Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

Alternatives: 13

Speedup: 1.0×

• 50.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|y\right| \leq 5 \cdot 10^{-18}:\\ \;\;\;\;\sin x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh \left(\left|y\right|\right) \cdot \sin x}{\left|y\right|}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (if (<= (fabs y) 5e-18)
  (* (sin x) 1.0)
  (/ (* (sinh (fabs y)) (sin x)) (fabs y))))

C

double code(double x, double y) {
	double tmp;
	if (fabs(y) <= 5e-18) {
		tmp = sin(x) * 1.0;
	} else {
		tmp = (sinh(fabs(y)) * sin(x)) / fabs(y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (abs(y) <= 5d-18) then
        tmp = sin(x) * 1.0d0
    else
        tmp = (sinh(abs(y)) * sin(x)) / abs(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.abs(y) <= 5e-18) {
		tmp = Math.sin(x) * 1.0;
	} else {
		tmp = (Math.sinh(Math.abs(y)) * Math.sin(x)) / Math.abs(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.fabs(y) <= 5e-18:
		tmp = math.sin(x) * 1.0
	else:
		tmp = (math.sinh(math.fabs(y)) * math.sin(x)) / math.fabs(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (abs(y) <= 5e-18)
		tmp = Float64(sin(x) * 1.0);
	else
		tmp = Float64(Float64(sinh(abs(y)) * sin(x)) / abs(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (abs(y) <= 5e-18)
		tmp = sin(x) * 1.0;
	else
		tmp = (sinh(abs(y)) * sin(x)) / abs(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Abs[y], $MachinePrecision], 5e-18], N[(N[Sin[x], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Sinh[N[Abs[y], $MachinePrecision]], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000004e-18

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.7%

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

   if 5.0000000000000004e-18 < y

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 89.2%

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

   3. Applied rewrites 89.2%

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

Alternative 2: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\left|x\right|\right)\\ t_1 := \frac{\sinh y}{y}\\ t_2 := t\_0 \cdot t\_1\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right| \cdot \left|x\right|, -0.16666666666666666 \cdot \left|x\right|, \left|x\right|\right) \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq 1:\\ \;\;\;\;t\_0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right| \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (sin (fabs x))) (t_1 (/ (sinh y) y)) (t_2 (* t_0 t_1)))
  (*
   (copysign 1.0 x)
   (if (<= t_2 (- INFINITY))
     (*
      (fma
       (* (fabs x) (fabs x))
       (* -0.16666666666666666 (fabs x))
       (fabs x))
      t_1)
     (if (<= t_2 1.0) (* t_0 1.0) (/ (* (fabs x) (sinh y)) y))))))

C

double code(double x, double y) {
	double t_0 = sin(fabs(x));
	double t_1 = sinh(y) / y;
	double t_2 = t_0 * t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma((fabs(x) * fabs(x)), (-0.16666666666666666 * fabs(x)), fabs(x)) * t_1;
	} else if (t_2 <= 1.0) {
		tmp = t_0 * 1.0;
	} else {
		tmp = (fabs(x) * sinh(y)) / y;
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x, y)
	t_0 = sin(abs(x))
	t_1 = Float64(sinh(y) / y)
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(fma(Float64(abs(x) * abs(x)), Float64(-0.16666666666666666 * abs(x)), abs(x)) * t_1);
	elseif (t_2 <= 1.0)
		tmp = Float64(t_0 * 1.0);
	else
		tmp = Float64(Float64(abs(x) * sinh(y)) / y);
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 1.0], N[(t$95$0 * 1.0), $MachinePrecision], N[(N[(N[Abs[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 62.9%

         \[\leadsto \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right)\right) \cdot \frac{\sinh y}{y} \]

   4. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 62.9%

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

   6. Applied rewrites 62.9%

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

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

      \[\leadsto \sin x \cdot \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.7%

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

   if 1 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 40.1%

         \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. mult-flip N/A

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

      10. lift--.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. rec-exp N/A

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

      15. sinh-def N/A

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

      16. lift-sinh.f64 N/A

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

      17. lower-*.f64 51.6%

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

   6. Applied rewrites 51.6%

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

Alternative 3: 75.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|x\right|\right) \cdot t\_0 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right| \cdot \left|x\right|, -0.16666666666666666 \cdot \left|x\right|, \left|x\right|\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\sinh y \cdot \left|x\right|} \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (/ (sinh y) y)))
  (*
   (copysign 1.0 x)
   (if (<= (* (sin (fabs x)) t_0) 5e-7)
     (*
      (fma
       (* (fabs x) (fabs x))
       (* -0.16666666666666666 (fabs x))
       (fabs x))
      t_0)
     (/ 1.0 (* (/ 1.0 (* (sinh y) (fabs x))) y))))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if ((sin(fabs(x)) * t_0) <= 5e-7) {
		tmp = fma((fabs(x) * fabs(x)), (-0.16666666666666666 * fabs(x)), fabs(x)) * t_0;
	} else {
		tmp = 1.0 / ((1.0 / (sinh(y) * fabs(x))) * y);
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (Float64(sin(abs(x)) * t_0) <= 5e-7)
		tmp = Float64(fma(Float64(abs(x) * abs(x)), Float64(-0.16666666666666666 * abs(x)), abs(x)) * t_0);
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 / Float64(sinh(y) * abs(x))) * y));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], 5e-7], N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 / N[(N[(1.0 / N[(N[Sinh[y], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 4.9999999999999998e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 62.9%

         \[\leadsto \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right)\right) \cdot \frac{\sinh y}{y} \]

   4. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 62.9%

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

   6. Applied rewrites 62.9%

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

   if 4.9999999999999998e-7 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 40.1%

         \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-unsound-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. mult-flip N/A

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

      12. lift--.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-exp.f64 N/A

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

      16. rec-exp N/A

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

      17. sinh-def N/A

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

      18. lift-sinh.f64 N/A

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

      19. lower-*.f64 51.5%

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

   6. Applied rewrites 51.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 50.9%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 50.9%

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

   8. Applied rewrites 50.9%

      \[\leadsto \frac{1}{\frac{1}{\sinh y \cdot x} \cdot \color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 72.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \sinh \left(\left|y\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|x\right|\right) \cdot \frac{t\_0}{\left|y\right|} \leq -0.02:\\ \;\;\;\;0.5 \cdot \frac{\left|x\right| \cdot \left(\left(1 + \left|y\right|\right) - \left(1 + \left|y\right| \cdot \left(0.5 \cdot \left|y\right| - 1\right)\right)\right)}{\left|y\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right|}{\frac{\left|y\right|}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (sinh (fabs y))))
  (*
   (copysign 1.0 x)
   (if (<= (* (sin (fabs x)) (/ t_0 (fabs y))) -0.02)
     (*
      0.5
      (/
       (*
        (fabs x)
        (-
         (+ 1.0 (fabs y))
         (+ 1.0 (* (fabs y) (- (* 0.5 (fabs y)) 1.0)))))
       (fabs y)))
     (/ (fabs x) (/ (fabs y) t_0))))))

C

double code(double x, double y) {
	double t_0 = sinh(fabs(y));
	double tmp;
	if ((sin(fabs(x)) * (t_0 / fabs(y))) <= -0.02) {
		tmp = 0.5 * ((fabs(x) * ((1.0 + fabs(y)) - (1.0 + (fabs(y) * ((0.5 * fabs(y)) - 1.0))))) / fabs(y));
	} else {
		tmp = fabs(x) / (fabs(y) / t_0);
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(Math.abs(y));
	double tmp;
	if ((Math.sin(Math.abs(x)) * (t_0 / Math.abs(y))) <= -0.02) {
		tmp = 0.5 * ((Math.abs(x) * ((1.0 + Math.abs(y)) - (1.0 + (Math.abs(y) * ((0.5 * Math.abs(y)) - 1.0))))) / Math.abs(y));
	} else {
		tmp = Math.abs(x) / (Math.abs(y) / t_0);
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(math.fabs(y))
	tmp = 0
	if (math.sin(math.fabs(x)) * (t_0 / math.fabs(y))) <= -0.02:
		tmp = 0.5 * ((math.fabs(x) * ((1.0 + math.fabs(y)) - (1.0 + (math.fabs(y) * ((0.5 * math.fabs(y)) - 1.0))))) / math.fabs(y))
	else:
		tmp = math.fabs(x) / (math.fabs(y) / t_0)
	return math.copysign(1.0, x) * tmp

Julia

function code(x, y)
	t_0 = sinh(abs(y))
	tmp = 0.0
	if (Float64(sin(abs(x)) * Float64(t_0 / abs(y))) <= -0.02)
		tmp = Float64(0.5 * Float64(Float64(abs(x) * Float64(Float64(1.0 + abs(y)) - Float64(1.0 + Float64(abs(y) * Float64(Float64(0.5 * abs(y)) - 1.0))))) / abs(y)));
	else
		tmp = Float64(abs(x) / Float64(abs(y) / t_0));
	end
	return Float64(copysign(1.0, x) * tmp)
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(abs(y));
	tmp = 0.0;
	if ((sin(abs(x)) * (t_0 / abs(y))) <= -0.02)
		tmp = 0.5 * ((abs(x) * ((1.0 + abs(y)) - (1.0 + (abs(y) * ((0.5 * abs(y)) - 1.0))))) / abs(y));
	else
		tmp = abs(x) / (abs(y) / t_0);
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sinh[N[Abs[y], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.02], N[(0.5 * N[(N[(N[Abs[x], $MachinePrecision] * N[(N[(1.0 + N[Abs[y], $MachinePrecision]), $MachinePrecision] - N[(1.0 + N[(N[Abs[y], $MachinePrecision] * N[(N[(0.5 * N[Abs[y], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] / N[(N[Abs[y], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.02

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 40.1%

         \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \frac{1}{e^{y}}\right)}{y} \]
   6. Step-by-step derivation

      1. lower-+.f64 24.7%

         \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \frac{1}{e^{y}}\right)}{y} \]

   7. Applied rewrites 24.7%

      \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \frac{1}{e^{y}}\right)}{y} \]

   8. Taylor expanded in y around 0

      \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \frac{1}{1 + y}\right)}{y} \]
   9. Step-by-step derivation

      1. lower-+.f64 9.5%

         \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \frac{1}{1 + y}\right)}{y} \]

   10. Applied rewrites 9.5%

       \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \frac{1}{1 + y}\right)}{y} \]

   11. Taylor expanded in y around 0

       \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \left(1 + y \cdot \left(\frac{1}{2} \cdot y - 1\right)\right)\right)}{y} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-*.f64 18.5%

          \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \left(1 + y \cdot \left(0.5 \cdot y - 1\right)\right)\right)}{y} \]

   13. Applied rewrites 18.5%

       \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \left(1 + y \cdot \left(0.5 \cdot y - 1\right)\right)\right)}{y} \]

   if -0.02 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 40.1%

         \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. mult-flip N/A

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

      9. lift--.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. rec-exp N/A

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

      14. sinh-def N/A

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

      15. lift-sinh.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. mult-flip-rev N/A

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 50.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. div-flip-rev N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. div-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 62.3%

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

   8. Applied rewrites 62.3%

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

Alternative 5: 72.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|x\right|\right) \cdot \frac{\sinh y}{y} \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{t\_0 \cdot t\_0}, -0.16666666666666666 \cdot \left|x\right|, \left|x\right|\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right|}{\frac{y}{\sinh y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x))))
  (*
   (copysign 1.0 x)
   (if (<= (* (sin (fabs x)) (/ (sinh y) y)) -0.02)
     (*
      (fma
       (sqrt (* t_0 t_0))
       (* -0.16666666666666666 (fabs x))
       (fabs x))
      1.0)
     (/ (fabs x) (/ y (sinh y)))))))

C

double code(double x, double y) {
	double t_0 = fabs(x) * fabs(x);
	double tmp;
	if ((sin(fabs(x)) * (sinh(y) / y)) <= -0.02) {
		tmp = fma(sqrt((t_0 * t_0)), (-0.16666666666666666 * fabs(x)), fabs(x)) * 1.0;
	} else {
		tmp = fabs(x) / (y / sinh(y));
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x, y)
	t_0 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (Float64(sin(abs(x)) * Float64(sinh(y) / y)) <= -0.02)
		tmp = Float64(fma(sqrt(Float64(t_0 * t_0)), Float64(-0.16666666666666666 * abs(x)), abs(x)) * 1.0);
	else
		tmp = Float64(abs(x) / Float64(y / sinh(y)));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(-0.16666666666666666 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] / N[(y / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.02

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 62.9%

         \[\leadsto \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right)\right) \cdot \frac{\sinh y}{y} \]

   4. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 62.9%

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

   6. Applied rewrites 62.9%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 33.8%

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

       1. rem-square-sqrt N/A

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

       2. sqrt-unprod N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-*.f64 34.8%

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

   11. Applied rewrites 34.8%

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

   if -0.02 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 40.1%

         \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. mult-flip N/A

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

      9. lift--.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. rec-exp N/A

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

      14. sinh-def N/A

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

      15. lift-sinh.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. mult-flip-rev N/A

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 50.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. div-flip-rev N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. div-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 62.3%

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

   8. Applied rewrites 62.3%

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

Alternative 6: 71.1% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|x\right|\right) \cdot \frac{\sinh y}{y} \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right| \cdot \left|x\right|, -0.16666666666666666 \cdot \left|x\right|, \left|x\right|\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right|}{\frac{y}{\sinh y}}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (*
 (copysign 1.0 x)
 (if (<= (* (sin (fabs x)) (/ (sinh y) y)) -0.02)
   (*
    (fma
     (* (fabs x) (fabs x))
     (* -0.16666666666666666 (fabs x))
     (fabs x))
    1.0)
   (/ (fabs x) (/ y (sinh y))))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(fabs(x)) * (sinh(y) / y)) <= -0.02) {
		tmp = fma((fabs(x) * fabs(x)), (-0.16666666666666666 * fabs(x)), fabs(x)) * 1.0;
	} else {
		tmp = fabs(x) / (y / sinh(y));
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(abs(x)) * Float64(sinh(y) / y)) <= -0.02)
		tmp = Float64(fma(Float64(abs(x) * abs(x)), Float64(-0.16666666666666666 * abs(x)), abs(x)) * 1.0);
	else
		tmp = Float64(abs(x) / Float64(y / sinh(y)));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_, y_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] / N[(y / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.02

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 62.9%

         \[\leadsto \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right)\right) \cdot \frac{\sinh y}{y} \]

   4. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 62.9%

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

   6. Applied rewrites 62.9%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 33.8%

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

   if -0.02 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 40.1%

         \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. mult-flip N/A

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

      9. lift--.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. rec-exp N/A

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

      14. sinh-def N/A

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

      15. lift-sinh.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. mult-flip-rev N/A

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 50.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. div-flip-rev N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. div-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 62.3%

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

   8. Applied rewrites 62.3%

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

Alternative 7: 70.7% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|x\right|\right) \cdot \frac{\sinh y}{y} \leq 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right| \cdot \left|x\right|, -0.16666666666666666 \cdot \left|x\right|, \left|x\right|\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right| \cdot \sinh y}{y}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (*
 (copysign 1.0 x)
 (if (<= (* (sin (fabs x)) (/ (sinh y) y)) 1e-12)
   (*
    (fma
     (* (fabs x) (fabs x))
     (* -0.16666666666666666 (fabs x))
     (fabs x))
    1.0)
   (/ (* (fabs x) (sinh y)) y))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(fabs(x)) * (sinh(y) / y)) <= 1e-12) {
		tmp = fma((fabs(x) * fabs(x)), (-0.16666666666666666 * fabs(x)), fabs(x)) * 1.0;
	} else {
		tmp = (fabs(x) * sinh(y)) / y;
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(abs(x)) * Float64(sinh(y) / y)) <= 1e-12)
		tmp = Float64(fma(Float64(abs(x) * abs(x)), Float64(-0.16666666666666666 * abs(x)), abs(x)) * 1.0);
	else
		tmp = Float64(Float64(abs(x) * sinh(y)) / y);
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_, y_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 1e-12], N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Abs[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 9.9999999999999998e-13

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 62.9%

         \[\leadsto \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right)\right) \cdot \frac{\sinh y}{y} \]

   4. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 62.9%

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

   6. Applied rewrites 62.9%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 33.8%

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

   if 9.9999999999999998e-13 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 40.1%

         \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. mult-flip N/A

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

      10. lift--.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-exp.f64 N/A

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

      14. rec-exp N/A

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

      15. sinh-def N/A

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

      16. lift-sinh.f64 N/A

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

      17. lower-*.f64 51.6%

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

   6. Applied rewrites 51.6%

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

Alternative 8: 59.1% accurate, 0.8× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|x\right|\right) \leq 4 \cdot 10^{-281}:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right| \cdot \left|x\right|, -0.16666666666666666 \cdot \left|x\right|, \left|x\right|\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \frac{\left|x\right|}{y}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (*
 (copysign 1.0 x)
 (if (<= (sin (fabs x)) 4e-281)
   (*
    (fma
     (* (fabs x) (fabs x))
     (* -0.16666666666666666 (fabs x))
     (fabs x))
    1.0)
   (* (sinh y) (/ (fabs x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (sin(fabs(x)) <= 4e-281) {
		tmp = fma((fabs(x) * fabs(x)), (-0.16666666666666666 * fabs(x)), fabs(x)) * 1.0;
	} else {
		tmp = sinh(y) * (fabs(x) / y);
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(abs(x)) <= 4e-281)
		tmp = Float64(fma(Float64(abs(x) * abs(x)), Float64(-0.16666666666666666 * abs(x)), abs(x)) * 1.0);
	else
		tmp = Float64(sinh(y) * Float64(abs(x) / y));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_, y_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision], 4e-281], N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 4.0000000000000001e-281

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 62.9%

         \[\leadsto \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right)\right) \cdot \frac{\sinh y}{y} \]

   4. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 62.9%

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

   6. Applied rewrites 62.9%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 33.8%

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

   if 4.0000000000000001e-281 < (sin.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 40.1%

         \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. mult-flip N/A

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

      9. lift--.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. rec-exp N/A

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

      14. sinh-def N/A

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

      15. lift-sinh.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. mult-flip-rev N/A

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 50.6%

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

Alternative 9: 40.6% accurate, 0.7× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|x\right|\right) \cdot \frac{\sinh y}{y} \leq 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right| \cdot \left|x\right|, -0.16666666666666666 \cdot \left|x\right|, \left|x\right|\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\left|x\right| \cdot \left(2 \cdot y\right)}{y}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (*
 (copysign 1.0 x)
 (if (<= (* (sin (fabs x)) (/ (sinh y) y)) 1e-12)
   (*
    (fma
     (* (fabs x) (fabs x))
     (* -0.16666666666666666 (fabs x))
     (fabs x))
    1.0)
   (* 0.5 (/ (* (fabs x) (* 2.0 y)) y)))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(fabs(x)) * (sinh(y) / y)) <= 1e-12) {
		tmp = fma((fabs(x) * fabs(x)), (-0.16666666666666666 * fabs(x)), fabs(x)) * 1.0;
	} else {
		tmp = 0.5 * ((fabs(x) * (2.0 * y)) / y);
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(abs(x)) * Float64(sinh(y) / y)) <= 1e-12)
		tmp = Float64(fma(Float64(abs(x) * abs(x)), Float64(-0.16666666666666666 * abs(x)), abs(x)) * 1.0);
	else
		tmp = Float64(0.5 * Float64(Float64(abs(x) * Float64(2.0 * y)) / y));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_, y_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 1e-12], N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(0.5 * N[(N[(N[Abs[x], $MachinePrecision] * N[(2.0 * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 9.9999999999999998e-13

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 62.9%

         \[\leadsto \left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right)\right) \cdot \frac{\sinh y}{y} \]

   4. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\right)} \cdot \frac{\sinh y}{y} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 62.9%

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

   6. Applied rewrites 62.9%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 33.8%

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

   if 9.9999999999999998e-13 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 40.1%

         \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \frac{1}{e^{y}}\right)}{y} \]
   6. Step-by-step derivation

      1. lower-+.f64 24.7%

         \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \frac{1}{e^{y}}\right)}{y} \]

   7. Applied rewrites 24.7%

      \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \frac{1}{e^{y}}\right)}{y} \]

   8. Taylor expanded in y around 0

      \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \frac{1}{1 + y}\right)}{y} \]
   9. Step-by-step derivation

      1. lower-+.f64 9.5%

         \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \frac{1}{1 + y}\right)}{y} \]

   10. Applied rewrites 9.5%

       \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \frac{1}{1 + y}\right)}{y} \]

   11. Taylor expanded in y around 0

       \[\leadsto 0.5 \cdot \frac{x \cdot \left(2 \cdot y\right)}{y} \]
   12. Step-by-step derivation

       1. lower-*.f64 21.1%

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

   13. Applied rewrites 21.1%

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

Alternative 10: 31.8% accurate, 0.7× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|x\right|\right) \cdot \frac{\sinh y}{y} \leq 10^{-12}:\\ \;\;\;\;\frac{\left|x\right|}{1}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\left|x\right| \cdot \left(2 \cdot y\right)}{y}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (*
 (copysign 1.0 x)
 (if (<= (* (sin (fabs x)) (/ (sinh y) y)) 1e-12)
   (/ (fabs x) 1.0)
   (* 0.5 (/ (* (fabs x) (* 2.0 y)) y)))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(fabs(x)) * (sinh(y) / y)) <= 1e-12) {
		tmp = fabs(x) / 1.0;
	} else {
		tmp = 0.5 * ((fabs(x) * (2.0 * y)) / y);
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sin(Math.abs(x)) * (Math.sinh(y) / y)) <= 1e-12) {
		tmp = Math.abs(x) / 1.0;
	} else {
		tmp = 0.5 * ((Math.abs(x) * (2.0 * y)) / y);
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sin(math.fabs(x)) * (math.sinh(y) / y)) <= 1e-12:
		tmp = math.fabs(x) / 1.0
	else:
		tmp = 0.5 * ((math.fabs(x) * (2.0 * y)) / y)
	return math.copysign(1.0, x) * tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(abs(x)) * Float64(sinh(y) / y)) <= 1e-12)
		tmp = Float64(abs(x) / 1.0);
	else
		tmp = Float64(0.5 * Float64(Float64(abs(x) * Float64(2.0 * y)) / y));
	end
	return Float64(copysign(1.0, x) * tmp)
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sin(abs(x)) * (sinh(y) / y)) <= 1e-12)
		tmp = abs(x) / 1.0;
	else
		tmp = 0.5 * ((abs(x) * (2.0 * y)) / y);
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 1e-12], N[(N[Abs[x], $MachinePrecision] / 1.0), $MachinePrecision], N[(0.5 * N[(N[(N[Abs[x], $MachinePrecision] * N[(2.0 * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 9.9999999999999998e-13

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 40.1%

         \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. mult-flip N/A

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

      9. lift--.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. rec-exp N/A

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

      14. sinh-def N/A

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

      15. lift-sinh.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. mult-flip-rev N/A

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 50.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. div-flip-rev N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. div-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 62.3%

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

   8. Applied rewrites 62.3%

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

   9. Taylor expanded in y around 0

      \[\leadsto \frac{x}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 26.1%

       \[\leadsto \frac{x}{1} \]

   if 9.9999999999999998e-13 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 40.1%

         \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \frac{1}{e^{y}}\right)}{y} \]
   6. Step-by-step derivation

      1. lower-+.f64 24.7%

         \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \frac{1}{e^{y}}\right)}{y} \]

   7. Applied rewrites 24.7%

      \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \frac{1}{e^{y}}\right)}{y} \]

   8. Taylor expanded in y around 0

      \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \frac{1}{1 + y}\right)}{y} \]
   9. Step-by-step derivation

      1. lower-+.f64 9.5%

         \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \frac{1}{1 + y}\right)}{y} \]

   10. Applied rewrites 9.5%

       \[\leadsto 0.5 \cdot \frac{x \cdot \left(\left(1 + y\right) - \frac{1}{1 + y}\right)}{y} \]

   11. Taylor expanded in y around 0

       \[\leadsto 0.5 \cdot \frac{x \cdot \left(2 \cdot y\right)}{y} \]
   12. Step-by-step derivation

       1. lower-*.f64 21.1%

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

   13. Applied rewrites 21.1%

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

Alternative 11: 31.8% accurate, 0.7× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|x\right|\right) \cdot \frac{\sinh y}{y} \leq 0.05:\\ \;\;\;\;\frac{\left|x\right|}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(\left|x\right| \cdot y\right)\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (*
 (copysign 1.0 x)
 (if (<= (* (sin (fabs x)) (/ (sinh y) y)) 0.05)
   (/ (fabs x) 1.0)
   (* (/ 1.0 y) (* (fabs x) y)))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(fabs(x)) * (sinh(y) / y)) <= 0.05) {
		tmp = fabs(x) / 1.0;
	} else {
		tmp = (1.0 / y) * (fabs(x) * y);
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sin(Math.abs(x)) * (Math.sinh(y) / y)) <= 0.05) {
		tmp = Math.abs(x) / 1.0;
	} else {
		tmp = (1.0 / y) * (Math.abs(x) * y);
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sin(math.fabs(x)) * (math.sinh(y) / y)) <= 0.05:
		tmp = math.fabs(x) / 1.0
	else:
		tmp = (1.0 / y) * (math.fabs(x) * y)
	return math.copysign(1.0, x) * tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(abs(x)) * Float64(sinh(y) / y)) <= 0.05)
		tmp = Float64(abs(x) / 1.0);
	else
		tmp = Float64(Float64(1.0 / y) * Float64(abs(x) * y));
	end
	return Float64(copysign(1.0, x) * tmp)
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sin(abs(x)) * (sinh(y) / y)) <= 0.05)
		tmp = abs(x) / 1.0;
	else
		tmp = (1.0 / y) * (abs(x) * y);
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.05], N[(N[Abs[x], $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.050000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 40.1%

         \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. mult-flip N/A

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

      9. lift--.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. rec-exp N/A

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

      14. sinh-def N/A

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

      15. lift-sinh.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. mult-flip-rev N/A

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 50.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. div-flip-rev N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. div-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 62.3%

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

   8. Applied rewrites 62.3%

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

   9. Taylor expanded in y around 0

      \[\leadsto \frac{x}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 26.1%

       \[\leadsto \frac{x}{1} \]

   if 0.050000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 40.1%

         \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]

   4. Applied rewrites 40.1%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. metadata-eval N/A

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

      8. mult-flip N/A

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

      9. lift--.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-exp.f64 N/A

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

      13. rec-exp N/A

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

      14. sinh-def N/A

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

      15. lift-sinh.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. mult-flip-rev N/A

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

      19. lift-/.f64 N/A

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

   6. Applied rewrites 50.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 51.5%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 51.5%

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

   8. Applied rewrites 51.5%

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

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 21.0%

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

   11. Applied rewrites 21.0%

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

Alternative 12: 26.1% accurate, 11.5× speedup

Math

\[\frac{x}{1} \]

FPCore

(FPCore (x y)
  :precision binary64
  (/ x 1.0))

C

double code(double x, double y) {
	return x / 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / 1.0d0
end function

Java

public static double code(double x, double y) {
	return x / 1.0;
}

Python

def code(x, y):
	return x / 1.0

Julia

function code(x, y)
	return Float64(x / 1.0)
end

MATLAB

function tmp = code(x, y)
	tmp = x / 1.0;
end

Wolfram

code[x_, y_] := N[(x / 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-exp.f64 40.1%

      \[\leadsto 0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y} \]

4. Applied rewrites 40.1%

   \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot \left(e^{y} - \frac{1}{e^{y}}\right)}{y}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. metadata-eval N/A

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

   8. mult-flip N/A

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

   9. lift--.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-exp.f64 N/A

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

   13. rec-exp N/A

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

   14. sinh-def N/A

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

   15. lift-sinh.f64 N/A

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

   16. *-commutative N/A

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

   17. associate-/l* N/A

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

   18. mult-flip-rev N/A

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

   19. lift-/.f64 N/A

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

6. Applied rewrites 50.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lift-*.f64 N/A

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

   6. div-flip-rev N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-/r* N/A

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

   10. div-flip-rev N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 62.3%

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

8. Applied rewrites 62.3%

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

9. Taylor expanded in y around 0

   \[\leadsto \frac{x}{1} \]
10. Step-by-step derivation

11. Applied rewrites 26.1%

    \[\leadsto \frac{x}{1} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))