Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

Alternatives: 11

Speedup: 1.0×

• 49.6% 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: 93.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(y \cdot y\right) \cdot y\right) \cdot y\\ \sin x \cdot \left(1 + 0.16666666666666666 \cdot \sqrt{\sqrt{t\_0 \cdot t\_0}}\right) \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* (* (* y y) y) y)))
  (*
   (sin x)
   (+ 1.0 (* 0.16666666666666666 (sqrt (sqrt (* t_0 t_0))))))))

C

double code(double x, double y) {
	double t_0 = ((y * y) * y) * y;
	return sin(x) * (1.0 + (0.16666666666666666 * sqrt(sqrt((t_0 * t_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
    real(8) :: t_0
    t_0 = ((y * y) * y) * y
    code = sin(x) * (1.0d0 + (0.16666666666666666d0 * sqrt(sqrt((t_0 * t_0)))))
end function

Java

public static double code(double x, double y) {
	double t_0 = ((y * y) * y) * y;
	return Math.sin(x) * (1.0 + (0.16666666666666666 * Math.sqrt(Math.sqrt((t_0 * t_0)))));
}

Python

def code(x, y):
	t_0 = ((y * y) * y) * y
	return math.sin(x) * (1.0 + (0.16666666666666666 * math.sqrt(math.sqrt((t_0 * t_0)))))

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(y * y) * y) * y)
	return Float64(sin(x) * Float64(1.0 + Float64(0.16666666666666666 * sqrt(sqrt(Float64(t_0 * t_0))))))
end

MATLAB

function tmp = code(x, y)
	t_0 = ((y * y) * y) * y;
	tmp = sin(x) * (1.0 + (0.16666666666666666 * sqrt(sqrt((t_0 * t_0)))));
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision]}, N[(N[Sin[x], $MachinePrecision] * N[(1.0 + N[(0.16666666666666666 * N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in y around 0

   \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
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 75.3%

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

4. Applied rewrites 75.3%

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

   1. rem-square-sqrt N/A

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

   2. sqrt-unprod N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 87.2%

      \[\leadsto \sin x \cdot \left(1 + 0.16666666666666666 \cdot \sqrt{{y}^{2} \cdot {y}^{2}}\right) \]

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 87.2%

      \[\leadsto \sin x \cdot \left(1 + 0.16666666666666666 \cdot \sqrt{\left(y \cdot y\right) \cdot {y}^{2}}\right) \]

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 87.2%

       \[\leadsto \sin x \cdot \left(1 + 0.16666666666666666 \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}\right) \]

6. Applied rewrites 87.2%

   \[\leadsto \sin x \cdot \left(1 + 0.16666666666666666 \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}\right) \]
7. Step-by-step derivation

   1. rem-square-sqrt N/A

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

   2. sqrt-unprod N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 93.5%

      \[\leadsto \sin x \cdot \left(1 + 0.16666666666666666 \cdot \sqrt{\sqrt{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}}\right) \]

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 93.5%

       \[\leadsto \sin x \cdot \left(1 + 0.16666666666666666 \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}}\right) \]

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 93.5%

       \[\leadsto \sin x \cdot \left(1 + 0.16666666666666666 \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right)}}\right) \]

8. Applied rewrites 93.5%

   \[\leadsto \sin x \cdot \left(1 + 0.16666666666666666 \cdot \sqrt{\sqrt{\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot \left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right)}}\right) \]
9. Add Preprocessing

Alternative 2: 87.2% accurate, 1.6× speedup

Math

\[\sin x \cdot \left(1 + 0.16666666666666666 \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}\right) \]

FPCore

(FPCore (x y)
  :precision binary64
  (* (sin x) (+ 1.0 (* 0.16666666666666666 (sqrt (* (* y y) (* y y)))))))

C

double code(double x, double y) {
	return sin(x) * (1.0 + (0.16666666666666666 * sqrt(((y * y) * (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) * (1.0d0 + (0.16666666666666666d0 * sqrt(((y * y) * (y * y)))))
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) * (1.0 + (0.16666666666666666 * Math.sqrt(((y * y) * (y * y)))));
}

Python

def code(x, y):
	return math.sin(x) * (1.0 + (0.16666666666666666 * math.sqrt(((y * y) * (y * y)))))

Julia

function code(x, y)
	return Float64(sin(x) * Float64(1.0 + Float64(0.16666666666666666 * sqrt(Float64(Float64(y * y) * Float64(y * y))))))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) * (1.0 + (0.16666666666666666 * sqrt(((y * y) * (y * y)))));
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * N[(1.0 + N[(0.16666666666666666 * N[Sqrt[N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in y around 0

   \[\leadsto \sin x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]
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 75.3%

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

4. Applied rewrites 75.3%

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

   1. rem-square-sqrt N/A

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

   2. sqrt-unprod N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 87.2%

      \[\leadsto \sin x \cdot \left(1 + 0.16666666666666666 \cdot \sqrt{{y}^{2} \cdot {y}^{2}}\right) \]

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 87.2%

      \[\leadsto \sin x \cdot \left(1 + 0.16666666666666666 \cdot \sqrt{\left(y \cdot y\right) \cdot {y}^{2}}\right) \]

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 87.2%

       \[\leadsto \sin x \cdot \left(1 + 0.16666666666666666 \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}\right) \]

6. Applied rewrites 87.2%

   \[\leadsto \sin x \cdot \left(1 + 0.16666666666666666 \cdot \sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}\right) \]
7. Add Preprocessing

Alternative 3: 80.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(y \cdot y\right) \cdot 0.16666666666666666 - -1\\ t_1 := \sin \left(\left|x\right|\right)\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \cdot \frac{\sinh y}{y} \leq -\infty:\\ \;\;\;\;\left(\left|x\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|x\right|\right)}^{2}\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- (* (* y y) 0.16666666666666666) -1.0))
       (t_1 (sin (fabs x))))
  (*
   (copysign 1.0 x)
   (if (<= (* t_1 (/ (sinh y) y)) (- INFINITY))
     (*
      (* (fabs x) (+ 1.0 (* -0.16666666666666666 (pow (fabs x) 2.0))))
      t_0)
     (* t_1 t_0)))))

C

double code(double x, double y) {
	double t_0 = ((y * y) * 0.16666666666666666) - -1.0;
	double t_1 = sin(fabs(x));
	double tmp;
	if ((t_1 * (sinh(y) / y)) <= -((double) INFINITY)) {
		tmp = (fabs(x) * (1.0 + (-0.16666666666666666 * pow(fabs(x), 2.0)))) * t_0;
	} else {
		tmp = t_1 * t_0;
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = ((y * y) * 0.16666666666666666) - -1.0;
	double t_1 = Math.sin(Math.abs(x));
	double tmp;
	if ((t_1 * (Math.sinh(y) / y)) <= -Double.POSITIVE_INFINITY) {
		tmp = (Math.abs(x) * (1.0 + (-0.16666666666666666 * Math.pow(Math.abs(x), 2.0)))) * t_0;
	} else {
		tmp = t_1 * t_0;
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x, y):
	t_0 = ((y * y) * 0.16666666666666666) - -1.0
	t_1 = math.sin(math.fabs(x))
	tmp = 0
	if (t_1 * (math.sinh(y) / y)) <= -math.inf:
		tmp = (math.fabs(x) * (1.0 + (-0.16666666666666666 * math.pow(math.fabs(x), 2.0)))) * t_0
	else:
		tmp = t_1 * t_0
	return math.copysign(1.0, x) * tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(y * y) * 0.16666666666666666) - -1.0)
	t_1 = sin(abs(x))
	tmp = 0.0
	if (Float64(t_1 * Float64(sinh(y) / y)) <= Float64(-Inf))
		tmp = Float64(Float64(abs(x) * Float64(1.0 + Float64(-0.16666666666666666 * (abs(x) ^ 2.0)))) * t_0);
	else
		tmp = Float64(t_1 * t_0);
	end
	return Float64(copysign(1.0, x) * tmp)
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((y * y) * 0.16666666666666666) - -1.0;
	t_1 = sin(abs(x));
	tmp = 0.0;
	if ((t_1 * (sinh(y) / y)) <= -Inf)
		tmp = (abs(x) * (1.0 + (-0.16666666666666666 * (abs(x) ^ 2.0)))) * t_0;
	else
		tmp = t_1 * t_0;
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$1 = N[Sin[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[(t$95$1 * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(N[Abs[x], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[Power[N[Abs[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$1 * t$95$0), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 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 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 75.3%

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

   4. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 75.3%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 75.3%

         \[\leadsto \sin x \cdot \left({y}^{2} \cdot 0.16666666666666666 - -1\right) \]

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 75.3%

          \[\leadsto \sin x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666 - -1\right) \]

   6. Applied rewrites 75.3%

      \[\leadsto \sin x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666 - \color{blue}{-1}\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)} \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666 - -1\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 49.7%

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

   9. Applied rewrites 49.7%

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

   if -inf.0 < (*.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 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 75.3%

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

   4. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 75.3%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 75.3%

         \[\leadsto \sin x \cdot \left({y}^{2} \cdot 0.16666666666666666 - -1\right) \]

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 75.3%

          \[\leadsto \sin x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666 - -1\right) \]

   6. Applied rewrites 75.3%

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

Alternative 4: 78.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x, y):
	t_0 = math.fabs(x) * math.fabs(x)
	t_1 = math.sin(math.fabs(x))
	tmp = 0
	if (t_1 * (math.sinh(y) / y)) <= -math.inf:
		tmp = (math.fabs(x) * (1.0 + (-0.16666666666666666 * math.sqrt((t_0 * t_0))))) * 1.0
	else:
		tmp = t_1 * (((y * y) * 0.16666666666666666) - -1.0)
	return math.copysign(1.0, x) * tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	t_0 = abs(x) * abs(x);
	t_1 = sin(abs(x));
	tmp = 0.0;
	if ((t_1 * (sinh(y) / y)) <= -Inf)
		tmp = (abs(x) * (1.0 + (-0.16666666666666666 * sqrt((t_0 * t_0))))) * 1.0;
	else
		tmp = t_1 * (((y * y) * 0.16666666666666666) - -1.0);
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 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 y around 0

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 33.8%

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

   7. Applied rewrites 33.8%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 34.8%

         \[\leadsto \left(x \cdot \left(1 + -0.16666666666666666 \cdot \sqrt{{x}^{2} \cdot {x}^{2}}\right)\right) \cdot 1 \]

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 34.8%

         \[\leadsto \left(x \cdot \left(1 + -0.16666666666666666 \cdot \sqrt{\left(x \cdot x\right) \cdot {x}^{2}}\right)\right) \cdot 1 \]

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 34.8%

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

   9. Applied rewrites 34.8%

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

   if -inf.0 < (*.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 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 75.3%

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

   4. Applied rewrites 75.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 75.3%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 75.3%

         \[\leadsto \sin x \cdot \left({y}^{2} \cdot 0.16666666666666666 - -1\right) \]

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 75.3%

          \[\leadsto \sin x \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666 - -1\right) \]

   6. Applied rewrites 75.3%

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

Alternative 5: 78.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x, y):
	t_0 = math.fabs(x) * math.fabs(x)
	t_1 = math.sin(math.fabs(x))
	tmp = 0
	if (t_1 * (math.sinh(y) / y)) <= -math.inf:
		tmp = (math.fabs(x) * (1.0 + (-0.16666666666666666 * math.sqrt((t_0 * t_0))))) * 1.0
	else:
		tmp = t_1 * (1.0 + ((0.16666666666666666 * y) * y))
	return math.copysign(1.0, x) * tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	t_0 = abs(x) * abs(x);
	t_1 = sin(abs(x));
	tmp = 0.0;
	if ((t_1 * (sinh(y) / y)) <= -Inf)
		tmp = (abs(x) * (1.0 + (-0.16666666666666666 * sqrt((t_0 * t_0))))) * 1.0;
	else
		tmp = t_1 * (1.0 + ((0.16666666666666666 * y) * y));
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 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 y around 0

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 33.8%

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

   7. Applied rewrites 33.8%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 34.8%

         \[\leadsto \left(x \cdot \left(1 + -0.16666666666666666 \cdot \sqrt{{x}^{2} \cdot {x}^{2}}\right)\right) \cdot 1 \]

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 34.8%

         \[\leadsto \left(x \cdot \left(1 + -0.16666666666666666 \cdot \sqrt{\left(x \cdot x\right) \cdot {x}^{2}}\right)\right) \cdot 1 \]

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 34.8%

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

   9. Applied rewrites 34.8%

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

   if -inf.0 < (*.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 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 75.3%

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

   4. Applied rewrites 75.3%

      \[\leadsto \sin x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   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 75.3%

         \[\leadsto \sin x \cdot \left(1 + \left(0.16666666666666666 \cdot y\right) \cdot y\right) \]

   6. Applied rewrites 75.3%

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

Alternative 6: 65.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y) {
	double t_0 = Math.abs(x) * Math.abs(x);
	double t_1 = Math.sin(Math.abs(x));
	double t_2 = t_1 * (Math.sinh(y) / y);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (Math.abs(x) * (1.0 + (-0.16666666666666666 * Math.sqrt((t_0 * t_0))))) * 1.0;
	} else if (t_2 <= 1.0) {
		tmp = t_1 * 1.0;
	} else {
		tmp = (Math.abs(x) * y) / y;
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x, y):
	t_0 = math.fabs(x) * math.fabs(x)
	t_1 = math.sin(math.fabs(x))
	t_2 = t_1 * (math.sinh(y) / y)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (math.fabs(x) * (1.0 + (-0.16666666666666666 * math.sqrt((t_0 * t_0))))) * 1.0
	elif t_2 <= 1.0:
		tmp = t_1 * 1.0
	else:
		tmp = (math.fabs(x) * y) / y
	return math.copysign(1.0, x) * tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	t_0 = abs(x) * abs(x);
	t_1 = sin(abs(x));
	t_2 = t_1 * (sinh(y) / y);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (abs(x) * (1.0 + (-0.16666666666666666 * sqrt((t_0 * t_0))))) * 1.0;
	elseif (t_2 <= 1.0)
		tmp = t_1 * 1.0;
	else
		tmp = (abs(x) * y) / y;
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

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

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 33.8%

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

   7. Applied rewrites 33.8%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 34.8%

         \[\leadsto \left(x \cdot \left(1 + -0.16666666666666666 \cdot \sqrt{{x}^{2} \cdot {x}^{2}}\right)\right) \cdot 1 \]

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 34.8%

         \[\leadsto \left(x \cdot \left(1 + -0.16666666666666666 \cdot \sqrt{\left(x \cdot x\right) \cdot {x}^{2}}\right)\right) \cdot 1 \]

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 34.8%

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

   9. Applied rewrites 34.8%

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

   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 50.0%

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

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

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. 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 89.0%

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

   3. Applied rewrites 89.0%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 39.0%

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

   6. Applied rewrites 39.0%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 20.5%

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

   9. Applied rewrites 20.5%

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

Alternative 7: 41.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|x\right|\right) \cdot \frac{\sinh y}{y} \leq 0.002:\\ \;\;\;\;\left(\left|x\right| \cdot \left(1 + -0.16666666666666666 \cdot \sqrt{t\_0 \cdot t\_0}\right)\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right| \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x))))
  (*
   (copysign 1.0 x)
   (if (<= (* (sin (fabs x)) (/ (sinh y) y)) 0.002)
     (*
      (* (fabs x) (+ 1.0 (* -0.16666666666666666 (sqrt (* t_0 t_0)))))
      1.0)
     (/ (* (fabs x) y) y)))))

C

double code(double x, double y) {
	double t_0 = fabs(x) * fabs(x);
	double tmp;
	if ((sin(fabs(x)) * (sinh(y) / y)) <= 0.002) {
		tmp = (fabs(x) * (1.0 + (-0.16666666666666666 * sqrt((t_0 * t_0))))) * 1.0;
	} else {
		tmp = (fabs(x) * y) / y;
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.abs(x) * Math.abs(x);
	double tmp;
	if ((Math.sin(Math.abs(x)) * (Math.sinh(y) / y)) <= 0.002) {
		tmp = (Math.abs(x) * (1.0 + (-0.16666666666666666 * Math.sqrt((t_0 * t_0))))) * 1.0;
	} else {
		tmp = (Math.abs(x) * y) / y;
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x, y):
	t_0 = math.fabs(x) * math.fabs(x)
	tmp = 0
	if (math.sin(math.fabs(x)) * (math.sinh(y) / y)) <= 0.002:
		tmp = (math.fabs(x) * (1.0 + (-0.16666666666666666 * math.sqrt((t_0 * t_0))))) * 1.0
	else:
		tmp = (math.fabs(x) * y) / y
	return math.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)) <= 0.002)
		tmp = Float64(Float64(abs(x) * Float64(1.0 + Float64(-0.16666666666666666 * sqrt(Float64(t_0 * t_0))))) * 1.0);
	else
		tmp = Float64(Float64(abs(x) * y) / y);
	end
	return Float64(copysign(1.0, x) * tmp)
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = abs(x) * abs(x);
	tmp = 0.0;
	if ((sin(abs(x)) * (sinh(y) / y)) <= 0.002)
		tmp = (abs(x) * (1.0 + (-0.16666666666666666 * sqrt((t_0 * t_0))))) * 1.0;
	else
		tmp = (abs(x) * y) / y;
	end
	tmp_2 = (sign(x) * abs(1.0)) * 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], 0.002], N[(N[(N[Abs[x], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Abs[x], $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 2e-3

   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 50.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 33.8%

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

   7. Applied rewrites 33.8%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 34.8%

         \[\leadsto \left(x \cdot \left(1 + -0.16666666666666666 \cdot \sqrt{{x}^{2} \cdot {x}^{2}}\right)\right) \cdot 1 \]

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 34.8%

         \[\leadsto \left(x \cdot \left(1 + -0.16666666666666666 \cdot \sqrt{\left(x \cdot x\right) \cdot {x}^{2}}\right)\right) \cdot 1 \]

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 34.8%

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

   9. Applied rewrites 34.8%

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

   if 2e-3 < (*.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. 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 89.0%

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

   3. Applied rewrites 89.0%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 39.0%

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

   6. Applied rewrites 39.0%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 20.5%

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

   9. Applied rewrites 20.5%

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

Alternative 8: 40.8% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|x\right|\right) \cdot \frac{\sinh y}{y} \leq 0.002:\\ \;\;\;\;\left(\left|x\right| - \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot 0.16666666666666666\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right| \cdot y}{y}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (*
 (copysign 1.0 x)
 (if (<= (* (sin (fabs x)) (/ (sinh y) y)) 0.002)
   (*
    (-
     (fabs x)
     (* (* (* (fabs x) (fabs x)) (fabs x)) 0.16666666666666666))
    1.0)
   (/ (* (fabs x) y) y))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(fabs(x)) * (sinh(y) / y)) <= 0.002) {
		tmp = (fabs(x) - (((fabs(x) * fabs(x)) * fabs(x)) * 0.16666666666666666)) * 1.0;
	} else {
		tmp = (fabs(x) * 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)) <= 0.002) {
		tmp = (Math.abs(x) - (((Math.abs(x) * Math.abs(x)) * Math.abs(x)) * 0.16666666666666666)) * 1.0;
	} else {
		tmp = (Math.abs(x) * 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)) <= 0.002:
		tmp = (math.fabs(x) - (((math.fabs(x) * math.fabs(x)) * math.fabs(x)) * 0.16666666666666666)) * 1.0
	else:
		tmp = (math.fabs(x) * 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)) <= 0.002)
		tmp = Float64(Float64(abs(x) - Float64(Float64(Float64(abs(x) * abs(x)) * abs(x)) * 0.16666666666666666)) * 1.0);
	else
		tmp = Float64(Float64(abs(x) * 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)) <= 0.002)
		tmp = (abs(x) - (((abs(x) * abs(x)) * abs(x)) * 0.16666666666666666)) * 1.0;
	else
		tmp = (abs(x) * 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], 0.002], N[(N[(N[Abs[x], $MachinePrecision] - N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Abs[x], $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 2e-3

   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 50.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 33.8%

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

   7. Applied rewrites 33.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. add-flip N/A

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

      5. *-lft-identity N/A

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

      6. *-commutative N/A

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

      7. lower--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. cube-unmult N/A

          \[\leadsto \left(x - {x}^{3} \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{6}}\right)\right)\right) \cdot 1 \]

      15. lower-pow.f32 N/A

          \[\leadsto \left(x - {x}^{3} \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{6}}\right)\right)\right) \cdot 1 \]

      16. lower-unsound-pow.f32 N/A

          \[\leadsto \left(x - {x}^{3} \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{6}}\right)\right)\right) \cdot 1 \]

      17. *-lft-identity N/A

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

      18. metadata-eval N/A

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

      19. lower-*.f64 N/A

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

   9. Applied rewrites 33.9%

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

   if 2e-3 < (*.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. 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 89.0%

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

   3. Applied rewrites 89.0%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 39.0%

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

   6. Applied rewrites 39.0%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 20.5%

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

   9. Applied rewrites 20.5%

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

Alternative 9: 31.5% accurate, 0.9× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\sin \left(\left|x\right|\right) \leq 10^{-13}:\\ \;\;\;\;\left(\left|x\right| \cdot 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right| \cdot y}{y}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (*
 (copysign 1.0 x)
 (if (<= (sin (fabs x)) 1e-13)
   (* (* (fabs x) 1.0) 1.0)
   (/ (* (fabs x) y) y))))

C

double code(double x, double y) {
	double tmp;
	if (sin(fabs(x)) <= 1e-13) {
		tmp = (fabs(x) * 1.0) * 1.0;
	} else {
		tmp = (fabs(x) * 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)) <= 1e-13) {
		tmp = (Math.abs(x) * 1.0) * 1.0;
	} else {
		tmp = (Math.abs(x) * y) / y;
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sin(math.fabs(x)) <= 1e-13:
		tmp = (math.fabs(x) * 1.0) * 1.0
	else:
		tmp = (math.fabs(x) * y) / y
	return math.copysign(1.0, x) * tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sin(abs(x)) <= 1e-13)
		tmp = Float64(Float64(abs(x) * 1.0) * 1.0);
	else
		tmp = Float64(Float64(abs(x) * 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)) <= 1e-13)
		tmp = (abs(x) * 1.0) * 1.0;
	else
		tmp = (abs(x) * 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[Sin[N[Abs[x], $MachinePrecision]], $MachinePrecision], 1e-13], N[(N[(N[Abs[x], $MachinePrecision] * 1.0), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Abs[x], $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 1e-13

   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 50.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 33.8%

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

   7. Applied rewrites 33.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \left(x \cdot 1\right) \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 25.7%

       \[\leadsto \left(x \cdot 1\right) \cdot 1 \]

   if 1e-13 < (sin.f64 x)

   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 89.0%

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

   3. Applied rewrites 89.0%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 39.0%

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

   6. Applied rewrites 39.0%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 20.5%

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

   9. Applied rewrites 20.5%

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

Alternative 10: 25.7% accurate, 19.7× speedup

Math

\[\left(x \cdot 1\right) \cdot 1 \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(x * 1.0), $MachinePrecision] * 1.0), $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 50.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 33.8%

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

7. Applied rewrites 33.8%

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

8. Taylor expanded in x around 0

   \[\leadsto \left(x \cdot 1\right) \cdot 1 \]
9. Step-by-step derivation

10. Applied rewrites 25.7%

    \[\leadsto \left(x \cdot 1\right) \cdot 1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))