Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 33.0s

Alternatives: 8

Speedup: 1.0×

• 50.9% 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: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \left|y\right| \cdot \left|y\right|\\ t_1 := \left(t\_0 \cdot \left|y\right|\right) \cdot \left(\frac{1}{36} \cdot \left|y\right|\right)\\ \mathbf{if}\;\left|y\right| \leq \frac{1261007895663739}{9007199254740992}:\\ \;\;\;\;\sin x \cdot \frac{\left(\frac{1}{36} \cdot t\_0\right) \cdot t\_0 - 1}{t\_0 \cdot \frac{1}{6} - 1}\\ \mathbf{elif}\;\left|y\right| \leq 649999999999999935800862526406656:\\ \;\;\;\;\left(\frac{1}{\left|y\right|} \cdot x\right) \cdot \sinh \left(\left|y\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(1 + \sqrt{\sqrt{t\_1 \cdot t\_1}}\right)\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* (fabs y) (fabs y)))
       (t_1 (* (* t_0 (fabs y)) (* 1/36 (fabs y)))))
  (if (<= (fabs y) 1261007895663739/9007199254740992)
    (* (sin x) (/ (- (* (* 1/36 t_0) t_0) 1) (- (* t_0 1/6) 1)))
    (if (<= (fabs y) 649999999999999935800862526406656)
      (* (* (/ 1 (fabs y)) x) (sinh (fabs y)))
      (* (sin x) (+ 1 (sqrt (sqrt (* t_1 t_1)))))))))

C

double code(double x, double y) {
	double t_0 = fabs(y) * fabs(y);
	double t_1 = (t_0 * fabs(y)) * (0.027777777777777776 * fabs(y));
	double tmp;
	if (fabs(y) <= 0.14) {
		tmp = sin(x) * ((((0.027777777777777776 * t_0) * t_0) - 1.0) / ((t_0 * 0.16666666666666666) - 1.0));
	} else if (fabs(y) <= 6.5e+32) {
		tmp = ((1.0 / fabs(y)) * x) * sinh(fabs(y));
	} else {
		tmp = sin(x) * (1.0 + sqrt(sqrt((t_1 * t_1))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs(y) * abs(y)
    t_1 = (t_0 * abs(y)) * (0.027777777777777776d0 * abs(y))
    if (abs(y) <= 0.14d0) then
        tmp = sin(x) * ((((0.027777777777777776d0 * t_0) * t_0) - 1.0d0) / ((t_0 * 0.16666666666666666d0) - 1.0d0))
    else if (abs(y) <= 6.5d+32) then
        tmp = ((1.0d0 / abs(y)) * x) * sinh(abs(y))
    else
        tmp = sin(x) * (1.0d0 + sqrt(sqrt((t_1 * t_1))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.abs(y) * Math.abs(y);
	double t_1 = (t_0 * Math.abs(y)) * (0.027777777777777776 * Math.abs(y));
	double tmp;
	if (Math.abs(y) <= 0.14) {
		tmp = Math.sin(x) * ((((0.027777777777777776 * t_0) * t_0) - 1.0) / ((t_0 * 0.16666666666666666) - 1.0));
	} else if (Math.abs(y) <= 6.5e+32) {
		tmp = ((1.0 / Math.abs(y)) * x) * Math.sinh(Math.abs(y));
	} else {
		tmp = Math.sin(x) * (1.0 + Math.sqrt(Math.sqrt((t_1 * t_1))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.fabs(y) * math.fabs(y)
	t_1 = (t_0 * math.fabs(y)) * (0.027777777777777776 * math.fabs(y))
	tmp = 0
	if math.fabs(y) <= 0.14:
		tmp = math.sin(x) * ((((0.027777777777777776 * t_0) * t_0) - 1.0) / ((t_0 * 0.16666666666666666) - 1.0))
	elif math.fabs(y) <= 6.5e+32:
		tmp = ((1.0 / math.fabs(y)) * x) * math.sinh(math.fabs(y))
	else:
		tmp = math.sin(x) * (1.0 + math.sqrt(math.sqrt((t_1 * t_1))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(abs(y) * abs(y))
	t_1 = Float64(Float64(t_0 * abs(y)) * Float64(0.027777777777777776 * abs(y)))
	tmp = 0.0
	if (abs(y) <= 0.14)
		tmp = Float64(sin(x) * Float64(Float64(Float64(Float64(0.027777777777777776 * t_0) * t_0) - 1.0) / Float64(Float64(t_0 * 0.16666666666666666) - 1.0)));
	elseif (abs(y) <= 6.5e+32)
		tmp = Float64(Float64(Float64(1.0 / abs(y)) * x) * sinh(abs(y)));
	else
		tmp = Float64(sin(x) * Float64(1.0 + sqrt(sqrt(Float64(t_1 * t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = abs(y) * abs(y);
	t_1 = (t_0 * abs(y)) * (0.027777777777777776 * abs(y));
	tmp = 0.0;
	if (abs(y) <= 0.14)
		tmp = sin(x) * ((((0.027777777777777776 * t_0) * t_0) - 1.0) / ((t_0 * 0.16666666666666666) - 1.0));
	elseif (abs(y) <= 6.5e+32)
		tmp = ((1.0 / abs(y)) * x) * sinh(abs(y));
	else
		tmp = sin(x) * (1.0 + sqrt(sqrt((t_1 * t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Abs[y], $MachinePrecision] * N[Abs[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[y], $MachinePrecision]), $MachinePrecision] * N[(1/36 * N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y], $MachinePrecision], 1261007895663739/9007199254740992], N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(N[(1/36 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] - 1), $MachinePrecision] / N[(N[(t$95$0 * 1/6), $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[y], $MachinePrecision], 649999999999999935800862526406656], N[(N[(N[(1 / N[Abs[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[Sinh[N[Abs[y], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(1 + N[Sqrt[N[Sqrt[N[(t$95$1 * t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.14000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 77.3%

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

   4. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 77.2%

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

   6. Applied rewrites 77.2%

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

   8. Applied rewrites 62.5%

      \[\leadsto \sin x \cdot \frac{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right) - 1}{\color{blue}{\left(y \cdot y\right) \cdot \frac{1}{6} - 1}} \]

   if 0.14000000000000001 < y < 6.4999999999999994e32

   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 88.3%

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

   3. Applied rewrites 88.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 51.2%

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

   6. Applied rewrites 51.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 51.1%

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

   8. Applied rewrites 51.1%

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

   if 6.4999999999999994e32 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 77.3%

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

   4. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 77.2%

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

   6. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. fabs-sqr N/A

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

      6. lift-*.f64 N/A

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

      7. fabs-mul N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. rem-sqrt-square-rev N/A

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

      11. lift-*.f64 N/A

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

      12. lower-sqrt.f64 88.4%

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. associate-*r* N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 88.4%

          \[\leadsto \sin x \cdot \left(1 + \sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)}\right) \]

   8. Applied rewrites 88.4%

      \[\leadsto \sin x \cdot \left(1 + \sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)}\right) \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sin x \cdot \left(1 + \sqrt{\sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)} \cdot \sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)}}\right) \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \sin x \cdot \left(1 + \sqrt{\sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)} \cdot \sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)}}\right) \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \sin x \cdot \left(1 + \sqrt{\sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)} \cdot \sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)}}\right) \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \sin x \cdot \left(1 + \sqrt{\sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)} \cdot \sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)}}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \sin x \cdot \left(1 + \sqrt{\sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)} \cdot \sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)}}\right) \]

      6. sqrt-fabs-rev N/A

         \[\leadsto \sin x \cdot \left(1 + \sqrt{\sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)} \cdot \left|\sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)}\right|}\right) \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \sin x \cdot \left(1 + \sqrt{\sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)} \cdot \left|\sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)}\right|}\right) \]

      8. rem-sqrt-square-rev N/A

         \[\leadsto \sin x \cdot \left(1 + \sqrt{\sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)} \cdot \sqrt{\sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)} \cdot \sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)}}}\right) \]

      9. sqrt-unprod N/A

         \[\leadsto \sin x \cdot \left(1 + \sqrt{\sqrt{\left(\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)} \cdot \sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)}\right)}}\right) \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \sin x \cdot \left(1 + \sqrt{\sqrt{\left(\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)} \cdot \sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)}\right)}}\right) \]

      11. lift-sqrt.f64 N/A

          \[\leadsto \sin x \cdot \left(1 + \sqrt{\sqrt{\left(\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)} \cdot \sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)}\right)}}\right) \]

      12. rem-square-sqrt N/A

          \[\leadsto \sin x \cdot \left(1 + \sqrt{\sqrt{\left(\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)\right)}}\right) \]

   10. Applied rewrites 94.1%

       \[\leadsto \sin x \cdot \left(1 + \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot \left(\frac{1}{36} \cdot y\right)\right) \cdot \left(\left(\left(y \cdot y\right) \cdot y\right) \cdot \left(\frac{1}{36} \cdot y\right)\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \left|y\right| \cdot \left|y\right|\\ t_1 := \left(\frac{1}{36} \cdot t\_0\right) \cdot t\_0\\ \mathbf{if}\;\left|y\right| \leq \frac{1261007895663739}{9007199254740992}:\\ \;\;\;\;\sin x \cdot \frac{t\_1 - 1}{t\_0 \cdot \frac{1}{6} - 1}\\ \mathbf{elif}\;\left|y\right| \leq 42000000000000001333750431124433612527765488976574808201089382533455513911296:\\ \;\;\;\;\left(\frac{1}{\left|y\right|} \cdot x\right) \cdot \sinh \left(\left|y\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(1 + \sqrt{t\_1}\right)\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* (fabs y) (fabs y))) (t_1 (* (* 1/36 t_0) t_0)))
  (if (<= (fabs y) 1261007895663739/9007199254740992)
    (* (sin x) (/ (- t_1 1) (- (* t_0 1/6) 1)))
    (if (<=
         (fabs y)
         42000000000000001333750431124433612527765488976574808201089382533455513911296)
      (* (* (/ 1 (fabs y)) x) (sinh (fabs y)))
      (* (sin x) (+ 1 (sqrt t_1)))))))

C

double code(double x, double y) {
	double t_0 = fabs(y) * fabs(y);
	double t_1 = (0.027777777777777776 * t_0) * t_0;
	double tmp;
	if (fabs(y) <= 0.14) {
		tmp = sin(x) * ((t_1 - 1.0) / ((t_0 * 0.16666666666666666) - 1.0));
	} else if (fabs(y) <= 4.2e+76) {
		tmp = ((1.0 / fabs(y)) * x) * sinh(fabs(y));
	} else {
		tmp = sin(x) * (1.0 + sqrt(t_1));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = abs(y) * abs(y)
    t_1 = (0.027777777777777776d0 * t_0) * t_0
    if (abs(y) <= 0.14d0) then
        tmp = sin(x) * ((t_1 - 1.0d0) / ((t_0 * 0.16666666666666666d0) - 1.0d0))
    else if (abs(y) <= 4.2d+76) then
        tmp = ((1.0d0 / abs(y)) * x) * sinh(abs(y))
    else
        tmp = sin(x) * (1.0d0 + sqrt(t_1))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.abs(y) * Math.abs(y);
	double t_1 = (0.027777777777777776 * t_0) * t_0;
	double tmp;
	if (Math.abs(y) <= 0.14) {
		tmp = Math.sin(x) * ((t_1 - 1.0) / ((t_0 * 0.16666666666666666) - 1.0));
	} else if (Math.abs(y) <= 4.2e+76) {
		tmp = ((1.0 / Math.abs(y)) * x) * Math.sinh(Math.abs(y));
	} else {
		tmp = Math.sin(x) * (1.0 + Math.sqrt(t_1));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.fabs(y) * math.fabs(y)
	t_1 = (0.027777777777777776 * t_0) * t_0
	tmp = 0
	if math.fabs(y) <= 0.14:
		tmp = math.sin(x) * ((t_1 - 1.0) / ((t_0 * 0.16666666666666666) - 1.0))
	elif math.fabs(y) <= 4.2e+76:
		tmp = ((1.0 / math.fabs(y)) * x) * math.sinh(math.fabs(y))
	else:
		tmp = math.sin(x) * (1.0 + math.sqrt(t_1))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(abs(y) * abs(y))
	t_1 = Float64(Float64(0.027777777777777776 * t_0) * t_0)
	tmp = 0.0
	if (abs(y) <= 0.14)
		tmp = Float64(sin(x) * Float64(Float64(t_1 - 1.0) / Float64(Float64(t_0 * 0.16666666666666666) - 1.0)));
	elseif (abs(y) <= 4.2e+76)
		tmp = Float64(Float64(Float64(1.0 / abs(y)) * x) * sinh(abs(y)));
	else
		tmp = Float64(sin(x) * Float64(1.0 + sqrt(t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = abs(y) * abs(y);
	t_1 = (0.027777777777777776 * t_0) * t_0;
	tmp = 0.0;
	if (abs(y) <= 0.14)
		tmp = sin(x) * ((t_1 - 1.0) / ((t_0 * 0.16666666666666666) - 1.0));
	elseif (abs(y) <= 4.2e+76)
		tmp = ((1.0 / abs(y)) * x) * sinh(abs(y));
	else
		tmp = sin(x) * (1.0 + sqrt(t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Abs[y], $MachinePrecision] * N[Abs[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1/36 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[y], $MachinePrecision], 1261007895663739/9007199254740992], N[(N[Sin[x], $MachinePrecision] * N[(N[(t$95$1 - 1), $MachinePrecision] / N[(N[(t$95$0 * 1/6), $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[y], $MachinePrecision], 42000000000000001333750431124433612527765488976574808201089382533455513911296], N[(N[(N[(1 / N[Abs[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[Sinh[N[Abs[y], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(1 + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.14000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 77.3%

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

   4. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 77.2%

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

   6. Applied rewrites 77.2%

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

   8. Applied rewrites 62.5%

      \[\leadsto \sin x \cdot \frac{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right) - 1}{\color{blue}{\left(y \cdot y\right) \cdot \frac{1}{6} - 1}} \]

   if 0.14000000000000001 < y < 4.2000000000000001e76

   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 88.3%

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

   3. Applied rewrites 88.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 51.2%

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

   6. Applied rewrites 51.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 51.1%

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

   8. Applied rewrites 51.1%

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

   if 4.2000000000000001e76 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 77.3%

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

   4. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 77.2%

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

   6. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. fabs-sqr N/A

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

      6. lift-*.f64 N/A

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

      7. fabs-mul N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. rem-sqrt-square-rev N/A

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

      11. lift-*.f64 N/A

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

      12. lower-sqrt.f64 88.4%

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. associate-*r* N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 88.4%

          \[\leadsto \sin x \cdot \left(1 + \sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)}\right) \]

   8. Applied rewrites 88.4%

      \[\leadsto \sin x \cdot \left(1 + \sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 96.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \left|y\right| \cdot \left|y\right|\\ \mathbf{if}\;\left|y\right| \leq \frac{1261007895663739}{9007199254740992}:\\ \;\;\;\;\sin x \cdot \left(1 + \left(\frac{1}{6} \cdot \left|y\right|\right) \cdot \left|y\right|\right)\\ \mathbf{elif}\;\left|y\right| \leq 42000000000000001333750431124433612527765488976574808201089382533455513911296:\\ \;\;\;\;\left(\frac{1}{\left|y\right|} \cdot x\right) \cdot \sinh \left(\left|y\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(1 + \sqrt{\left(\frac{1}{36} \cdot t\_0\right) \cdot t\_0}\right)\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* (fabs y) (fabs y))))
  (if (<= (fabs y) 1261007895663739/9007199254740992)
    (* (sin x) (+ 1 (* (* 1/6 (fabs y)) (fabs y))))
    (if (<=
         (fabs y)
         42000000000000001333750431124433612527765488976574808201089382533455513911296)
      (* (* (/ 1 (fabs y)) x) (sinh (fabs y)))
      (* (sin x) (+ 1 (sqrt (* (* 1/36 t_0) t_0))))))))

C

double code(double x, double y) {
	double t_0 = fabs(y) * fabs(y);
	double tmp;
	if (fabs(y) <= 0.14) {
		tmp = sin(x) * (1.0 + ((0.16666666666666666 * fabs(y)) * fabs(y)));
	} else if (fabs(y) <= 4.2e+76) {
		tmp = ((1.0 / fabs(y)) * x) * sinh(fabs(y));
	} else {
		tmp = sin(x) * (1.0 + sqrt(((0.027777777777777776 * t_0) * t_0)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(y) * abs(y)
    if (abs(y) <= 0.14d0) then
        tmp = sin(x) * (1.0d0 + ((0.16666666666666666d0 * abs(y)) * abs(y)))
    else if (abs(y) <= 4.2d+76) then
        tmp = ((1.0d0 / abs(y)) * x) * sinh(abs(y))
    else
        tmp = sin(x) * (1.0d0 + sqrt(((0.027777777777777776d0 * t_0) * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.abs(y) * Math.abs(y);
	double tmp;
	if (Math.abs(y) <= 0.14) {
		tmp = Math.sin(x) * (1.0 + ((0.16666666666666666 * Math.abs(y)) * Math.abs(y)));
	} else if (Math.abs(y) <= 4.2e+76) {
		tmp = ((1.0 / Math.abs(y)) * x) * Math.sinh(Math.abs(y));
	} else {
		tmp = Math.sin(x) * (1.0 + Math.sqrt(((0.027777777777777776 * t_0) * t_0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.fabs(y) * math.fabs(y)
	tmp = 0
	if math.fabs(y) <= 0.14:
		tmp = math.sin(x) * (1.0 + ((0.16666666666666666 * math.fabs(y)) * math.fabs(y)))
	elif math.fabs(y) <= 4.2e+76:
		tmp = ((1.0 / math.fabs(y)) * x) * math.sinh(math.fabs(y))
	else:
		tmp = math.sin(x) * (1.0 + math.sqrt(((0.027777777777777776 * t_0) * t_0)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(abs(y) * abs(y))
	tmp = 0.0
	if (abs(y) <= 0.14)
		tmp = Float64(sin(x) * Float64(1.0 + Float64(Float64(0.16666666666666666 * abs(y)) * abs(y))));
	elseif (abs(y) <= 4.2e+76)
		tmp = Float64(Float64(Float64(1.0 / abs(y)) * x) * sinh(abs(y)));
	else
		tmp = Float64(sin(x) * Float64(1.0 + sqrt(Float64(Float64(0.027777777777777776 * t_0) * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = abs(y) * abs(y);
	tmp = 0.0;
	if (abs(y) <= 0.14)
		tmp = sin(x) * (1.0 + ((0.16666666666666666 * abs(y)) * abs(y)));
	elseif (abs(y) <= 4.2e+76)
		tmp = ((1.0 / abs(y)) * x) * sinh(abs(y));
	else
		tmp = sin(x) * (1.0 + sqrt(((0.027777777777777776 * t_0) * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Abs[y], $MachinePrecision] * N[Abs[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y], $MachinePrecision], 1261007895663739/9007199254740992], N[(N[Sin[x], $MachinePrecision] * N[(1 + N[(N[(1/6 * N[Abs[y], $MachinePrecision]), $MachinePrecision] * N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[y], $MachinePrecision], 42000000000000001333750431124433612527765488976574808201089382533455513911296], N[(N[(N[(1 / N[Abs[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[Sinh[N[Abs[y], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(1 + N[Sqrt[N[(N[(1/36 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.14000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 77.3%

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

   4. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 77.2%

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

   6. Applied rewrites 77.2%

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

   if 0.14000000000000001 < y < 4.2000000000000001e76

   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 88.3%

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

   3. Applied rewrites 88.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 51.2%

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

   6. Applied rewrites 51.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 51.1%

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

   8. Applied rewrites 51.1%

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

   if 4.2000000000000001e76 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 77.3%

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

   4. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 77.2%

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

   6. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. fabs-sqr N/A

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

      6. lift-*.f64 N/A

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

      7. fabs-mul N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. rem-sqrt-square-rev N/A

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

      11. lift-*.f64 N/A

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

      12. lower-sqrt.f64 88.4%

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. associate-*r* N/A

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

      21. lower-*.f64 N/A

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

      22. metadata-eval 88.4%

          \[\leadsto \sin x \cdot \left(1 + \sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)}\right) \]

   8. Applied rewrites 88.4%

      \[\leadsto \sin x \cdot \left(1 + \sqrt{\left(\frac{1}{36} \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 91.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \sin x \cdot \left(1 + \left(\frac{1}{6} \cdot \left|y\right|\right) \cdot \left|y\right|\right)\\ \mathbf{if}\;\left|y\right| \leq \frac{1261007895663739}{9007199254740992}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\left|y\right| \leq 22500000000000000459178351141959369210816864978487727427761875070431876369408262113249838982777431993744833480349327934346847195128257057415137899157913600:\\ \;\;\;\;\left(\frac{1}{\left|y\right|} \cdot x\right) \cdot \sinh \left(\left|y\right|\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* (sin x) (+ 1 (* (* 1/6 (fabs y)) (fabs y))))))
  (if (<= (fabs y) 1261007895663739/9007199254740992)
    t_0
    (if (<=
         (fabs y)
         22500000000000000459178351141959369210816864978487727427761875070431876369408262113249838982777431993744833480349327934346847195128257057415137899157913600)
      (* (* (/ 1 (fabs y)) x) (sinh (fabs y)))
      t_0))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (1.0 + ((0.16666666666666666 * fabs(y)) * fabs(y)));
	double tmp;
	if (fabs(y) <= 0.14) {
		tmp = t_0;
	} else if (fabs(y) <= 2.25e+154) {
		tmp = ((1.0 / fabs(y)) * x) * sinh(fabs(y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(x) * (1.0d0 + ((0.16666666666666666d0 * abs(y)) * abs(y)))
    if (abs(y) <= 0.14d0) then
        tmp = t_0
    else if (abs(y) <= 2.25d+154) then
        tmp = ((1.0d0 / abs(y)) * x) * sinh(abs(y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(x) * (1.0 + ((0.16666666666666666 * Math.abs(y)) * Math.abs(y)));
	double tmp;
	if (Math.abs(y) <= 0.14) {
		tmp = t_0;
	} else if (Math.abs(y) <= 2.25e+154) {
		tmp = ((1.0 / Math.abs(y)) * x) * Math.sinh(Math.abs(y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(x) * (1.0 + ((0.16666666666666666 * math.fabs(y)) * math.fabs(y)))
	tmp = 0
	if math.fabs(y) <= 0.14:
		tmp = t_0
	elif math.fabs(y) <= 2.25e+154:
		tmp = ((1.0 / math.fabs(y)) * x) * math.sinh(math.fabs(y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(1.0 + Float64(Float64(0.16666666666666666 * abs(y)) * abs(y))))
	tmp = 0.0
	if (abs(y) <= 0.14)
		tmp = t_0;
	elseif (abs(y) <= 2.25e+154)
		tmp = Float64(Float64(Float64(1.0 / abs(y)) * x) * sinh(abs(y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) * (1.0 + ((0.16666666666666666 * abs(y)) * abs(y)));
	tmp = 0.0;
	if (abs(y) <= 0.14)
		tmp = t_0;
	elseif (abs(y) <= 2.25e+154)
		tmp = ((1.0 / abs(y)) * x) * sinh(abs(y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[(1 + N[(N[(1/6 * N[Abs[y], $MachinePrecision]), $MachinePrecision] * N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y], $MachinePrecision], 1261007895663739/9007199254740992], t$95$0, If[LessEqual[N[Abs[y], $MachinePrecision], 22500000000000000459178351141959369210816864978487727427761875070431876369408262113249838982777431993744833480349327934346847195128257057415137899157913600], N[(N[(N[(1 / N[Abs[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[Sinh[N[Abs[y], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 0.14000000000000001 or 2.25e154 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 77.3%

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

   4. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 77.2%

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

   6. Applied rewrites 77.2%

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

   if 0.14000000000000001 < y < 2.25e154

   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 88.3%

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

   3. Applied rewrites 88.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 51.2%

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

   6. Applied rewrites 51.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 51.1%

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

   8. Applied rewrites 51.1%

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

Alternative 5: 91.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \sin x \cdot \left(1 + \left(\frac{1}{6} \cdot \left|y\right|\right) \cdot \left|y\right|\right)\\ \mathbf{if}\;\left|y\right| \leq \frac{1261007895663739}{9007199254740992}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\left|y\right| \leq 22500000000000000459178351141959369210816864978487727427761875070431876369408262113249838982777431993744833480349327934346847195128257057415137899157913600:\\ \;\;\;\;\frac{x}{\left|y\right|} \cdot \sinh \left(\left|y\right|\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* (sin x) (+ 1 (* (* 1/6 (fabs y)) (fabs y))))))
  (if (<= (fabs y) 1261007895663739/9007199254740992)
    t_0
    (if (<=
         (fabs y)
         22500000000000000459178351141959369210816864978487727427761875070431876369408262113249838982777431993744833480349327934346847195128257057415137899157913600)
      (* (/ x (fabs y)) (sinh (fabs y)))
      t_0))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (1.0 + ((0.16666666666666666 * fabs(y)) * fabs(y)));
	double tmp;
	if (fabs(y) <= 0.14) {
		tmp = t_0;
	} else if (fabs(y) <= 2.25e+154) {
		tmp = (x / fabs(y)) * sinh(fabs(y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(x) * (1.0d0 + ((0.16666666666666666d0 * abs(y)) * abs(y)))
    if (abs(y) <= 0.14d0) then
        tmp = t_0
    else if (abs(y) <= 2.25d+154) then
        tmp = (x / abs(y)) * sinh(abs(y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(x) * (1.0 + ((0.16666666666666666 * Math.abs(y)) * Math.abs(y)));
	double tmp;
	if (Math.abs(y) <= 0.14) {
		tmp = t_0;
	} else if (Math.abs(y) <= 2.25e+154) {
		tmp = (x / Math.abs(y)) * Math.sinh(Math.abs(y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sin(x) * (1.0 + ((0.16666666666666666 * math.fabs(y)) * math.fabs(y)))
	tmp = 0
	if math.fabs(y) <= 0.14:
		tmp = t_0
	elif math.fabs(y) <= 2.25e+154:
		tmp = (x / math.fabs(y)) * math.sinh(math.fabs(y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(1.0 + Float64(Float64(0.16666666666666666 * abs(y)) * abs(y))))
	tmp = 0.0
	if (abs(y) <= 0.14)
		tmp = t_0;
	elseif (abs(y) <= 2.25e+154)
		tmp = Float64(Float64(x / abs(y)) * sinh(abs(y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(x) * (1.0 + ((0.16666666666666666 * abs(y)) * abs(y)));
	tmp = 0.0;
	if (abs(y) <= 0.14)
		tmp = t_0;
	elseif (abs(y) <= 2.25e+154)
		tmp = (x / abs(y)) * sinh(abs(y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[(1 + N[(N[(1/6 * N[Abs[y], $MachinePrecision]), $MachinePrecision] * N[Abs[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y], $MachinePrecision], 1261007895663739/9007199254740992], t$95$0, If[LessEqual[N[Abs[y], $MachinePrecision], 22500000000000000459178351141959369210816864978487727427761875070431876369408262113249838982777431993744833480349327934346847195128257057415137899157913600], N[(N[(x / N[Abs[y], $MachinePrecision]), $MachinePrecision] * N[Sinh[N[Abs[y], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 0.14000000000000001 or 2.25e154 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 77.3%

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

   4. Applied rewrites 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 77.2%

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

   6. Applied rewrites 77.2%

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

   if 0.14000000000000001 < y < 2.25e154

   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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 88.3%

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

   3. Applied rewrites 88.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 51.2%

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

   6. Applied rewrites 51.2%

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

Alternative 6: 75.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (sin (fabs x))))
  (*
   (copysign 1 x)
   (if (<= (* t_0 (/ (sinh y) y)) 5)
     (* t_0 1)
     (* (/ (fabs x) y) (sinh y))))))

C

double code(double x, double y) {
	double t_0 = sin(fabs(x));
	double tmp;
	if ((t_0 * (sinh(y) / y)) <= 5.0) {
		tmp = t_0 * 1.0;
	} else {
		tmp = (fabs(x) / y) * sinh(y);
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.sin(Math.abs(x));
	double tmp;
	if ((t_0 * (Math.sinh(y) / y)) <= 5.0) {
		tmp = t_0 * 1.0;
	} else {
		tmp = (Math.abs(x) / y) * Math.sinh(y);
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x, y):
	t_0 = math.sin(math.fabs(x))
	tmp = 0
	if (t_0 * (math.sinh(y) / y)) <= 5.0:
		tmp = t_0 * 1.0
	else:
		tmp = (math.fabs(x) / y) * math.sinh(y)
	return math.copysign(1.0, x) * tmp

Julia

function code(x, y)
	t_0 = sin(abs(x))
	tmp = 0.0
	if (Float64(t_0 * Float64(sinh(y) / y)) <= 5.0)
		tmp = Float64(t_0 * 1.0);
	else
		tmp = Float64(Float64(abs(x) / y) * sinh(y));
	end
	return Float64(copysign(1.0, x) * tmp)
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sin(abs(x));
	tmp = 0.0;
	if ((t_0 * (sinh(y) / y)) <= 5.0)
		tmp = t_0 * 1.0;
	else
		tmp = (abs(x) / y) * sinh(y);
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 5

   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 51.5%

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

   if 5 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) 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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 88.3%

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

   3. Applied rewrites 88.3%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 51.2%

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

   6. Applied rewrites 51.2%

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

Alternative 7: 51.5% accurate, 2.0× speedup

Math

\[\sin x \cdot 1 \]

FPCore

(FPCore (x y)
  :precision binary64
  (* (sin x) 1))

C

double code(double x, double y) {
	return sin(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 = sin(x) * 1.0d0
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) * 1.0;
}

Python

def code(x, y):
	return math.sin(x) * 1.0

Julia

function code(x, y)
	return Float64(sin(x) * 1.0)
end

MATLAB

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

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * 1), $MachinePrecision]

Derivation

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 51.5%

   \[\leadsto \sin x \cdot \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))