2sin (example 3.3)

Percentage Accurate: 62.6% → 99.9%

Time: 12.5s

Alternatives: 12

Speedup: 34.5×

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: 62.6% 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: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.3%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. associate--l+ N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{\color{blue}{0} + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

   17. cos-neg-rev N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   18. lower-cos.f64 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   19. distribute-neg-frac2 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\frac{\left(x + \varepsilon\right) + x}{\mathsf{neg}\left(2\right)}\right)} \]

   20. lower-/.f64 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\frac{\left(x + \varepsilon\right) + x}{\mathsf{neg}\left(2\right)}\right)} \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right)} \]

5. Final simplification 99.9%

   \[\leadsto \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot 2\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]
6. Add Preprocessing

Alternative 2: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 63.3%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. associate--l+ N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{\color{blue}{0} + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

   17. cos-neg-rev N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   18. lower-cos.f64 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   19. distribute-neg-frac2 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\frac{\left(x + \varepsilon\right) + x}{\mathsf{neg}\left(2\right)}\right)} \]

   20. lower-/.f64 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\frac{\left(x + \varepsilon\right) + x}{\mathsf{neg}\left(2\right)}\right)} \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right)} \]
5. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \color{blue}{\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right)} \]

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

      \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   3. lower-sin.f64 N/A

      \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   4. lift-/.f64 N/A

      \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \sin \left(\color{blue}{\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   5. frac-2neg N/A

      \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \sin \left(\color{blue}{\frac{\mathsf{neg}\left(\mathsf{fma}\left(2, x, \varepsilon\right)\right)}{\mathsf{neg}\left(-2\right)}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\mathsf{fma}\left(2, x, \varepsilon\right)\right)}{\color{blue}{2}} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   7. div-add-rev N/A

      \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \sin \color{blue}{\left(\frac{\left(\mathsf{neg}\left(\mathsf{fma}\left(2, x, \varepsilon\right)\right)\right) + \mathsf{PI}\left(\right)}{2}\right)} \]

   8. lower-/.f64 N/A

      \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \sin \color{blue}{\left(\frac{\left(\mathsf{neg}\left(\mathsf{fma}\left(2, x, \varepsilon\right)\right)\right) + \mathsf{PI}\left(\right)}{2}\right)} \]

   9. lower-+.f64 N/A

      \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \sin \left(\frac{\color{blue}{\left(\mathsf{neg}\left(\mathsf{fma}\left(2, x, \varepsilon\right)\right)\right) + \mathsf{PI}\left(\right)}}{2}\right) \]

   10. lift-fma.f64 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \sin \left(\frac{\left(\mathsf{neg}\left(\color{blue}{\left(2 \cdot x + \varepsilon\right)}\right)\right) + \mathsf{PI}\left(\right)}{2}\right) \]

   11. count-2-rev N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \sin \left(\frac{\left(\mathsf{neg}\left(\left(\color{blue}{\left(x + x\right)} + \varepsilon\right)\right)\right) + \mathsf{PI}\left(\right)}{2}\right) \]

   12. associate-+r+ N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \sin \left(\frac{\left(\mathsf{neg}\left(\color{blue}{\left(x + \left(x + \varepsilon\right)\right)}\right)\right) + \mathsf{PI}\left(\right)}{2}\right) \]

   13. +-commutative N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \sin \left(\frac{\left(\mathsf{neg}\left(\left(x + \color{blue}{\left(\varepsilon + x\right)}\right)\right)\right) + \mathsf{PI}\left(\right)}{2}\right) \]

   14. lift-+.f64 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \sin \left(\frac{\left(\mathsf{neg}\left(\left(x + \color{blue}{\left(\varepsilon + x\right)}\right)\right)\right) + \mathsf{PI}\left(\right)}{2}\right) \]

   15. lower-neg.f64 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \sin \left(\frac{\color{blue}{\left(-\left(x + \left(\varepsilon + x\right)\right)\right)} + \mathsf{PI}\left(\right)}{2}\right) \]

   16. lift-+.f64 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \sin \left(\frac{\left(-\left(x + \color{blue}{\left(\varepsilon + x\right)}\right)\right) + \mathsf{PI}\left(\right)}{2}\right) \]

   17. +-commutative N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \sin \left(\frac{\left(-\left(x + \color{blue}{\left(x + \varepsilon\right)}\right)\right) + \mathsf{PI}\left(\right)}{2}\right) \]

   18. associate-+r+ N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \sin \left(\frac{\left(-\color{blue}{\left(\left(x + x\right) + \varepsilon\right)}\right) + \mathsf{PI}\left(\right)}{2}\right) \]

   19. count-2-rev N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \sin \left(\frac{\left(-\left(\color{blue}{2 \cdot x} + \varepsilon\right)\right) + \mathsf{PI}\left(\right)}{2}\right) \]

   20. *-commutative N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \sin \left(\frac{\left(-\left(\color{blue}{x \cdot 2} + \varepsilon\right)\right) + \mathsf{PI}\left(\right)}{2}\right) \]

   21. lower-fma.f64 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \sin \left(\frac{\left(-\color{blue}{\mathsf{fma}\left(x, 2, \varepsilon\right)}\right) + \mathsf{PI}\left(\right)}{2}\right) \]

   22. lower-PI.f64 99.9

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \sin \left(\frac{\left(-\mathsf{fma}\left(x, 2, \varepsilon\right)\right) + \color{blue}{\mathsf{PI}\left(\right)}}{2}\right) \]

6. Applied rewrites 99.9%

   \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \color{blue}{\sin \left(\frac{\left(-\mathsf{fma}\left(x, 2, \varepsilon\right)\right) + \mathsf{PI}\left(\right)}{2}\right)} \]

7. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

9. Applied rewrites 99.9%

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

10. Taylor expanded in x around inf

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

    1. associate-*r* N/A

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

    2. lower-*.f64 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 N/A

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

    5. lower-sin.f64 N/A

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

    6. lower-*.f64 N/A

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

    7. lower-sin.f64 N/A

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

    8. +-commutative N/A

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

    9. associate--r+ N/A

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

    10. metadata-eval N/A

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

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

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

    12. *-lft-identity N/A

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

    13. metadata-eval N/A

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

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

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

    15. mul-1-neg N/A

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

    16. +-commutative N/A

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

    17. associate-+l+ N/A

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

12. Applied rewrites 99.9%

    \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \varepsilon\right) \cdot 2\right) \cdot \sin \left(\left(\mathsf{PI}\left(\right) - \varepsilon\right) \cdot 0.5 - x\right)} \]
13. Add Preprocessing

Alternative 3: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(-3.1001984126984127 \cdot 10^{-6}, \varepsilon \cdot \varepsilon, 0.0005208333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.041666666666666664, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (*
   (fma
    (-
     (*
      (fma -3.1001984126984127e-6 (* eps eps) 0.0005208333333333333)
      (* eps eps))
     0.041666666666666664)
    (* eps eps)
    1.0)
   eps)
  (cos (fma 0.5 eps x))))

C

double code(double x, double eps) {
	return (fma(((fma(-3.1001984126984127e-6, (eps * eps), 0.0005208333333333333) * (eps * eps)) - 0.041666666666666664), (eps * eps), 1.0) * eps) * cos(fma(0.5, eps, x));
}

Julia

function code(x, eps)
	return Float64(Float64(fma(Float64(Float64(fma(-3.1001984126984127e-6, Float64(eps * eps), 0.0005208333333333333) * Float64(eps * eps)) - 0.041666666666666664), Float64(eps * eps), 1.0) * eps) * cos(fma(0.5, eps, x)))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(-3.1001984126984127e-6 * N[(eps * eps), $MachinePrecision] + 0.0005208333333333333), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision] * N[Cos[N[(0.5 * eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.3%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. associate--l+ N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{\color{blue}{0} + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

   17. cos-neg-rev N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   18. lower-cos.f64 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   19. distribute-neg-frac2 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\frac{\left(x + \varepsilon\right) + x}{\mathsf{neg}\left(2\right)}\right)} \]

   20. lower-/.f64 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\frac{\left(x + \varepsilon\right) + x}{\mathsf{neg}\left(2\right)}\right)} \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right)} \]

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}\right)\right)\right)} \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}\right)\right) \cdot \varepsilon\right)} \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}\right)\right) \cdot \varepsilon\right)} \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   3. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}\right) + 1\right)} \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   4. *-commutative N/A

      \[\leadsto \left(\left(\color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}\right) \cdot {\varepsilon}^{2}} + 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left({\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}, {\varepsilon}^{2}, 1\right)} \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   6. lower--.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   7. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2}} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2}} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   9. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{322560} \cdot {\varepsilon}^{2} + \frac{1}{1920}\right)} \cdot {\varepsilon}^{2} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{322560}, {\varepsilon}^{2}, \frac{1}{1920}\right)} \cdot {\varepsilon}^{2} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   11. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{1920}\right) \cdot {\varepsilon}^{2} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{1920}\right) \cdot {\varepsilon}^{2} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   13. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   15. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{24}, \color{blue}{\varepsilon \cdot \varepsilon}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   16. lower-*.f64 99.7

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-3.1001984126984127 \cdot 10^{-6}, \varepsilon \cdot \varepsilon, 0.0005208333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.041666666666666664, \color{blue}{\varepsilon \cdot \varepsilon}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

7. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-3.1001984126984127 \cdot 10^{-6}, \varepsilon \cdot \varepsilon, 0.0005208333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.041666666666666664, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right)} \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

8. Taylor expanded in x around inf

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{24}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)} \]
9. Step-by-step derivation

   1. cos-neg-rev N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{24}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)} \]

   2. lower-cos.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{24}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right)\right)} \]

   3. distribute-rgt-in N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{24}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\varepsilon \cdot \frac{-1}{2} + \left(2 \cdot x\right) \cdot \frac{-1}{2}\right)}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{24}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\mathsf{neg}\left(\left(\color{blue}{\frac{-1}{2} \cdot \varepsilon} + \left(2 \cdot x\right) \cdot \frac{-1}{2}\right)\right)\right) \]

   5. distribute-neg-in N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{24}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2} \cdot \varepsilon\right)\right) + \left(\mathsf{neg}\left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right)\right)\right)} \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{24}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \varepsilon} + \left(\mathsf{neg}\left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{24}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\color{blue}{\frac{1}{2}} \cdot \varepsilon + \left(\mathsf{neg}\left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{24}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \varepsilon + \left(\mathsf{neg}\left(\color{blue}{\frac{-1}{2} \cdot \left(2 \cdot x\right)}\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{24}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \varepsilon + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-1}{2} \cdot 2\right) \cdot x}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{24}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \varepsilon + \left(\mathsf{neg}\left(\color{blue}{-1} \cdot x\right)\right)\right) \]

   11. mul-1-neg N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{24}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \varepsilon + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)\right)\right) \]

   12. remove-double-neg N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{24}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{1}{2} \cdot \varepsilon + \color{blue}{x}\right) \]

   13. lower-fma.f64 99.7

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-3.1001984126984127 \cdot 10^{-6}, \varepsilon \cdot \varepsilon, 0.0005208333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.041666666666666664, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right)} \]

10. Applied rewrites 99.7%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-3.1001984126984127 \cdot 10^{-6}, \varepsilon \cdot \varepsilon, 0.0005208333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.041666666666666664, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \color{blue}{\cos \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right)} \]
11. Add Preprocessing

Alternative 4: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(0.0005208333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.041666666666666664, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (*
   (fma
    (- (* 0.0005208333333333333 (* eps eps)) 0.041666666666666664)
    (* eps eps)
    1.0)
   eps)
  (cos (/ (fma 2.0 x eps) -2.0))))

C

double code(double x, double eps) {
	return (fma(((0.0005208333333333333 * (eps * eps)) - 0.041666666666666664), (eps * eps), 1.0) * eps) * cos((fma(2.0, x, eps) / -2.0));
}

Julia

function code(x, eps)
	return Float64(Float64(fma(Float64(Float64(0.0005208333333333333 * Float64(eps * eps)) - 0.041666666666666664), Float64(eps * eps), 1.0) * eps) * cos(Float64(fma(2.0, x, eps) / -2.0)))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(0.0005208333333333333 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision] * N[Cos[N[(N[(2.0 * x + eps), $MachinePrecision] / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.3%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. associate--l+ N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{\color{blue}{0} + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

   17. cos-neg-rev N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   18. lower-cos.f64 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   19. distribute-neg-frac2 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\frac{\left(x + \varepsilon\right) + x}{\mathsf{neg}\left(2\right)}\right)} \]

   20. lower-/.f64 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\frac{\left(x + \varepsilon\right) + x}{\mathsf{neg}\left(2\right)}\right)} \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right)} \]

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{1920} \cdot {\varepsilon}^{2} - \frac{1}{24}\right)\right)\right)} \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{1920} \cdot {\varepsilon}^{2} - \frac{1}{24}\right)\right) \cdot \varepsilon\right)} \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{1920} \cdot {\varepsilon}^{2} - \frac{1}{24}\right)\right) \cdot \varepsilon\right)} \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   3. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{1920} \cdot {\varepsilon}^{2} - \frac{1}{24}\right) + 1\right)} \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   4. *-commutative N/A

      \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{1920} \cdot {\varepsilon}^{2} - \frac{1}{24}\right) \cdot {\varepsilon}^{2}} + 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{1}{1920} \cdot {\varepsilon}^{2} - \frac{1}{24}, {\varepsilon}^{2}, 1\right)} \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   6. lower--.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{1}{1920} \cdot {\varepsilon}^{2} - \frac{1}{24}}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{1}{1920} \cdot {\varepsilon}^{2}} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   8. unpow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{1920} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{1920} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   10. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{1}{1920} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{24}, \color{blue}{\varepsilon \cdot \varepsilon}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   11. lower-*.f64 99.6

       \[\leadsto \left(\mathsf{fma}\left(0.0005208333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.041666666666666664, \color{blue}{\varepsilon \cdot \varepsilon}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

7. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.0005208333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.041666666666666664, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right)} \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]
8. Add Preprocessing

Alternative 5: 99.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.3%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. associate--l+ N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{\color{blue}{0} + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

   17. cos-neg-rev N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   18. lower-cos.f64 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   19. distribute-neg-frac2 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\frac{\left(x + \varepsilon\right) + x}{\mathsf{neg}\left(2\right)}\right)} \]

   20. lower-/.f64 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\frac{\left(x + \varepsilon\right) + x}{\mathsf{neg}\left(2\right)}\right)} \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right)} \]

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}\right)\right)\right)} \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}\right)\right) \cdot \varepsilon\right)} \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}\right)\right) \cdot \varepsilon\right)} \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   3. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}\right) + 1\right)} \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   4. *-commutative N/A

      \[\leadsto \left(\left(\color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}\right) \cdot {\varepsilon}^{2}} + 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left({\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}, {\varepsilon}^{2}, 1\right)} \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   6. lower--.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   7. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2}} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2}} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   9. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{322560} \cdot {\varepsilon}^{2} + \frac{1}{1920}\right)} \cdot {\varepsilon}^{2} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{322560}, {\varepsilon}^{2}, \frac{1}{1920}\right)} \cdot {\varepsilon}^{2} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   11. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{1920}\right) \cdot {\varepsilon}^{2} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{1920}\right) \cdot {\varepsilon}^{2} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   13. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   15. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{24}, \color{blue}{\varepsilon \cdot \varepsilon}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   16. lower-*.f64 99.7

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-3.1001984126984127 \cdot 10^{-6}, \varepsilon \cdot \varepsilon, 0.0005208333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.041666666666666664, \color{blue}{\varepsilon \cdot \varepsilon}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

7. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-3.1001984126984127 \cdot 10^{-6}, \varepsilon \cdot \varepsilon, 0.0005208333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.041666666666666664, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right)} \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

8. Taylor expanded in eps around 0

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

10. Applied rewrites 99.4%

    \[\leadsto \left(\mathsf{fma}\left(-0.041666666666666664, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

11. Taylor expanded in x around inf

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

    1. cos-neg-rev N/A

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

    2. lower-cos.f64 N/A

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

    3. distribute-rgt-in N/A

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

    4. *-commutative N/A

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

    5. distribute-neg-in N/A

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

    6. distribute-lft-neg-in N/A

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

    7. metadata-eval N/A

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

    8. *-commutative N/A

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

    9. associate-*r* N/A

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

    10. metadata-eval N/A

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

    11. mul-1-neg N/A

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

    12. remove-double-neg N/A

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

    13. lower-fma.f64 99.4

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

13. Applied rewrites 99.4%

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

Alternative 6: 99.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.3%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. associate--l+ N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{\color{blue}{0} + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

   17. cos-neg-rev N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   18. lower-cos.f64 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   19. distribute-neg-frac2 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\frac{\left(x + \varepsilon\right) + x}{\mathsf{neg}\left(2\right)}\right)} \]

   20. lower-/.f64 N/A

       \[\leadsto \left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\frac{\left(x + \varepsilon\right) + x}{\mathsf{neg}\left(2\right)}\right)} \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot 2\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right)} \]

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}\right)\right)\right)} \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}\right)\right) \cdot \varepsilon\right)} \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}\right)\right) \cdot \varepsilon\right)} \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   3. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}\right) + 1\right)} \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   4. *-commutative N/A

      \[\leadsto \left(\left(\color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}\right) \cdot {\varepsilon}^{2}} + 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left({\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}, {\varepsilon}^{2}, 1\right)} \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   6. lower--.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) - \frac{1}{24}}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   7. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2}} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{1920} + \frac{-1}{322560} \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2}} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   9. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{322560} \cdot {\varepsilon}^{2} + \frac{1}{1920}\right)} \cdot {\varepsilon}^{2} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{322560}, {\varepsilon}^{2}, \frac{1}{1920}\right)} \cdot {\varepsilon}^{2} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   11. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{1920}\right) \cdot {\varepsilon}^{2} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{1920}\right) \cdot {\varepsilon}^{2} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   13. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} - \frac{1}{24}, {\varepsilon}^{2}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   15. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{322560}, \varepsilon \cdot \varepsilon, \frac{1}{1920}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{24}, \color{blue}{\varepsilon \cdot \varepsilon}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

   16. lower-*.f64 99.7

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-3.1001984126984127 \cdot 10^{-6}, \varepsilon \cdot \varepsilon, 0.0005208333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.041666666666666664, \color{blue}{\varepsilon \cdot \varepsilon}, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

7. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-3.1001984126984127 \cdot 10^{-6}, \varepsilon \cdot \varepsilon, 0.0005208333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.041666666666666664, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right)} \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

8. Taylor expanded in eps around 0

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

10. Applied rewrites 99.4%

    \[\leadsto \left(\mathsf{fma}\left(-0.041666666666666664, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{-2}\right) \]

11. Taylor expanded in eps around 0

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

    1. mul-1-neg N/A

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

    2. cos-neg-rev N/A

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

    3. lower-cos.f64 98.7

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

13. Applied rewrites 98.7%

    \[\leadsto \left(\mathsf{fma}\left(-0.041666666666666664, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right) \cdot \color{blue}{\cos x} \]
14. Add Preprocessing

Alternative 7: 99.1% 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 63.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 98.7

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

5. Applied rewrites 98.7%

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

Alternative 8: 98.4% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 99.0

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

5. Applied rewrites 99.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.0%

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

Alternative 9: 98.4% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 99.0

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

5. Applied rewrites 99.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.0%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 97.9%

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

Alternative 10: 97.8% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 99.0

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

5. Applied rewrites 99.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.5%

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

Alternative 11: 97.9% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. lower-sin.f64 97.5

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

5. Applied rewrites 97.5%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 97.5%

   \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.16666666666666666, 1\right) \cdot \color{blue}{\varepsilon} \]
9. Add Preprocessing

Alternative 12: 97.8% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \varepsilon \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.3%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 99.0

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

5. Applied rewrites 99.0%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 97.5%

   \[\leadsto 1 \cdot \varepsilon \]
9. Add Preprocessing

Developer Target 1: 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 2024350 
(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 (* (cos (* 1/2 (- eps (* -2 x)))) (sin (* 1/2 eps)) 2))

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