2sin (example 3.3)

Percentage Accurate: 62.5% → 100.0%

Time: 7.6s

Alternatives: 20

Speedup: 207.0×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

Math

\[\begin{array}{l} \\ \sin \left(x + \varepsilon\right) - \sin x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))

C

double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = sin((x + eps)) - sin(x)
end function

Java

public static double code(double x, double eps) {
	return Math.sin((x + eps)) - Math.sin(x);
}

Python

def code(x, eps):
	return math.sin((x + eps)) - math.sin(x)

Julia

function code(x, eps)
	return Float64(sin(Float64(x + eps)) - sin(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = sin((x + eps)) - sin(x);
end

Wolfram

code[x_, eps_] := N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]

Initial Program: 62.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin \left(x + \varepsilon\right) - \sin x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))

C

double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = sin((x + eps)) - sin(x)
end function

Java

public static double code(double x, double eps) {
	return Math.sin((x + eps)) - Math.sin(x);
}

Python

def code(x, eps):
	return math.sin((x + eps)) - math.sin(x)

Julia

function code(x, eps)
	return Float64(sin(Float64(x + eps)) - sin(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = sin((x + eps)) - sin(x);
end

Wolfram

code[x_, eps_] := N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ \left(\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(2 \cdot x\right) \cdot 0.5\right) - t\_0 \cdot \sin x\right) \cdot t\_0\right) \cdot 2 \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 eps))))
   (*
    (* (- (* (cos (* 0.5 eps)) (cos (* (* 2.0 x) 0.5))) (* t_0 (sin x))) t_0)
    2.0)))

C

double code(double x, double eps) {
	double t_0 = sin((0.5 * eps));
	return (((cos((0.5 * eps)) * cos(((2.0 * x) * 0.5))) - (t_0 * sin(x))) * t_0) * 2.0;
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = sin((0.5d0 * eps))
    code = (((cos((0.5d0 * eps)) * cos(((2.0d0 * x) * 0.5d0))) - (t_0 * sin(x))) * t_0) * 2.0d0
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((0.5 * eps));
	return (((Math.cos((0.5 * eps)) * Math.cos(((2.0 * x) * 0.5))) - (t_0 * Math.sin(x))) * t_0) * 2.0;
}

Python

def code(x, eps):
	t_0 = math.sin((0.5 * eps))
	return (((math.cos((0.5 * eps)) * math.cos(((2.0 * x) * 0.5))) - (t_0 * math.sin(x))) * t_0) * 2.0

Julia

function code(x, eps)
	t_0 = sin(Float64(0.5 * eps))
	return Float64(Float64(Float64(Float64(cos(Float64(0.5 * eps)) * cos(Float64(Float64(2.0 * x) * 0.5))) - Float64(t_0 * sin(x))) * t_0) * 2.0)
end

MATLAB

function tmp = code(x, eps)
	t_0 = sin((0.5 * eps));
	tmp = (((cos((0.5 * eps)) * cos(((2.0 * x) * 0.5))) - (t_0 * sin(x))) * t_0) * 2.0;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(N[Cos[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(2.0 * x), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right) - \sin x} \]

   2. lift-+.f64 N/A

      \[\leadsto \sin \color{blue}{\left(x + \varepsilon\right)} - \sin x \]

   3. lift-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right)} - \sin x \]

   4. lift-sin.f64 N/A

      \[\leadsto \sin \left(x + \varepsilon\right) - \color{blue}{\sin x} \]

   5. diff-sin N/A

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2} \]

3. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \sin \left(\frac{\left(\varepsilon + x\right) - x}{2}\right)\right) \cdot 2} \]

4. Taylor expanded in x around inf

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

   1. fp-cancel-sign-sub-inv N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - \left(\mathsf{neg}\left(2\right)\right) \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   2. metadata-eval N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot 2 \]

   4. lower-cos.f64 N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot 2 \]

   5. metadata-eval N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - \left(\mathsf{neg}\left(2\right)\right) \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   6. fp-cancel-sign-sub-inv N/A

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

   7. count-2-rev N/A

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

   8. associate-+l+ N/A

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

   9. *-commutative N/A

      \[\leadsto \left(\cos \left(\left(\left(\varepsilon + x\right) + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \varepsilon\right)\right) \cdot 2 \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\cos \left(\left(\left(\varepsilon + x\right) + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \varepsilon\right)\right) \cdot 2 \]

   11. associate-+l+ N/A

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

   12. count-2-rev N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

       \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   15. lower-sin.f64 N/A

       \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   16. lower-*.f64 99.9

       \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

6. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)} \cdot 2 \]
7. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot 2 \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \varepsilon\right)\right) \cdot 2 \]

   3. lift-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. count-2-rev N/A

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

   6. associate-+l+ N/A

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

   7. *-commutative N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\left(\varepsilon + x\right) + x\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \varepsilon\right)\right) \cdot 2 \]

   8. associate-+l+ N/A

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

   9. count-2-rev N/A

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

   10. fp-cancel-sign-sub-inv N/A

       \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - \left(\mathsf{neg}\left(2\right)\right) \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   11. metadata-eval N/A

       \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   12. metadata-eval N/A

       \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - \left(\mathsf{neg}\left(2\right)\right) \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   13. fp-cancel-sign-sub-inv N/A

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

   14. distribute-rgt-in N/A

       \[\leadsto \left(\cos \left(\varepsilon \cdot \frac{1}{2} + \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \varepsilon\right)\right) \cdot 2 \]

   15. *-commutative N/A

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

   16. cos-sum N/A

       \[\leadsto \left(\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot 2 \]

   17. lower--.f64 N/A

       \[\leadsto \left(\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot 2 \]

8. Applied rewrites 100.0%

   \[\leadsto \left(\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(2 \cdot x\right) \cdot 0.5\right) - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot 0.5\right)\right) \cdot \sin \color{blue}{\left(0.5 \cdot \varepsilon\right)}\right) \cdot 2 \]

9. Taylor expanded in x around 0

   \[\leadsto \left(\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]
10. Step-by-step derivation

11. Applied rewrites 100.0%

    \[\leadsto \left(\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\left(2 \cdot x\right) \cdot 0.5\right) - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin x\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]
12. Add Preprocessing

Alternative 2: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (* (cos (* (fma 2.0 x eps) 0.5)) (sin (* 0.5 eps))) 2.0))

C

double code(double x, double eps) {
	return (cos((fma(2.0, x, eps) * 0.5)) * sin((0.5 * eps))) * 2.0;
}

Julia

function code(x, eps)
	return Float64(Float64(cos(Float64(fma(2.0, x, eps) * 0.5)) * sin(Float64(0.5 * eps))) * 2.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[Cos[N[(N[(2.0 * x + eps), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right) - \sin x} \]

   2. lift-+.f64 N/A

      \[\leadsto \sin \color{blue}{\left(x + \varepsilon\right)} - \sin x \]

   3. lift-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right)} - \sin x \]

   4. lift-sin.f64 N/A

      \[\leadsto \sin \left(x + \varepsilon\right) - \color{blue}{\sin x} \]

   5. diff-sin N/A

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2} \]

3. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \sin \left(\frac{\left(\varepsilon + x\right) - x}{2}\right)\right) \cdot 2} \]

4. Taylor expanded in x around inf

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

   1. fp-cancel-sign-sub-inv N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - \left(\mathsf{neg}\left(2\right)\right) \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   2. metadata-eval N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot 2 \]

   4. lower-cos.f64 N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot 2 \]

   5. metadata-eval N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - \left(\mathsf{neg}\left(2\right)\right) \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   6. fp-cancel-sign-sub-inv N/A

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

   7. count-2-rev N/A

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

   8. associate-+l+ N/A

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

   9. *-commutative N/A

      \[\leadsto \left(\cos \left(\left(\left(\varepsilon + x\right) + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \varepsilon\right)\right) \cdot 2 \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\cos \left(\left(\left(\varepsilon + x\right) + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \varepsilon\right)\right) \cdot 2 \]

   11. associate-+l+ N/A

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

   12. count-2-rev N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

       \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   15. lower-sin.f64 N/A

       \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   16. lower-*.f64 99.9

       \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

6. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)} \cdot 2 \]
7. Add Preprocessing

Alternative 3: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2} + \frac{\pi}{2}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-1.5500992063492063 \cdot 10^{-6}, \varepsilon \cdot \varepsilon, 0.00026041666666666666\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (*
   (sin (+ (/ (fma 2.0 x eps) 2.0) (/ PI 2.0)))
   (*
    (fma
     (-
      (*
       (fma -1.5500992063492063e-6 (* eps eps) 0.00026041666666666666)
       (* eps eps))
      0.020833333333333332)
     (* eps eps)
     0.5)
    eps))
  2.0))

C

double code(double x, double eps) {
	return (sin(((fma(2.0, x, eps) / 2.0) + (((double) M_PI) / 2.0))) * (fma(((fma(-1.5500992063492063e-6, (eps * eps), 0.00026041666666666666) * (eps * eps)) - 0.020833333333333332), (eps * eps), 0.5) * eps)) * 2.0;
}

Julia

function code(x, eps)
	return Float64(Float64(sin(Float64(Float64(fma(2.0, x, eps) / 2.0) + Float64(pi / 2.0))) * Float64(fma(Float64(Float64(fma(-1.5500992063492063e-6, Float64(eps * eps), 0.00026041666666666666) * Float64(eps * eps)) - 0.020833333333333332), Float64(eps * eps), 0.5) * eps)) * 2.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[Sin[N[(N[(N[(2.0 * x + eps), $MachinePrecision] / 2.0), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(N[(-1.5500992063492063e-6 * N[(eps * eps), $MachinePrecision] + 0.00026041666666666666), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 0.020833333333333332), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 0.5), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right) - \sin x} \]

   2. lift-+.f64 N/A

      \[\leadsto \sin \color{blue}{\left(x + \varepsilon\right)} - \sin x \]

   3. lift-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right)} - \sin x \]

   4. lift-sin.f64 N/A

      \[\leadsto \sin \left(x + \varepsilon\right) - \color{blue}{\sin x} \]

   5. diff-sin N/A

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2} \]

3. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \sin \left(\frac{\left(\varepsilon + x\right) - x}{2}\right)\right) \cdot 2} \]

4. Taylor expanded in eps around 0

   \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right)}\right) \cdot 2 \]
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{\varepsilon}\right)\right) \cdot 2 \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{\varepsilon}\right)\right) \cdot 2 \]

   3. +-commutative N/A

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\left(\frac{-1}{48} \cdot {\varepsilon}^{2} + \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   4. lower-fma.f64 N/A

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   5. unpow2 N/A

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   6. lower-*.f64 99.7

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot 2 \]

6. Applied rewrites 99.7%

   \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)}\right) \cdot 2 \]
7. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \left(\color{blue}{\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   2. lift-/.f64 N/A

      \[\leadsto \left(\cos \color{blue}{\left(\frac{\left(\varepsilon + x\right) + x}{2}\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   3. lift-+.f64 N/A

      \[\leadsto \left(\cos \left(\frac{\color{blue}{\left(\varepsilon + x\right) + x}}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   4. lift-+.f64 N/A

      \[\leadsto \left(\cos \left(\frac{\color{blue}{\left(\varepsilon + x\right)} + x}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   5. sin-+PI/2-rev N/A

      \[\leadsto \left(\color{blue}{\sin \left(\frac{\left(\varepsilon + x\right) + x}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   6. lower-sin.f64 N/A

      \[\leadsto \left(\color{blue}{\sin \left(\frac{\left(\varepsilon + x\right) + x}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   7. lower-+.f64 N/A

      \[\leadsto \left(\sin \color{blue}{\left(\frac{\left(\varepsilon + x\right) + x}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   8. lower-/.f64 N/A

      \[\leadsto \left(\sin \left(\color{blue}{\frac{\left(\varepsilon + x\right) + x}{2}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   9. associate-+l+ N/A

      \[\leadsto \left(\sin \left(\frac{\color{blue}{\varepsilon + \left(x + x\right)}}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   10. count-2-rev N/A

       \[\leadsto \left(\sin \left(\frac{\varepsilon + \color{blue}{2 \cdot x}}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   11. +-commutative N/A

       \[\leadsto \left(\sin \left(\frac{\color{blue}{2 \cdot x + \varepsilon}}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   12. lift-fma.f64 N/A

       \[\leadsto \left(\sin \left(\frac{\color{blue}{\mathsf{fma}\left(2, x, \varepsilon\right)}}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   13. lower-/.f64 N/A

       \[\leadsto \left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2} + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   14. lower-PI.f64 99.7

       \[\leadsto \left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2} + \frac{\color{blue}{\pi}}{2}\right) \cdot \left(\mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot 2 \]

8. Applied rewrites 99.7%

   \[\leadsto \left(\color{blue}{\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2} + \frac{\pi}{2}\right)} \cdot \left(\mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot 2 \]

9. Taylor expanded in eps around 0

   \[\leadsto \left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2} + \frac{\pi}{2}\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right)\right)}\right) \cdot 2 \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2} + \frac{\pi}{2}\right) \cdot \left(\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right) \cdot \color{blue}{\varepsilon}\right)\right) \cdot 2 \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2} + \frac{\pi}{2}\right) \cdot \left(\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right) \cdot \color{blue}{\varepsilon}\right)\right) \cdot 2 \]

11. Applied rewrites 99.8%

    \[\leadsto \left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2} + \frac{\pi}{2}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.5500992063492063 \cdot 10^{-6}, \varepsilon \cdot \varepsilon, 0.00026041666666666666\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)}\right) \cdot 2 \]
12. Add Preprocessing

Alternative 4: 99.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-1.5500992063492063 \cdot 10^{-6}, \varepsilon \cdot \varepsilon, 0.00026041666666666666\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (*
   (cos (* (fma 2.0 x eps) 0.5))
   (*
    (fma
     (-
      (*
       (fma -1.5500992063492063e-6 (* eps eps) 0.00026041666666666666)
       (* eps eps))
      0.020833333333333332)
     (* eps eps)
     0.5)
    eps))
  2.0))

C

double code(double x, double eps) {
	return (cos((fma(2.0, x, eps) * 0.5)) * (fma(((fma(-1.5500992063492063e-6, (eps * eps), 0.00026041666666666666) * (eps * eps)) - 0.020833333333333332), (eps * eps), 0.5) * eps)) * 2.0;
}

Julia

function code(x, eps)
	return Float64(Float64(cos(Float64(fma(2.0, x, eps) * 0.5)) * Float64(fma(Float64(Float64(fma(-1.5500992063492063e-6, Float64(eps * eps), 0.00026041666666666666) * Float64(eps * eps)) - 0.020833333333333332), Float64(eps * eps), 0.5) * eps)) * 2.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[Cos[N[(N[(2.0 * x + eps), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(N[(-1.5500992063492063e-6 * N[(eps * eps), $MachinePrecision] + 0.00026041666666666666), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 0.020833333333333332), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 0.5), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right) - \sin x} \]

   2. lift-+.f64 N/A

      \[\leadsto \sin \color{blue}{\left(x + \varepsilon\right)} - \sin x \]

   3. lift-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right)} - \sin x \]

   4. lift-sin.f64 N/A

      \[\leadsto \sin \left(x + \varepsilon\right) - \color{blue}{\sin x} \]

   5. diff-sin N/A

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2} \]

3. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \sin \left(\frac{\left(\varepsilon + x\right) - x}{2}\right)\right) \cdot 2} \]

4. Taylor expanded in x around inf

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

   1. fp-cancel-sign-sub-inv N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - \left(\mathsf{neg}\left(2\right)\right) \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   2. metadata-eval N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot 2 \]

   4. lower-cos.f64 N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot 2 \]

   5. metadata-eval N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - \left(\mathsf{neg}\left(2\right)\right) \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   6. fp-cancel-sign-sub-inv N/A

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

   7. count-2-rev N/A

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

   8. associate-+l+ N/A

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

   9. *-commutative N/A

      \[\leadsto \left(\cos \left(\left(\left(\varepsilon + x\right) + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \varepsilon\right)\right) \cdot 2 \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\cos \left(\left(\left(\varepsilon + x\right) + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \varepsilon\right)\right) \cdot 2 \]

   11. associate-+l+ N/A

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

   12. count-2-rev N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

       \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   15. lower-sin.f64 N/A

       \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   16. lower-*.f64 99.9

       \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

6. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)} \cdot 2 \]

7. Taylor expanded in eps around 0

   \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right)}\right)\right) \cdot 2 \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right) \cdot \varepsilon\right)\right) \cdot 2 \]

9. Applied rewrites 99.8%

   \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-1.5500992063492063 \cdot 10^{-6}, \varepsilon \cdot \varepsilon, 0.00026041666666666666\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \color{blue}{\varepsilon}\right)\right) \cdot 2 \]
10. Add Preprocessing

Alternative 5: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(0.00026041666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (*
   (cos (* (fma 2.0 x eps) 0.5))
   (*
    (fma
     (- (* 0.00026041666666666666 (* eps eps)) 0.020833333333333332)
     (* eps eps)
     0.5)
    eps))
  2.0))

C

double code(double x, double eps) {
	return (cos((fma(2.0, x, eps) * 0.5)) * (fma(((0.00026041666666666666 * (eps * eps)) - 0.020833333333333332), (eps * eps), 0.5) * eps)) * 2.0;
}

Julia

function code(x, eps)
	return Float64(Float64(cos(Float64(fma(2.0, x, eps) * 0.5)) * Float64(fma(Float64(Float64(0.00026041666666666666 * Float64(eps * eps)) - 0.020833333333333332), Float64(eps * eps), 0.5) * eps)) * 2.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[Cos[N[(N[(2.0 * x + eps), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(0.00026041666666666666 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 0.020833333333333332), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 0.5), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right) - \sin x} \]

   2. lift-+.f64 N/A

      \[\leadsto \sin \color{blue}{\left(x + \varepsilon\right)} - \sin x \]

   3. lift-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right)} - \sin x \]

   4. lift-sin.f64 N/A

      \[\leadsto \sin \left(x + \varepsilon\right) - \color{blue}{\sin x} \]

   5. diff-sin N/A

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2} \]

3. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \sin \left(\frac{\left(\varepsilon + x\right) - x}{2}\right)\right) \cdot 2} \]

4. Taylor expanded in x around inf

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

   1. fp-cancel-sign-sub-inv N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - \left(\mathsf{neg}\left(2\right)\right) \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   2. metadata-eval N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot 2 \]

   4. lower-cos.f64 N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot 2 \]

   5. metadata-eval N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - \left(\mathsf{neg}\left(2\right)\right) \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   6. fp-cancel-sign-sub-inv N/A

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

   7. count-2-rev N/A

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

   8. associate-+l+ N/A

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

   9. *-commutative N/A

      \[\leadsto \left(\cos \left(\left(\left(\varepsilon + x\right) + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \varepsilon\right)\right) \cdot 2 \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\cos \left(\left(\left(\varepsilon + x\right) + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \varepsilon\right)\right) \cdot 2 \]

   11. associate-+l+ N/A

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

   12. count-2-rev N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

       \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   15. lower-sin.f64 N/A

       \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   16. lower-*.f64 99.9

       \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

6. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)} \cdot 2 \]

7. Taylor expanded in eps around 0

   \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)}\right)\right) \cdot 2 \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   3. +-commutative N/A

      \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \left(\left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right) + \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   4. *-commutative N/A

      \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \left(\left(\left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right) \cdot {\varepsilon}^{2} + \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}, {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   6. lower--.f64 N/A

      \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}, {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}, {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   8. pow2 N/A

      \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{1}{3840} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{48}, {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{1}{3840} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{48}, {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   10. pow2 N/A

       \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{1}{3840} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   11. lift-*.f64 99.7

       \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(0.00026041666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot 2 \]

9. Applied rewrites 99.7%

   \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right) \cdot \left(\mathsf{fma}\left(0.00026041666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \color{blue}{\varepsilon}\right)\right) \cdot 2 \]
10. Add Preprocessing

Alternative 6: 99.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\sin \left(\mathsf{fma}\left(0.5, \varepsilon + \pi, x\right)\right) \cdot \left(\mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (*
   (sin (fma 0.5 (+ eps PI) x))
   (* (fma -0.020833333333333332 (* eps eps) 0.5) eps))
  2.0))

C

double code(double x, double eps) {
	return (sin(fma(0.5, (eps + ((double) M_PI)), x)) * (fma(-0.020833333333333332, (eps * eps), 0.5) * eps)) * 2.0;
}

Julia

function code(x, eps)
	return Float64(Float64(sin(fma(0.5, Float64(eps + pi), x)) * Float64(fma(-0.020833333333333332, Float64(eps * eps), 0.5) * eps)) * 2.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[Sin[N[(0.5 * N[(eps + Pi), $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision] * N[(N[(-0.020833333333333332 * N[(eps * eps), $MachinePrecision] + 0.5), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right) - \sin x} \]

   2. lift-+.f64 N/A

      \[\leadsto \sin \color{blue}{\left(x + \varepsilon\right)} - \sin x \]

   3. lift-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right)} - \sin x \]

   4. lift-sin.f64 N/A

      \[\leadsto \sin \left(x + \varepsilon\right) - \color{blue}{\sin x} \]

   5. diff-sin N/A

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2} \]

3. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \sin \left(\frac{\left(\varepsilon + x\right) - x}{2}\right)\right) \cdot 2} \]

4. Taylor expanded in eps around 0

   \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right)}\right) \cdot 2 \]
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{\varepsilon}\right)\right) \cdot 2 \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{\varepsilon}\right)\right) \cdot 2 \]

   3. +-commutative N/A

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\left(\frac{-1}{48} \cdot {\varepsilon}^{2} + \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   4. lower-fma.f64 N/A

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   5. unpow2 N/A

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   6. lower-*.f64 99.7

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot 2 \]

6. Applied rewrites 99.7%

   \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)}\right) \cdot 2 \]
7. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \left(\color{blue}{\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   2. lift-/.f64 N/A

      \[\leadsto \left(\cos \color{blue}{\left(\frac{\left(\varepsilon + x\right) + x}{2}\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   3. lift-+.f64 N/A

      \[\leadsto \left(\cos \left(\frac{\color{blue}{\left(\varepsilon + x\right) + x}}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   4. lift-+.f64 N/A

      \[\leadsto \left(\cos \left(\frac{\color{blue}{\left(\varepsilon + x\right)} + x}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   5. sin-+PI/2-rev N/A

      \[\leadsto \left(\color{blue}{\sin \left(\frac{\left(\varepsilon + x\right) + x}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   6. lower-sin.f64 N/A

      \[\leadsto \left(\color{blue}{\sin \left(\frac{\left(\varepsilon + x\right) + x}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   7. lower-+.f64 N/A

      \[\leadsto \left(\sin \color{blue}{\left(\frac{\left(\varepsilon + x\right) + x}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   8. lower-/.f64 N/A

      \[\leadsto \left(\sin \left(\color{blue}{\frac{\left(\varepsilon + x\right) + x}{2}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   9. associate-+l+ N/A

      \[\leadsto \left(\sin \left(\frac{\color{blue}{\varepsilon + \left(x + x\right)}}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   10. count-2-rev N/A

       \[\leadsto \left(\sin \left(\frac{\varepsilon + \color{blue}{2 \cdot x}}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   11. +-commutative N/A

       \[\leadsto \left(\sin \left(\frac{\color{blue}{2 \cdot x + \varepsilon}}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   12. lift-fma.f64 N/A

       \[\leadsto \left(\sin \left(\frac{\color{blue}{\mathsf{fma}\left(2, x, \varepsilon\right)}}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   13. lower-/.f64 N/A

       \[\leadsto \left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2} + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   14. lower-PI.f64 99.7

       \[\leadsto \left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2} + \frac{\color{blue}{\pi}}{2}\right) \cdot \left(\mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot 2 \]

8. Applied rewrites 99.7%

   \[\leadsto \left(\color{blue}{\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2} + \frac{\pi}{2}\right)} \cdot \left(\mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot 2 \]

9. Taylor expanded in x around 0

   \[\leadsto \left(\sin \color{blue}{\left(x + \left(\frac{1}{2} \cdot \varepsilon + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \left(\sin \left(\left(\frac{1}{2} \cdot \varepsilon + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{x}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

    2. distribute-lft-out N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon + \mathsf{PI}\left(\right)\right) + x\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

    3. lower-fma.f64 N/A

       \[\leadsto \left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\varepsilon + \mathsf{PI}\left(\right)}, x\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

    4. lower-+.f64 N/A

       \[\leadsto \left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon + \color{blue}{\mathsf{PI}\left(\right)}, x\right)\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

    5. lift-PI.f64 99.7

       \[\leadsto \left(\sin \left(\mathsf{fma}\left(0.5, \varepsilon + \pi, x\right)\right) \cdot \left(\mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot 2 \]

11. Applied rewrites 99.7%

    \[\leadsto \left(\sin \color{blue}{\left(\mathsf{fma}\left(0.5, \varepsilon + \pi, x\right)\right)} \cdot \left(\mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot 2 \]
12. Add Preprocessing

Alternative 7: 99.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\cos \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right) \cdot \left(\mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (* (cos (fma 0.5 eps x)) (* (fma -0.020833333333333332 (* eps eps) 0.5) eps))
  2.0))

C

double code(double x, double eps) {
	return (cos(fma(0.5, eps, x)) * (fma(-0.020833333333333332, (eps * eps), 0.5) * eps)) * 2.0;
}

Julia

function code(x, eps)
	return Float64(Float64(cos(fma(0.5, eps, x)) * Float64(fma(-0.020833333333333332, Float64(eps * eps), 0.5) * eps)) * 2.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[Cos[N[(0.5 * eps + x), $MachinePrecision]], $MachinePrecision] * N[(N[(-0.020833333333333332 * N[(eps * eps), $MachinePrecision] + 0.5), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right) - \sin x} \]

   2. lift-+.f64 N/A

      \[\leadsto \sin \color{blue}{\left(x + \varepsilon\right)} - \sin x \]

   3. lift-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right)} - \sin x \]

   4. lift-sin.f64 N/A

      \[\leadsto \sin \left(x + \varepsilon\right) - \color{blue}{\sin x} \]

   5. diff-sin N/A

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2} \]

3. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \sin \left(\frac{\left(\varepsilon + x\right) - x}{2}\right)\right) \cdot 2} \]

4. Taylor expanded in eps around 0

   \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right)}\right) \cdot 2 \]
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{\varepsilon}\right)\right) \cdot 2 \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{\varepsilon}\right)\right) \cdot 2 \]

   3. +-commutative N/A

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\left(\frac{-1}{48} \cdot {\varepsilon}^{2} + \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   4. lower-fma.f64 N/A

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   5. unpow2 N/A

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   6. lower-*.f64 99.7

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot 2 \]

6. Applied rewrites 99.7%

   \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)}\right) \cdot 2 \]

7. Taylor expanded in x around 0

   \[\leadsto \left(\cos \color{blue}{\left(x + \frac{1}{2} \cdot \varepsilon\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \varepsilon + \color{blue}{x}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   2. lower-fma.f64 99.7

      \[\leadsto \left(\cos \left(\mathsf{fma}\left(0.5, \color{blue}{\varepsilon}, x\right)\right) \cdot \left(\mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot 2 \]

9. Applied rewrites 99.7%

   \[\leadsto \left(\cos \color{blue}{\left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right)} \cdot \left(\mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot 2 \]
10. Add Preprocessing

Alternative 8: 99.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right) \cdot \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (* (cos (* (fma 2.0 x eps) 0.5)) (* 0.5 eps)) 2.0))

C

double code(double x, double eps) {
	return (cos((fma(2.0, x, eps) * 0.5)) * (0.5 * eps)) * 2.0;
}

Julia

function code(x, eps)
	return Float64(Float64(cos(Float64(fma(2.0, x, eps) * 0.5)) * Float64(0.5 * eps)) * 2.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[Cos[N[(N[(2.0 * x + eps), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(0.5 * eps), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right) - \sin x} \]

   2. lift-+.f64 N/A

      \[\leadsto \sin \color{blue}{\left(x + \varepsilon\right)} - \sin x \]

   3. lift-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right)} - \sin x \]

   4. lift-sin.f64 N/A

      \[\leadsto \sin \left(x + \varepsilon\right) - \color{blue}{\sin x} \]

   5. diff-sin N/A

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2} \]

3. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \sin \left(\frac{\left(\varepsilon + x\right) - x}{2}\right)\right) \cdot 2} \]

4. Taylor expanded in x around inf

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

   1. fp-cancel-sign-sub-inv N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - \left(\mathsf{neg}\left(2\right)\right) \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   2. metadata-eval N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot 2 \]

   4. lower-cos.f64 N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot 2 \]

   5. metadata-eval N/A

      \[\leadsto \left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon - \left(\mathsf{neg}\left(2\right)\right) \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   6. fp-cancel-sign-sub-inv N/A

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

   7. count-2-rev N/A

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

   8. associate-+l+ N/A

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

   9. *-commutative N/A

      \[\leadsto \left(\cos \left(\left(\left(\varepsilon + x\right) + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \varepsilon\right)\right) \cdot 2 \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\cos \left(\left(\left(\varepsilon + x\right) + x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \varepsilon\right)\right) \cdot 2 \]

   11. associate-+l+ N/A

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

   12. count-2-rev N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

       \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   15. lower-sin.f64 N/A

       \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot 2 \]

   16. lower-*.f64 99.9

       \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

6. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)} \cdot 2 \]

7. Taylor expanded in eps around 0

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

   1. lift-*.f64 99.4

      \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right) \cdot \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

9. Applied rewrites 99.4%

   \[\leadsto \left(\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right) \cdot \left(0.5 \cdot \color{blue}{\varepsilon}\right)\right) \cdot 2 \]
10. Add Preprocessing

Alternative 9: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sin \left(\mathsf{fma}\left(\pi, 0.5, x\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (sin (fma PI 0.5 x)) eps))

C

double code(double x, double eps) {
	return sin(fma(((double) M_PI), 0.5, x)) * eps;
}

Julia

function code(x, eps)
	return Float64(sin(fma(pi, 0.5, x)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[Sin[N[(Pi * 0.5 + x), $MachinePrecision]], $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right) - \sin x} \]

   2. lift-+.f64 N/A

      \[\leadsto \sin \color{blue}{\left(x + \varepsilon\right)} - \sin x \]

   3. lift-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right)} - \sin x \]

   4. lift-sin.f64 N/A

      \[\leadsto \sin \left(x + \varepsilon\right) - \color{blue}{\sin x} \]

   5. diff-sin N/A

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2} \]

3. Applied rewrites 62.5%

   \[\leadsto \color{blue}{\left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \sin \left(\frac{\left(\varepsilon + x\right) - x}{2}\right)\right) \cdot 2} \]

4. Taylor expanded in eps around 0

   \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right)}\right) \cdot 2 \]
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{\varepsilon}\right)\right) \cdot 2 \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{\varepsilon}\right)\right) \cdot 2 \]

   3. +-commutative N/A

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\left(\frac{-1}{48} \cdot {\varepsilon}^{2} + \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   4. lower-fma.f64 N/A

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   5. unpow2 N/A

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   6. lower-*.f64 99.7

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(\mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot 2 \]

6. Applied rewrites 99.7%

   \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)}\right) \cdot 2 \]
7. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \left(\color{blue}{\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   2. lift-/.f64 N/A

      \[\leadsto \left(\cos \color{blue}{\left(\frac{\left(\varepsilon + x\right) + x}{2}\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   3. lift-+.f64 N/A

      \[\leadsto \left(\cos \left(\frac{\color{blue}{\left(\varepsilon + x\right) + x}}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   4. lift-+.f64 N/A

      \[\leadsto \left(\cos \left(\frac{\color{blue}{\left(\varepsilon + x\right)} + x}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   5. sin-+PI/2-rev N/A

      \[\leadsto \left(\color{blue}{\sin \left(\frac{\left(\varepsilon + x\right) + x}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   6. lower-sin.f64 N/A

      \[\leadsto \left(\color{blue}{\sin \left(\frac{\left(\varepsilon + x\right) + x}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   7. lower-+.f64 N/A

      \[\leadsto \left(\sin \color{blue}{\left(\frac{\left(\varepsilon + x\right) + x}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   8. lower-/.f64 N/A

      \[\leadsto \left(\sin \left(\color{blue}{\frac{\left(\varepsilon + x\right) + x}{2}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   9. associate-+l+ N/A

      \[\leadsto \left(\sin \left(\frac{\color{blue}{\varepsilon + \left(x + x\right)}}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   10. count-2-rev N/A

       \[\leadsto \left(\sin \left(\frac{\varepsilon + \color{blue}{2 \cdot x}}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   11. +-commutative N/A

       \[\leadsto \left(\sin \left(\frac{\color{blue}{2 \cdot x + \varepsilon}}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   12. lift-fma.f64 N/A

       \[\leadsto \left(\sin \left(\frac{\color{blue}{\mathsf{fma}\left(2, x, \varepsilon\right)}}{2} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   13. lower-/.f64 N/A

       \[\leadsto \left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2} + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{48}, \varepsilon \cdot \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot 2 \]

   14. lower-PI.f64 99.7

       \[\leadsto \left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2} + \frac{\color{blue}{\pi}}{2}\right) \cdot \left(\mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot 2 \]

8. Applied rewrites 99.7%

   \[\leadsto \left(\color{blue}{\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2} + \frac{\pi}{2}\right)} \cdot \left(\mathsf{fma}\left(-0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot 2 \]

9. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \sin \left(x + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
10. Step-by-step derivation

    1. associate-*l* N/A

       \[\leadsto \color{blue}{\varepsilon} \cdot \sin \left(x + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

    2. sin-+PI/2 N/A

       \[\leadsto \varepsilon \cdot \sin \left(x + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

    3. +-commutative N/A

       \[\leadsto \varepsilon \cdot \sin \left(x + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

    4. count-2-rev N/A

       \[\leadsto \varepsilon \cdot \sin \left(x + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

    5. associate-+l+ N/A

       \[\leadsto \varepsilon \cdot \sin \left(x + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

    6. associate-*l* N/A

       \[\leadsto \color{blue}{\varepsilon} \cdot \sin \left(x + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \]

    7. *-commutative N/A

       \[\leadsto \sin \left(x + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\varepsilon} \]

    8. lower-*.f64 N/A

       \[\leadsto \sin \left(x + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\varepsilon} \]

    9. lower-sin.f64 N/A

       \[\leadsto \sin \left(x + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \varepsilon \]

    10. +-commutative N/A

        \[\leadsto \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + x\right) \cdot \varepsilon \]

    11. *-commutative N/A

        \[\leadsto \sin \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + x\right) \cdot \varepsilon \]

    12. lower-fma.f64 N/A

        \[\leadsto \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, x\right)\right) \cdot \varepsilon \]

    13. lift-PI.f64 99.0

        \[\leadsto \sin \left(\mathsf{fma}\left(\pi, 0.5, x\right)\right) \cdot \varepsilon \]

11. Applied rewrites 99.0%

    \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(\pi, 0.5, x\right)\right) \cdot \varepsilon} \]
12. Add Preprocessing

Alternative 10: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (cos x) eps))

C

double code(double x, double eps) {
	return cos(x) * eps;
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = cos(x) * eps
end function

Java

public static double code(double x, double eps) {
	return Math.cos(x) * eps;
}

Python

def code(x, eps):
	return math.cos(x) * eps

Julia

function code(x, eps)
	return Float64(cos(x) * eps)
end

MATLAB

function tmp = code(x, eps)
	tmp = cos(x) * eps;
end

Wolfram

code[x_, eps_] := N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

      \[\leadsto \cos x \cdot \color{blue}{\varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{\varepsilon} \]

   3. lower-cos.f64 99.0

      \[\leadsto \cos x \cdot \varepsilon \]

4. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\cos x \cdot \varepsilon} \]
5. Add Preprocessing

Alternative 11: 98.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006944444444444444, \varepsilon \cdot \varepsilon, 0.08333333333333333\right) \cdot x, \varepsilon, 0.08333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.5, x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) + 1\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+
   (fma
    (fma
     (-
      (fma
       (* (fma -0.006944444444444444 (* eps eps) 0.08333333333333333) x)
       eps
       (* 0.08333333333333333 (* eps eps)))
      0.5)
     x
     (* (- (* (* eps eps) 0.041666666666666664) 0.5) eps))
    x
    (* (* eps eps) -0.16666666666666666))
   1.0)
  eps))

C

double code(double x, double eps) {
	return (fma(fma((fma((fma(-0.006944444444444444, (eps * eps), 0.08333333333333333) * x), eps, (0.08333333333333333 * (eps * eps))) - 0.5), x, ((((eps * eps) * 0.041666666666666664) - 0.5) * eps)), x, ((eps * eps) * -0.16666666666666666)) + 1.0) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(Float64(fma(Float64(fma(-0.006944444444444444, Float64(eps * eps), 0.08333333333333333) * x), eps, Float64(0.08333333333333333 * Float64(eps * eps))) - 0.5), x, Float64(Float64(Float64(Float64(eps * eps) * 0.041666666666666664) - 0.5) * eps)), x, Float64(Float64(eps * eps) * -0.16666666666666666)) + 1.0) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(N[(N[(-0.006944444444444444 * N[(eps * eps), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * x), $MachinePrecision] * eps + N[(0.08333333333333333 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * x + N[(N[(N[(N[(eps * eps), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \sin x \cdot \varepsilon, \cos x \cdot -0.16666666666666666\right), \varepsilon, -0.5 \cdot \sin x\right), \varepsilon, \cos x\right) \cdot \varepsilon} \]

5. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\left(\frac{1}{12} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{144} \cdot {\varepsilon}^{2}\right)\right)\right) - \frac{1}{2}\right)\right)\right)\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\left(\frac{1}{12} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{144} \cdot {\varepsilon}^{2}\right)\right)\right) - \frac{1}{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\left(\frac{1}{12} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{144} \cdot {\varepsilon}^{2}\right)\right)\right) - \frac{1}{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]

7. Applied rewrites 98.4%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.006944444444444444, \varepsilon \cdot \varepsilon, 0.08333333333333333\right) \cdot x, \varepsilon, 0.08333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.5, x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) + 1\right) \cdot \varepsilon \]
8. Add Preprocessing

Alternative 12: 98.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(0.08333333333333333 \cdot \left(x + \varepsilon\right)\right) \cdot \varepsilon - 0.5\right) \cdot \varepsilon, x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.16666666666666666, 1\right) \cdot \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   (* (- (* (* 0.08333333333333333 (+ x eps)) eps) 0.5) eps)
   x
   (* (- (* (* eps eps) 0.041666666666666664) 0.5) (* eps eps)))
  x
  (* (fma (* eps eps) -0.16666666666666666 1.0) eps)))

C

double code(double x, double eps) {
	return fma(fma(((((0.08333333333333333 * (x + eps)) * eps) - 0.5) * eps), x, ((((eps * eps) * 0.041666666666666664) - 0.5) * (eps * eps))), x, (fma((eps * eps), -0.16666666666666666, 1.0) * eps));
}

Julia

function code(x, eps)
	return fma(fma(Float64(Float64(Float64(Float64(0.08333333333333333 * Float64(x + eps)) * eps) - 0.5) * eps), x, Float64(Float64(Float64(Float64(eps * eps) * 0.041666666666666664) - 0.5) * Float64(eps * eps))), x, Float64(fma(Float64(eps * eps), -0.16666666666666666, 1.0) * eps))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(0.08333333333333333 * N[(x + eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] - 0.5), $MachinePrecision] * eps), $MachinePrecision] * x + N[(N[(N[(N[(eps * eps), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + N[(N[(N[(eps * eps), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \sin x \cdot \varepsilon, \cos x \cdot -0.16666666666666666\right), \varepsilon, -0.5 \cdot \sin x\right), \varepsilon, \cos x\right) \cdot \varepsilon} \]

5. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right)\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]

7. Applied rewrites 98.4%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5, x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) + 1\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \color{blue}{x \cdot \left(x \cdot \left(\varepsilon \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + {\varepsilon}^{2} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{144} \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \left(\varepsilon \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + {\varepsilon}^{2} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{144} \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right) + \varepsilon \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(x \cdot \left(\varepsilon \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + {\varepsilon}^{2} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{144} \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right) \cdot x + \varepsilon \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \left(\varepsilon \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + {\varepsilon}^{2} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{144} \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right), x, \varepsilon \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right) \]

10. Applied rewrites 98.4%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot x, \mathsf{fma}\left(-0.006944444444444444, \varepsilon \cdot \varepsilon, 0.08333333333333333\right), \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.08333333333333333 - 0.5\right) \cdot \varepsilon\right), x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \color{blue}{x}, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.16666666666666666, 1\right) \cdot \varepsilon\right) \]

11. Taylor expanded in eps around 0

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{12} \cdot \varepsilon + \frac{1}{12} \cdot x\right) - \frac{1}{2}\right), x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{24} - \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\varepsilon \cdot \left(\frac{1}{12} \cdot \varepsilon + \frac{1}{12} \cdot x\right) - \frac{1}{2}\right) \cdot \varepsilon, x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{24} - \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]

    2. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\varepsilon \cdot \left(\frac{1}{12} \cdot \varepsilon + \frac{1}{12} \cdot x\right) - \frac{1}{2}\right) \cdot \varepsilon, x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{24} - \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]

    3. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\varepsilon \cdot \left(\frac{1}{12} \cdot \varepsilon + \frac{1}{12} \cdot x\right) - \frac{1}{2}\right) \cdot \varepsilon, x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{24} - \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\frac{1}{12} \cdot \varepsilon + \frac{1}{12} \cdot x\right) \cdot \varepsilon - \frac{1}{2}\right) \cdot \varepsilon, x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{24} - \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]

    5. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\frac{1}{12} \cdot \varepsilon + \frac{1}{12} \cdot x\right) \cdot \varepsilon - \frac{1}{2}\right) \cdot \varepsilon, x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{24} - \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]

    6. distribute-lft-out N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\frac{1}{12} \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon - \frac{1}{2}\right) \cdot \varepsilon, x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{24} - \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]

    7. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\frac{1}{12} \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon - \frac{1}{2}\right) \cdot \varepsilon, x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{24} - \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]

    8. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\frac{1}{12} \cdot \left(x + \varepsilon\right)\right) \cdot \varepsilon - \frac{1}{2}\right) \cdot \varepsilon, x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{24} - \frac{1}{2}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]

    9. lower-+.f64 98.4

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(0.08333333333333333 \cdot \left(x + \varepsilon\right)\right) \cdot \varepsilon - 0.5\right) \cdot \varepsilon, x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.16666666666666666, 1\right) \cdot \varepsilon\right) \]

13. Applied rewrites 98.4%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(0.08333333333333333 \cdot \left(x + \varepsilon\right)\right) \cdot \varepsilon - 0.5\right) \cdot \varepsilon, x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.16666666666666666, 1\right) \cdot \varepsilon\right) \]
14. Add Preprocessing

Alternative 13: 98.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333 \cdot \mathsf{fma}\left(x, \varepsilon, x \cdot x\right) - 0.5, \varepsilon, -0.5 \cdot x\right) \cdot \varepsilon, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.16666666666666666, 1\right) \cdot \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (*
   (fma (- (* 0.08333333333333333 (fma x eps (* x x))) 0.5) eps (* -0.5 x))
   eps)
  x
  (* (fma (* eps eps) -0.16666666666666666 1.0) eps)))

C

double code(double x, double eps) {
	return fma((fma(((0.08333333333333333 * fma(x, eps, (x * x))) - 0.5), eps, (-0.5 * x)) * eps), x, (fma((eps * eps), -0.16666666666666666, 1.0) * eps));
}

Julia

function code(x, eps)
	return fma(Float64(fma(Float64(Float64(0.08333333333333333 * fma(x, eps, Float64(x * x))) - 0.5), eps, Float64(-0.5 * x)) * eps), x, Float64(fma(Float64(eps * eps), -0.16666666666666666, 1.0) * eps))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(0.08333333333333333 * N[(x * eps + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * eps + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * x + N[(N[(N[(eps * eps), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \sin x \cdot \varepsilon, \cos x \cdot -0.16666666666666666\right), \varepsilon, -0.5 \cdot \sin x\right), \varepsilon, \cos x\right) \cdot \varepsilon} \]

5. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right)\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]

7. Applied rewrites 98.4%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5, x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) + 1\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \color{blue}{x \cdot \left(x \cdot \left(\varepsilon \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + {\varepsilon}^{2} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{144} \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \left(\varepsilon \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + {\varepsilon}^{2} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{144} \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right) + \varepsilon \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(x \cdot \left(\varepsilon \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + {\varepsilon}^{2} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{144} \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right) \cdot x + \varepsilon \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \left(\varepsilon \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + {\varepsilon}^{2} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{144} \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right), x, \varepsilon \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right) \]

10. Applied rewrites 98.4%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot x, \mathsf{fma}\left(-0.006944444444444444, \varepsilon \cdot \varepsilon, 0.08333333333333333\right), \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.08333333333333333 - 0.5\right) \cdot \varepsilon\right), x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \color{blue}{x}, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.16666666666666666, 1\right) \cdot \varepsilon\right) \]

11. Taylor expanded in eps around 0

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

    1. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{2} \cdot x + \varepsilon \cdot \left(\left(\frac{1}{12} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{12} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right) \cdot \varepsilon, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]

    2. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{2} \cdot x + \varepsilon \cdot \left(\left(\frac{1}{12} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{12} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right) \cdot \varepsilon, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]

13. Applied rewrites 98.4%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333 \cdot \mathsf{fma}\left(x, \varepsilon, x \cdot x\right) - 0.5, \varepsilon, -0.5 \cdot x\right) \cdot \varepsilon, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.16666666666666666, 1\right) \cdot \varepsilon\right) \]
14. Add Preprocessing

Alternative 14: 98.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.08333333333333333 - 0.5, \varepsilon, -0.5 \cdot x\right) \cdot \varepsilon, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.16666666666666666, 1\right) \cdot \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (* (fma (- (* (* x x) 0.08333333333333333) 0.5) eps (* -0.5 x)) eps)
  x
  (* (fma (* eps eps) -0.16666666666666666 1.0) eps)))

C

double code(double x, double eps) {
	return fma((fma((((x * x) * 0.08333333333333333) - 0.5), eps, (-0.5 * x)) * eps), x, (fma((eps * eps), -0.16666666666666666, 1.0) * eps));
}

Julia

function code(x, eps)
	return fma(Float64(fma(Float64(Float64(Float64(x * x) * 0.08333333333333333) - 0.5), eps, Float64(-0.5 * x)) * eps), x, Float64(fma(Float64(eps * eps), -0.16666666666666666, 1.0) * eps))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] - 0.5), $MachinePrecision] * eps + N[(-0.5 * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * x + N[(N[(N[(eps * eps), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \sin x \cdot \varepsilon, \cos x \cdot -0.16666666666666666\right), \varepsilon, -0.5 \cdot \sin x\right), \varepsilon, \cos x\right) \cdot \varepsilon} \]

5. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right)\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]

7. Applied rewrites 98.4%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5, x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) + 1\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \color{blue}{x \cdot \left(x \cdot \left(\varepsilon \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + {\varepsilon}^{2} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{144} \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \left(\varepsilon \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + {\varepsilon}^{2} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{144} \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right) + \varepsilon \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(x \cdot \left(\varepsilon \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + {\varepsilon}^{2} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{144} \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right) \cdot x + \varepsilon \cdot \left(\color{blue}{1} + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \left(\varepsilon \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + {\varepsilon}^{2} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{144} \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right), x, \varepsilon \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right) \]

10. Applied rewrites 98.4%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot x, \mathsf{fma}\left(-0.006944444444444444, \varepsilon \cdot \varepsilon, 0.08333333333333333\right), \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.08333333333333333 - 0.5\right) \cdot \varepsilon\right), x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \color{blue}{x}, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.16666666666666666, 1\right) \cdot \varepsilon\right) \]

11. Taylor expanded in eps around 0

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

    1. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{2} \cdot x + \varepsilon \cdot \left(\frac{1}{12} \cdot {x}^{2} - \frac{1}{2}\right)\right) \cdot \varepsilon, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]

    2. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{2} \cdot x + \varepsilon \cdot \left(\frac{1}{12} \cdot {x}^{2} - \frac{1}{2}\right)\right) \cdot \varepsilon, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]

    3. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \left(\frac{1}{12} \cdot {x}^{2} - \frac{1}{2}\right) + \frac{-1}{2} \cdot x\right) \cdot \varepsilon, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(\left(\frac{1}{12} \cdot {x}^{2} - \frac{1}{2}\right) \cdot \varepsilon + \frac{-1}{2} \cdot x\right) \cdot \varepsilon, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]

    5. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} \cdot {x}^{2} - \frac{1}{2}, \varepsilon, \frac{-1}{2} \cdot x\right) \cdot \varepsilon, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]

    6. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} \cdot {x}^{2} - \frac{1}{2}, \varepsilon, \frac{-1}{2} \cdot x\right) \cdot \varepsilon, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({x}^{2} \cdot \frac{1}{12} - \frac{1}{2}, \varepsilon, \frac{-1}{2} \cdot x\right) \cdot \varepsilon, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]

    8. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({x}^{2} \cdot \frac{1}{12} - \frac{1}{2}, \varepsilon, \frac{-1}{2} \cdot x\right) \cdot \varepsilon, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]

    9. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{12} - \frac{1}{2}, \varepsilon, \frac{-1}{2} \cdot x\right) \cdot \varepsilon, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]

    10. lift-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{12} - \frac{1}{2}, \varepsilon, \frac{-1}{2} \cdot x\right) \cdot \varepsilon, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{6}, 1\right) \cdot \varepsilon\right) \]

    11. lower-*.f64 98.4

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.08333333333333333 - 0.5, \varepsilon, -0.5 \cdot x\right) \cdot \varepsilon, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.16666666666666666, 1\right) \cdot \varepsilon\right) \]

13. Applied rewrites 98.4%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.08333333333333333 - 0.5, \varepsilon, -0.5 \cdot x\right) \cdot \varepsilon, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.16666666666666666, 1\right) \cdot \varepsilon\right) \]
14. Add Preprocessing

Alternative 15: 98.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5, x, -0.5 \cdot \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) + 1\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+
   (fma
    (fma (- (* 0.08333333333333333 (* eps eps)) 0.5) x (* -0.5 eps))
    x
    (* (* eps eps) -0.16666666666666666))
   1.0)
  eps))

C

double code(double x, double eps) {
	return (fma(fma(((0.08333333333333333 * (eps * eps)) - 0.5), x, (-0.5 * eps)), x, ((eps * eps) * -0.16666666666666666)) + 1.0) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(Float64(Float64(0.08333333333333333 * Float64(eps * eps)) - 0.5), x, Float64(-0.5 * eps)), x, Float64(Float64(eps * eps) * -0.16666666666666666)) + 1.0) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(0.08333333333333333 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * x + N[(-0.5 * eps), $MachinePrecision]), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \sin x \cdot \varepsilon, \cos x \cdot -0.16666666666666666\right), \varepsilon, -0.5 \cdot \sin x\right), \varepsilon, \cos x\right) \cdot \varepsilon} \]

5. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right)\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]

7. Applied rewrites 98.4%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5, x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) + 1\right) \cdot \varepsilon \]

8. Taylor expanded in eps around 0

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

10. Applied rewrites 98.4%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5, x, -0.5 \cdot \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) + 1\right) \cdot \varepsilon \]
11. Add Preprocessing

Alternative 16: 98.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(-0.5 \cdot \left(x + \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) + 1\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+ (fma (* -0.5 (+ x eps)) x (* (* eps eps) -0.16666666666666666)) 1.0)
  eps))

C

double code(double x, double eps) {
	return (fma((-0.5 * (x + eps)), x, ((eps * eps) * -0.16666666666666666)) + 1.0) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(Float64(-0.5 * Float64(x + eps)), x, Float64(Float64(eps * eps) * -0.16666666666666666)) + 1.0) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(-0.5 * N[(x + eps), $MachinePrecision]), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \sin x \cdot \varepsilon, \cos x \cdot -0.16666666666666666\right), \varepsilon, -0.5 \cdot \sin x\right), \varepsilon, \cos x\right) \cdot \varepsilon} \]

5. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right)\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]

7. Applied rewrites 98.4%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5, x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) + 1\right) \cdot \varepsilon \]

8. Taylor expanded in eps around 0

   \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{2} \cdot x, x, \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{-1}{6}\right) + 1\right) \cdot \varepsilon \]
9. Step-by-step derivation

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 98.4

      \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \left(x + \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) + 1\right) \cdot \varepsilon \]

10. Applied rewrites 98.4%

    \[\leadsto \left(\mathsf{fma}\left(-0.5 \cdot \left(x + \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) + 1\right) \cdot \varepsilon \]
11. Add Preprocessing

Alternative 17: 98.4% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664 - 0.5, x \cdot x, 1\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (fma (- (* (* x x) 0.041666666666666664) 0.5) (* x x) 1.0) eps))

C

double code(double x, double eps) {
	return fma((((x * x) * 0.041666666666666664) - 0.5), (x * x), 1.0) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(Float64(Float64(Float64(x * x) * 0.041666666666666664) - 0.5), Float64(x * x), 1.0) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

      \[\leadsto \cos x \cdot \color{blue}{\varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{\varepsilon} \]

   3. lower-cos.f64 99.0

      \[\leadsto \cos x \cdot \varepsilon \]

4. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\cos x \cdot \varepsilon} \]

5. Taylor expanded in x around 0

   \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right) \cdot \varepsilon \]

   2. *-commutative N/A

      \[\leadsto \left(\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2} + 1\right) \cdot \varepsilon \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \cdot \varepsilon \]

   4. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right) \cdot \varepsilon \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \frac{1}{24} - \frac{1}{2}, {x}^{2}, 1\right) \cdot \varepsilon \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \frac{1}{24} - \frac{1}{2}, {x}^{2}, 1\right) \cdot \varepsilon \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{24} - \frac{1}{2}, {x}^{2}, 1\right) \cdot \varepsilon \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{24} - \frac{1}{2}, {x}^{2}, 1\right) \cdot \varepsilon \]

   9. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{24} - \frac{1}{2}, x \cdot x, 1\right) \cdot \varepsilon \]

   10. lower-*.f64 98.4

       \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664 - 0.5, x \cdot x, 1\right) \cdot \varepsilon \]

7. Applied rewrites 98.4%

   \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664 - 0.5, x \cdot x, 1\right) \cdot \varepsilon \]
8. Add Preprocessing

Alternative 18: 98.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \mathsf{fma}\left(x, \varepsilon, x \cdot x\right), 1\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (fma -0.5 (fma x eps (* x x)) 1.0) eps))

C

double code(double x, double eps) {
	return fma(-0.5, fma(x, eps, (x * x)), 1.0) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(-0.5, fma(x, eps, Float64(x * x)), 1.0) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(-0.5 * N[(x * eps + N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \sin x \cdot \varepsilon, \cos x \cdot -0.16666666666666666\right), \varepsilon, -0.5 \cdot \sin x\right), \varepsilon, \cos x\right) \cdot \varepsilon} \]

5. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right)\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]

7. Applied rewrites 98.4%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5, x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) + 1\right) \cdot \varepsilon \]

8. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 98.4

      \[\leadsto \mathsf{fma}\left(-0.5, \mathsf{fma}\left(x, \varepsilon, x \cdot x\right), 1\right) \cdot \varepsilon \]

10. Applied rewrites 98.4%

    \[\leadsto \mathsf{fma}\left(-0.5, \mathsf{fma}\left(x, \varepsilon, x \cdot x\right), 1\right) \cdot \varepsilon \]
11. Add Preprocessing

Alternative 19: 98.3% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, -0.5, 1\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (fma (* x x) -0.5 1.0) eps))

C

double code(double x, double eps) {
	return fma((x * x), -0.5, 1.0) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(Float64(x * x), -0.5, 1.0) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \sin x \cdot \varepsilon, \cos x \cdot -0.16666666666666666\right), \varepsilon, -0.5 \cdot \sin x\right), \varepsilon, \cos x\right) \cdot \varepsilon} \]

5. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right)\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]

   2. lower-+.f64 N/A

      \[\leadsto \left(\left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{12} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]

7. Applied rewrites 98.4%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5, x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666\right) + 1\right) \cdot \varepsilon \]

8. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 98.3

      \[\leadsto \mathsf{fma}\left(x \cdot x, -0.5, 1\right) \cdot \varepsilon \]

10. Applied rewrites 98.3%

    \[\leadsto \mathsf{fma}\left(x \cdot x, -0.5, 1\right) \cdot \varepsilon \]
11. Add Preprocessing

Alternative 20: 97.9% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 eps)

C

double code(double x, double eps) {
	return eps;
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps
end function

Java

public static double code(double x, double eps) {
	return eps;
}

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

function tmp = code(x, eps)
	tmp = eps;
end

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]

2. Taylor expanded in x around 0

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

   1. lower-sin.f64 97.9

      \[\leadsto \sin \varepsilon \]

4. Applied rewrites 97.9%

   \[\leadsto \color{blue}{\sin \varepsilon} \]

5. Taylor expanded in eps around 0

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

7. Applied rewrites 97.9%

   \[\leadsto \varepsilon \]
8. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot \cos \left(x + \frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (* 2.0 (cos (+ x (/ eps 2.0)))) (sin (/ eps 2.0))))

C

double code(double x, double eps) {
	return (2.0 * cos((x + (eps / 2.0)))) * sin((eps / 2.0));
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (2.0d0 * cos((x + (eps / 2.0d0)))) * sin((eps / 2.0d0))
end function

Java

public static double code(double x, double eps) {
	return (2.0 * Math.cos((x + (eps / 2.0)))) * Math.sin((eps / 2.0));
}

Python

def code(x, eps):
	return (2.0 * math.cos((x + (eps / 2.0)))) * math.sin((eps / 2.0))

Julia

function code(x, eps)
	return Float64(Float64(2.0 * cos(Float64(x + Float64(eps / 2.0)))) * sin(Float64(eps / 2.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = (2.0 * cos((x + (eps / 2.0)))) * sin((eps / 2.0));
end

Wolfram

code[x_, eps_] := N[(N[(2.0 * N[Cos[N[(x + N[(eps / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Developer Target 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \left(\cos \varepsilon - 1\right) + \cos x \cdot \sin \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+ (* (sin x) (- (cos eps) 1.0)) (* (cos x) (sin eps))))

C

double code(double x, double eps) {
	return (sin(x) * (cos(eps) - 1.0)) + (cos(x) * sin(eps));
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (sin(x) * (cos(eps) - 1.0d0)) + (cos(x) * sin(eps))
end function

Java

public static double code(double x, double eps) {
	return (Math.sin(x) * (Math.cos(eps) - 1.0)) + (Math.cos(x) * Math.sin(eps));
}

Python

def code(x, eps):
	return (math.sin(x) * (math.cos(eps) - 1.0)) + (math.cos(x) * math.sin(eps))

Julia

function code(x, eps)
	return Float64(Float64(sin(x) * Float64(cos(eps) - 1.0)) + Float64(cos(x) * sin(eps)))
end

MATLAB

function tmp = code(x, eps)
	tmp = (sin(x) * (cos(eps) - 1.0)) + (cos(x) * sin(eps));
end

Wolfram

code[x_, eps_] := N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Developer Target 3: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(\cos \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (* (cos (* 0.5 (- eps (* -2.0 x)))) (sin (* 0.5 eps))) 2.0))

C

double code(double x, double eps) {
	return (cos((0.5 * (eps - (-2.0 * x)))) * sin((0.5 * eps))) * 2.0;
}

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, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (cos((0.5d0 * (eps - ((-2.0d0) * x)))) * sin((0.5d0 * eps))) * 2.0d0
end function

Java

public static double code(double x, double eps) {
	return (Math.cos((0.5 * (eps - (-2.0 * x)))) * Math.sin((0.5 * eps))) * 2.0;
}

Python

def code(x, eps):
	return (math.cos((0.5 * (eps - (-2.0 * x)))) * math.sin((0.5 * eps))) * 2.0

Julia

function code(x, eps)
	return Float64(Float64(cos(Float64(0.5 * Float64(eps - Float64(-2.0 * x)))) * sin(Float64(0.5 * eps))) * 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (cos((0.5 * (eps - (-2.0 * x)))) * sin((0.5 * eps))) * 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(N[Cos[N[(0.5 * N[(eps - N[(-2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Reproduce

herbie shell --seed 2025106 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (* 2 (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  :alt
  (! :herbie-platform default (+ (* (sin x) (- (cos eps) 1)) (* (cos x) (sin eps))))

  :alt
  (! :herbie-platform default (* (cos (* 1/2 (- eps (* -2 x)))) (sin (* 1/2 eps)) 2))

  (- (sin (+ x eps)) (sin x)))