2e-213:
		tmp = ((((x * x) * -0.16666666666666666) * x) * y_m) / x
	elif t_0 <= 0.0:
		tmp = (((y_m + y_m) / x) * 0.5) * x
	else:
		tmp = (2.0 * math.sinh(y_m)) * 0.5
	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 <= -2e-213)
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * -0.16666666666666666) * x) * y_m) / x);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(y_m + y_m) / x) * 0.5) * x);
	else
		tmp = Float64(Float64(2.0 * sinh(y_m)) * 0.5);
	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)
	t_0 = (sin(x) * sinh(y_m)) / x;
	tmp = 0.0;
	if (t_0 <= -2e-213)
		tmp = ((((x * x) * -0.16666666666666666) * x) * y_m) / x;
	elseif (t_0 <= 0.0)
		tmp = (((y_m + y_m) / x) * 0.5) * x;
	else
		tmp = (2.0 * sinh(y_m)) * 0.5;
	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_] := 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, -2e-213], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * y$95$m), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(y$95$m + y$95$m), $MachinePrecision] / x), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision], N[(N[(2.0 * N[Sinh[y$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-213

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 81.8

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

   4. Applied rewrites 81.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 68.1%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 68.1

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

   10. Applied rewrites 68.1%

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

   if -1.9999999999999999e-213 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.8%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 84.9

         \[\leadsto \left(\frac{2 \cdot \sinh y}{x} \cdot 0.5\right) \cdot x \]

   9. Applied rewrites 84.9%

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

   10. Taylor expanded in y around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. count-2-rev N/A

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

       4. lift-+.f64 84.9

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

   12. Applied rewrites 84.9%

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

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

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 89.2

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

   4. Applied rewrites 89.2%

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

Alternative 6: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y\_m}{x} \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot x\right) \cdot y\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y\_m}{x} \cdot x\\ \end{array} \end{array} \]

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) -2e-213)
    (/ (* (* (* (* x x) -0.16666666666666666) x) y_m) x)
    (* (/ (sinh y_m) x) x))))

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) <= -2e-213) {
		tmp = ((((x * x) * -0.16666666666666666) * x) * y_m) / x;
	} else {
		tmp = (sinh(y_m) / x) * x;
	}
	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) <= (-2d-213)) then
        tmp = ((((x * x) * (-0.16666666666666666d0)) * x) * y_m) / x
    else
        tmp = (sinh(y_m) / x) * x
    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) <= -2e-213) {
		tmp = ((((x * x) * -0.16666666666666666) * x) * y_m) / x;
	} else {
		tmp = (Math.sinh(y_m) / x) * x;
	}
	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) <= -2e-213:
		tmp = ((((x * x) * -0.16666666666666666) * x) * y_m) / x
	else:
		tmp = (math.sinh(y_m) / x) * x
	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) <= -2e-213)
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * -0.16666666666666666) * x) * y_m) / x);
	else
		tmp = Float64(Float64(sinh(y_m) / x) * x);
	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) <= -2e-213)
		tmp = ((((x * x) * -0.16666666666666666) * x) * y_m) / x;
	else
		tmp = (sinh(y_m) / x) * x;
	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], -2e-213], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * y$95$m), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[Sinh[y$95$m], $MachinePrecision] / x), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-213

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 81.8

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

   4. Applied rewrites 81.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 68.1%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 68.1

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

   10. Applied rewrites 68.1%

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

   if -1.9999999999999999e-213 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 86.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 87.7%

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

Alternative 7: 76.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y\_m}{x} \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot x\right) \cdot y\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(y\_m \cdot y\_m, 0.16666666666666666, 1\right)}{x} \cdot y\_m\right) \cdot x\\ \end{array} \end{array} \]

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) -2e-213)
    (/ (* (* (* (* x x) -0.16666666666666666) x) y_m) x)
    (* (* (/ (fma (* y_m y_m) 0.16666666666666666 1.0) x) y_m) x))))

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) <= -2e-213) {
		tmp = ((((x * x) * -0.16666666666666666) * x) * y_m) / x;
	} else {
		tmp = ((fma((y_m * y_m), 0.16666666666666666, 1.0) / x) * y_m) * x;
	}
	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) <= -2e-213)
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * -0.16666666666666666) * x) * y_m) / x);
	else
		tmp = Float64(Float64(Float64(fma(Float64(y_m * y_m), 0.16666666666666666, 1.0) / x) * y_m) * x);
	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], -2e-213], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * y$95$m), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / x), $MachinePrecision] * y$95$m), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-213

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 81.8

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

   4. Applied rewrites 81.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 68.1%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 68.1

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

   10. Applied rewrites 68.1%

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

   if -1.9999999999999999e-213 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 86.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 87.7%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 78.2

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

   9. Applied rewrites 78.2%

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

Alternative 8: 72.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y\_m}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot x\right) \cdot y\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\frac{y\_m + y\_m}{x} \cdot 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y\_m \cdot y\_m, 0.16666666666666666, 1\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

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 -2e-213)
      (/ (* (* (* (* x x) -0.16666666666666666) x) y_m) x)
      (if (<= t_0 0.0)
        (* (* (/ (+ y_m y_m) x) 0.5) x)
        (* (fma (* y_m y_m) 0.16666666666666666 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) {
	double t_0 = (sin(x) * sinh(y_m)) / x;
	double tmp;
	if (t_0 <= -2e-213) {
		tmp = ((((x * x) * -0.16666666666666666) * x) * y_m) / x;
	} else if (t_0 <= 0.0) {
		tmp = (((y_m + y_m) / x) * 0.5) * x;
	} else {
		tmp = fma((y_m * y_m), 0.16666666666666666, 1.0) * 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 <= -2e-213)
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * -0.16666666666666666) * x) * y_m) / x);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(y_m + y_m) / x) * 0.5) * x);
	else
		tmp = Float64(fma(Float64(y_m * y_m), 0.16666666666666666, 1.0) * 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, -2e-213], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * y$95$m), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(y$95$m + y$95$m), $MachinePrecision] / x), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-213

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 81.8

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

   4. Applied rewrites 81.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 68.1%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 68.1

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

   10. Applied rewrites 68.1%

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

   if -1.9999999999999999e-213 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.8%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 84.9

         \[\leadsto \left(\frac{2 \cdot \sinh y}{x} \cdot 0.5\right) \cdot x \]

   9. Applied rewrites 84.9%

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

   10. Taylor expanded in y around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. count-2-rev N/A

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

       4. lift-+.f64 84.9

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

   12. Applied rewrites 84.9%

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

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

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 89.2

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 68.2

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

   7. Applied rewrites 68.2%

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

Alternative 9: 71.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y\_m}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\frac{y\_m + y\_m}{x} \cdot 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y\_m \cdot y\_m, 0.16666666666666666, 1\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

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 -2e-213)
      (* (* (* x x) -0.16666666666666666) y_m)
      (if (<= t_0 0.0)
        (* (* (/ (+ y_m y_m) x) 0.5) x)
        (* (fma (* y_m y_m) 0.16666666666666666 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) {
	double t_0 = (sin(x) * sinh(y_m)) / x;
	double tmp;
	if (t_0 <= -2e-213) {
		tmp = ((x * x) * -0.16666666666666666) * y_m;
	} else if (t_0 <= 0.0) {
		tmp = (((y_m + y_m) / x) * 0.5) * x;
	} else {
		tmp = fma((y_m * y_m), 0.16666666666666666, 1.0) * 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 <= -2e-213)
		tmp = Float64(Float64(Float64(x * x) * -0.16666666666666666) * y_m);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(y_m + y_m) / x) * 0.5) * x);
	else
		tmp = Float64(fma(Float64(y_m * y_m), 0.16666666666666666, 1.0) * 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, -2e-213], N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(y$95$m + y$95$m), $MachinePrecision] / x), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-213

   1. Initial program 99.8%

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

   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 20.8

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

   4. Applied rewrites 20.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 63.2

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

   7. Applied rewrites 63.2%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 63.2

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

   10. Applied rewrites 63.2%

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

   if -1.9999999999999999e-213 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.8%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. rec-exp N/A

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

      5. sinh-undef N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 84.9

         \[\leadsto \left(\frac{2 \cdot \sinh y}{x} \cdot 0.5\right) \cdot x \]

   9. Applied rewrites 84.9%

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

   10. Taylor expanded in y around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. count-2-rev N/A

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

       4. lift-+.f64 84.9

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

   12. Applied rewrites 84.9%

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

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

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 89.2

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 68.2

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

   7. Applied rewrites 68.2%

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

Alternative 10: 71.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y\_m}{x}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{y\_m}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y\_m \cdot y\_m, 0.16666666666666666, 1\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

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 -2e-213)
      (* (* (* x x) -0.16666666666666666) y_m)
      (if (<= t_0 0.0)
        (* (/ y_m x) x)
        (* (fma (* y_m y_m) 0.16666666666666666 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) {
	double t_0 = (sin(x) * sinh(y_m)) / x;
	double tmp;
	if (t_0 <= -2e-213) {
		tmp = ((x * x) * -0.16666666666666666) * y_m;
	} else if (t_0 <= 0.0) {
		tmp = (y_m / x) * x;
	} else {
		tmp = fma((y_m * y_m), 0.16666666666666666, 1.0) * 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 <= -2e-213)
		tmp = Float64(Float64(Float64(x * x) * -0.16666666666666666) * y_m);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(y_m / x) * x);
	else
		tmp = Float64(fma(Float64(y_m * y_m), 0.16666666666666666, 1.0) * 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, -2e-213], N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(y$95$m / x), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-213

   1. Initial program 99.8%

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

   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 20.8

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

   4. Applied rewrites 20.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 63.2

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

   7. Applied rewrites 63.2%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 63.2

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

   10. Applied rewrites 63.2%

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

   if -1.9999999999999999e-213 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0

   1. Initial program 59.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 84.8%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 84.8%

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

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

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. rec-exp N/A

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

      4. sinh-undef N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sinh.f64 89.2

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 68.2

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

   7. Applied rewrites 68.2%

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

Alternative 11: 60.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y\_m}{x} \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x} \cdot x\\ \end{array} \end{array} \]

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) -2e-213)
    (* (* (* x x) -0.16666666666666666) y_m)
    (* (/ y_m x) x))))

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) <= -2e-213) {
		tmp = ((x * x) * -0.16666666666666666) * y_m;
	} else {
		tmp = (y_m / x) * x;
	}
	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) <= (-2d-213)) then
        tmp = ((x * x) * (-0.16666666666666666d0)) * y_m
    else
        tmp = (y_m / x) * x
    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) <= -2e-213) {
		tmp = ((x * x) * -0.16666666666666666) * y_m;
	} else {
		tmp = (y_m / x) * x;
	}
	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) <= -2e-213:
		tmp = ((x * x) * -0.16666666666666666) * y_m
	else:
		tmp = (y_m / x) * x
	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) <= -2e-213)
		tmp = Float64(Float64(Float64(x * x) * -0.16666666666666666) * y_m);
	else
		tmp = Float64(Float64(y_m / x) * x);
	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) <= -2e-213)
		tmp = ((x * x) * -0.16666666666666666) * y_m;
	else
		tmp = (y_m / x) * x;
	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], -2e-213], N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(y$95$m / x), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-213

   1. Initial program 99.8%

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

   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 20.8

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

   4. Applied rewrites 20.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 63.2

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

   7. Applied rewrites 63.2%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 63.2

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

   10. Applied rewrites 63.2%

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

   if -1.9999999999999999e-213 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 86.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sinh.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sinh.f64 N/A

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

      10. lift-sin.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 87.7%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 60.0%

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

Alternative 12: 51.1% accurate, 7.0× speedup

Math

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

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 (* (/ y_m x) 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 * ((y_m / x) * 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 * ((y_m / x) * 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 * ((y_m / x) * x);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	return y_s * ((y_m / x) * 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(y_m / x) * 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 * ((y_m / x) * 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[(y$95$m / x), $MachinePrecision] * x), $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. lower-/.f64 N/A

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

   9. lift-sinh.f64 N/A

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

   10. lift-sin.f64 99.8

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 74.6%

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

7. Taylor expanded in y around 0

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

9. Applied rewrites 51.1%

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

Alternative 13: 28.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(\left(y\_m + y\_m\right) \cdot 0.5\right) \end{array} \]

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 (* (+ y_m y_m) 0.5)))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m) {
	return y_s * ((y_m + y_m) * 0.5);
}

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 * ((y_m + y_m) * 0.5d0)
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 * ((y_m + y_m) * 0.5);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m):
	return y_s * ((y_m + y_m) * 0.5)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	return Float64(y_s * Float64(Float64(y_m + y_m) * 0.5))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m)
	tmp = y_s * ((y_m + y_m) * 0.5);
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[(y$95$m + y$95$m), $MachinePrecision] * 0.5), $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. lower-*.f64 N/A

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

   3. rec-exp N/A

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

   4. sinh-undef N/A

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

   5. lower-*.f64 N/A

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

   6. lift-sinh.f64 64.4

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

4. Applied rewrites 64.4%

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

5. Taylor expanded in y around 0

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

   1. count-2-rev N/A

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

   2. lower-+.f64 28.7

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

7. Applied rewrites 28.7%

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

Alternative 14: 28.7% accurate, 51.3× speedup

Math

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

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 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 * 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 * 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 * 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 * y_m

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m)
	return Float64(y_s * 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 * 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 * y$95$m), $MachinePrecision]

Derivation

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 51.6

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

4. Applied rewrites 51.6%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 28.7%

   \[\leadsto y \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025120 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64
  (/ (* (sin x) (sinh y)) x))