2sin (example 3.3)

Percentage Accurate: 62.5% → 99.9%

Time: 6.5s

Alternatives: 11

Speedup: 69.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \mathsf{fma}\left(\cos \left(\varepsilon \cdot 0.5\right), \cos \left(-1 \cdot x\right), \sin \left(--1 \cdot x\right) \cdot \left(\varepsilon \cdot \left(0.020833333333333332 \cdot {\varepsilon}^{2} - 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (* (sin (* 0.5 (- eps 0.0))) 2.0)
  (fma
   (cos (* eps 0.5))
   (cos (* -1.0 x))
   (*
    (sin (- (* -1.0 x)))
    (* eps (- (* 0.020833333333333332 (pow eps 2.0)) 0.5))))))

C

double code(double x, double eps) {
	return (sin((0.5 * (eps - 0.0))) * 2.0) * fma(cos((eps * 0.5)), cos((-1.0 * x)), (sin(-(-1.0 * x)) * (eps * ((0.020833333333333332 * pow(eps, 2.0)) - 0.5))));
}

Julia

function code(x, eps)
	return Float64(Float64(sin(Float64(0.5 * Float64(eps - 0.0))) * 2.0) * fma(cos(Float64(eps * 0.5)), cos(Float64(-1.0 * x)), Float64(sin(Float64(-Float64(-1.0 * x))) * Float64(eps * Float64(Float64(0.020833333333333332 * (eps ^ 2.0)) - 0.5)))))
end

Wolfram

code[x_, eps_] := N[(N[(N[Sin[N[(0.5 * N[(eps - 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-1.0 * x), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[(-N[(-1.0 * x), $MachinePrecision])], $MachinePrecision] * N[(eps * N[(N[(0.020833333333333332 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

   1. lift--.f64 N/A

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

   2. lift-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. mult-flip N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. add-flip N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 N/A

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

   20. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - \color{blue}{0}\right)\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right)} \]
4. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{-1}{2}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{-1}{2}\right)} \]

   3. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\frac{-1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)} \]

   4. lift-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\frac{-1}{2} \cdot \color{blue}{\left(2 \cdot x + \varepsilon\right)}\right) \]

   5. distribute-rgt-in N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\left(2 \cdot x\right) \cdot \frac{-1}{2} + \varepsilon \cdot \frac{-1}{2}\right)} \]

   6. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\left(2 \cdot x\right) \cdot \frac{-1}{2} + \color{blue}{\frac{-1}{2} \cdot \varepsilon}\right) \]

   7. cos-sum N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right)} \]

   8. lower--.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\cos \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right)} - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   10. lower-cos.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\cos \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right)} \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \color{blue}{\left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right)} \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   12. count-2-rev N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\color{blue}{\left(x + x\right)} \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   13. lower-+.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\color{blue}{\left(x + x\right)} \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   14. lower-cos.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \varepsilon\right)} - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon\right)} - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \color{blue}{\sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)}\right) \]

   17. lower-sin.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \color{blue}{\sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right)} \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   18. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \color{blue}{\left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right)} \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   19. count-2-rev N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\color{blue}{\left(x + x\right)} \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   20. lower-+.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\color{blue}{\left(x + x\right)} \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   21. lower-sin.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \color{blue}{\sin \left(\frac{-1}{2} \cdot \varepsilon\right)}\right) \]

5. Applied rewrites 100.0%

   \[\leadsto \left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(\left(x + x\right) \cdot -0.5\right) \cdot \cos \left(-0.5 \cdot \varepsilon\right) - \sin \left(\left(x + x\right) \cdot -0.5\right) \cdot \sin \left(-0.5 \cdot \varepsilon\right)\right)} \]
6. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \color{blue}{\sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)}\right) \]

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

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) + \left(\mathsf{neg}\left(\sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right)} + \left(\mathsf{neg}\left(\sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\cos \left(\frac{-1}{2} \cdot \varepsilon\right) \cdot \cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right)} + \left(\mathsf{neg}\left(\sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \varepsilon\right), \cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right), \left(\mathsf{neg}\left(\sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right)} \]

7. Applied rewrites 100.0%

   \[\leadsto \left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\varepsilon \cdot 0.5\right), \cos \left(-1 \cdot x\right), \sin \left(--1 \cdot x\right) \cdot \sin \left(-0.5 \cdot \varepsilon\right)\right)} \]

8. Taylor expanded in eps around 0

   \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \mathsf{fma}\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(-1 \cdot x\right), \sin \left(--1 \cdot x\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{48} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)}\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \mathsf{fma}\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(-1 \cdot x\right), \sin \left(--1 \cdot x\right) \cdot \left(\varepsilon \cdot \color{blue}{\left(\frac{1}{48} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)}\right)\right) \]

   2. lower--.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \mathsf{fma}\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(-1 \cdot x\right), \sin \left(--1 \cdot x\right) \cdot \left(\varepsilon \cdot \left(\frac{1}{48} \cdot {\varepsilon}^{2} - \color{blue}{\frac{1}{2}}\right)\right)\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \mathsf{fma}\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(-1 \cdot x\right), \sin \left(--1 \cdot x\right) \cdot \left(\varepsilon \cdot \left(\frac{1}{48} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right) \]

   4. lower-pow.f64 99.8

      \[\leadsto \left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \mathsf{fma}\left(\cos \left(\varepsilon \cdot 0.5\right), \cos \left(-1 \cdot x\right), \sin \left(--1 \cdot x\right) \cdot \left(\varepsilon \cdot \left(0.020833333333333332 \cdot {\varepsilon}^{2} - 0.5\right)\right)\right) \]

10. Applied rewrites 99.8%

    \[\leadsto \left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \mathsf{fma}\left(\cos \left(\varepsilon \cdot 0.5\right), \cos \left(-1 \cdot x\right), \sin \left(--1 \cdot x\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(0.020833333333333332 \cdot {\varepsilon}^{2} - 0.5\right)\right)}\right) \]
11. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + x\right) \cdot -0.5\\ \left(\varepsilon \cdot \left(1 + -0.041666666666666664 \cdot {\varepsilon}^{2}\right)\right) \cdot \left(\cos t\_0 \cdot \cos \left(-0.5 \cdot \varepsilon\right) - \sin t\_0 \cdot \sin \left(-0.5 \cdot \varepsilon\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (+ x x) -0.5)))
   (*
    (* eps (+ 1.0 (* -0.041666666666666664 (pow eps 2.0))))
    (- (* (cos t_0) (cos (* -0.5 eps))) (* (sin t_0) (sin (* -0.5 eps)))))))

C

double code(double x, double eps) {
	double t_0 = (x + x) * -0.5;
	return (eps * (1.0 + (-0.041666666666666664 * pow(eps, 2.0)))) * ((cos(t_0) * cos((-0.5 * eps))) - (sin(t_0) * sin((-0.5 * 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
    real(8) :: t_0
    t_0 = (x + x) * (-0.5d0)
    code = (eps * (1.0d0 + ((-0.041666666666666664d0) * (eps ** 2.0d0)))) * ((cos(t_0) * cos(((-0.5d0) * eps))) - (sin(t_0) * sin(((-0.5d0) * eps))))
end function

Java

public static double code(double x, double eps) {
	double t_0 = (x + x) * -0.5;
	return (eps * (1.0 + (-0.041666666666666664 * Math.pow(eps, 2.0)))) * ((Math.cos(t_0) * Math.cos((-0.5 * eps))) - (Math.sin(t_0) * Math.sin((-0.5 * eps))));
}

Python

def code(x, eps):
	t_0 = (x + x) * -0.5
	return (eps * (1.0 + (-0.041666666666666664 * math.pow(eps, 2.0)))) * ((math.cos(t_0) * math.cos((-0.5 * eps))) - (math.sin(t_0) * math.sin((-0.5 * eps))))

Julia

function code(x, eps)
	t_0 = Float64(Float64(x + x) * -0.5)
	return Float64(Float64(eps * Float64(1.0 + Float64(-0.041666666666666664 * (eps ^ 2.0)))) * Float64(Float64(cos(t_0) * cos(Float64(-0.5 * eps))) - Float64(sin(t_0) * sin(Float64(-0.5 * eps)))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = (x + x) * -0.5;
	tmp = (eps * (1.0 + (-0.041666666666666664 * (eps ^ 2.0)))) * ((cos(t_0) * cos((-0.5 * eps))) - (sin(t_0) * sin((-0.5 * eps))));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x + x), $MachinePrecision] * -0.5), $MachinePrecision]}, N[(N[(eps * N[(1.0 + N[(-0.041666666666666664 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[t$95$0], $MachinePrecision] * N[Cos[N[(-0.5 * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t$95$0], $MachinePrecision] * N[Sin[N[(-0.5 * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.5%

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

   1. lift--.f64 N/A

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

   2. lift-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. mult-flip N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. add-flip N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 N/A

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

   20. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - \color{blue}{0}\right)\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right)} \]
4. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{-1}{2}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{-1}{2}\right)} \]

   3. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\frac{-1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)} \]

   4. lift-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\frac{-1}{2} \cdot \color{blue}{\left(2 \cdot x + \varepsilon\right)}\right) \]

   5. distribute-rgt-in N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\left(2 \cdot x\right) \cdot \frac{-1}{2} + \varepsilon \cdot \frac{-1}{2}\right)} \]

   6. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\left(2 \cdot x\right) \cdot \frac{-1}{2} + \color{blue}{\frac{-1}{2} \cdot \varepsilon}\right) \]

   7. cos-sum N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right)} \]

   8. lower--.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\cos \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right)} - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   10. lower-cos.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\cos \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right)} \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \color{blue}{\left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right)} \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   12. count-2-rev N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\color{blue}{\left(x + x\right)} \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   13. lower-+.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\color{blue}{\left(x + x\right)} \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   14. lower-cos.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \varepsilon\right)} - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon\right)} - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \color{blue}{\sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)}\right) \]

   17. lower-sin.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \color{blue}{\sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right)} \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   18. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \color{blue}{\left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right)} \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   19. count-2-rev N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\color{blue}{\left(x + x\right)} \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   20. lower-+.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\color{blue}{\left(x + x\right)} \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   21. lower-sin.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \color{blue}{\sin \left(\frac{-1}{2} \cdot \varepsilon\right)}\right) \]

5. Applied rewrites 100.0%

   \[\leadsto \left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(\left(x + x\right) \cdot -0.5\right) \cdot \cos \left(-0.5 \cdot \varepsilon\right) - \sin \left(\left(x + x\right) \cdot -0.5\right) \cdot \sin \left(-0.5 \cdot \varepsilon\right)\right)} \]

6. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 99.7

      \[\leadsto \left(\varepsilon \cdot \left(1 + -0.041666666666666664 \cdot {\varepsilon}^{\color{blue}{2}}\right)\right) \cdot \left(\cos \left(\left(x + x\right) \cdot -0.5\right) \cdot \cos \left(-0.5 \cdot \varepsilon\right) - \sin \left(\left(x + x\right) \cdot -0.5\right) \cdot \sin \left(-0.5 \cdot \varepsilon\right)\right) \]

8. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(1 + -0.041666666666666664 \cdot {\varepsilon}^{2}\right)\right)} \cdot \left(\cos \left(\left(x + x\right) \cdot -0.5\right) \cdot \cos \left(-0.5 \cdot \varepsilon\right) - \sin \left(\left(x + x\right) \cdot -0.5\right) \cdot \sin \left(-0.5 \cdot \varepsilon\right)\right) \]
9. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\varepsilon \cdot \left(1 + -0.041666666666666664 \cdot {\varepsilon}^{2}\right)\right) \cdot \mathsf{fma}\left(\cos \left(\varepsilon \cdot 0.5\right), \cos \left(-1 \cdot x\right), \sin \left(--1 \cdot x\right) \cdot \sin \left(-0.5 \cdot \varepsilon\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (* eps (+ 1.0 (* -0.041666666666666664 (pow eps 2.0))))
  (fma
   (cos (* eps 0.5))
   (cos (* -1.0 x))
   (* (sin (- (* -1.0 x))) (sin (* -0.5 eps))))))

C

double code(double x, double eps) {
	return (eps * (1.0 + (-0.041666666666666664 * pow(eps, 2.0)))) * fma(cos((eps * 0.5)), cos((-1.0 * x)), (sin(-(-1.0 * x)) * sin((-0.5 * eps))));
}

Julia

function code(x, eps)
	return Float64(Float64(eps * Float64(1.0 + Float64(-0.041666666666666664 * (eps ^ 2.0)))) * fma(cos(Float64(eps * 0.5)), cos(Float64(-1.0 * x)), Float64(sin(Float64(-Float64(-1.0 * x))) * sin(Float64(-0.5 * eps)))))
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(1.0 + N[(-0.041666666666666664 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-1.0 * x), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[(-N[(-1.0 * x), $MachinePrecision])], $MachinePrecision] * N[Sin[N[(-0.5 * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

   1. lift--.f64 N/A

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

   2. lift-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. mult-flip N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. add-flip N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 N/A

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

   20. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - \color{blue}{0}\right)\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right)} \]
4. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{-1}{2}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{-1}{2}\right)} \]

   3. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\frac{-1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)} \]

   4. lift-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\frac{-1}{2} \cdot \color{blue}{\left(2 \cdot x + \varepsilon\right)}\right) \]

   5. distribute-rgt-in N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\left(2 \cdot x\right) \cdot \frac{-1}{2} + \varepsilon \cdot \frac{-1}{2}\right)} \]

   6. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\left(2 \cdot x\right) \cdot \frac{-1}{2} + \color{blue}{\frac{-1}{2} \cdot \varepsilon}\right) \]

   7. cos-sum N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right)} \]

   8. lower--.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\cos \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right)} - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   10. lower-cos.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\cos \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right)} \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \color{blue}{\left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right)} \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   12. count-2-rev N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\color{blue}{\left(x + x\right)} \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   13. lower-+.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\color{blue}{\left(x + x\right)} \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   14. lower-cos.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \varepsilon\right)} - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon\right)} - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \color{blue}{\sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)}\right) \]

   17. lower-sin.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \color{blue}{\sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right)} \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   18. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \color{blue}{\left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right)} \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   19. count-2-rev N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\color{blue}{\left(x + x\right)} \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   20. lower-+.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\color{blue}{\left(x + x\right)} \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   21. lower-sin.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \color{blue}{\sin \left(\frac{-1}{2} \cdot \varepsilon\right)}\right) \]

5. Applied rewrites 100.0%

   \[\leadsto \left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(\left(x + x\right) \cdot -0.5\right) \cdot \cos \left(-0.5 \cdot \varepsilon\right) - \sin \left(\left(x + x\right) \cdot -0.5\right) \cdot \sin \left(-0.5 \cdot \varepsilon\right)\right)} \]
6. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \color{blue}{\sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)}\right) \]

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

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) + \left(\mathsf{neg}\left(\sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right)} + \left(\mathsf{neg}\left(\sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\cos \left(\frac{-1}{2} \cdot \varepsilon\right) \cdot \cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right)} + \left(\mathsf{neg}\left(\sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \varepsilon\right), \cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right), \left(\mathsf{neg}\left(\sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right)} \]

7. Applied rewrites 100.0%

   \[\leadsto \left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\varepsilon \cdot 0.5\right), \cos \left(-1 \cdot x\right), \sin \left(--1 \cdot x\right) \cdot \sin \left(-0.5 \cdot \varepsilon\right)\right)} \]

8. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 99.7

      \[\leadsto \left(\varepsilon \cdot \left(1 + -0.041666666666666664 \cdot {\varepsilon}^{\color{blue}{2}}\right)\right) \cdot \mathsf{fma}\left(\cos \left(\varepsilon \cdot 0.5\right), \cos \left(-1 \cdot x\right), \sin \left(--1 \cdot x\right) \cdot \sin \left(-0.5 \cdot \varepsilon\right)\right) \]

10. Applied rewrites 99.7%

    \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(1 + -0.041666666666666664 \cdot {\varepsilon}^{2}\right)\right)} \cdot \mathsf{fma}\left(\cos \left(\varepsilon \cdot 0.5\right), \cos \left(-1 \cdot x\right), \sin \left(--1 \cdot x\right) \cdot \sin \left(-0.5 \cdot \varepsilon\right)\right) \]
11. Add Preprocessing

Alternative 4: 99.7% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (fma eps (fma (sin x) -0.5 (* (* (cos x) eps) -0.16666666666666666)) (cos x))
  eps))

C

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

Julia

function code(x, eps)
	return Float64(fma(eps, fma(sin(x), -0.5, Float64(Float64(cos(x) * eps) * -0.16666666666666666)), cos(x)) * eps)
end

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 99.7

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

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \mathsf{fma}\left(-0.5, \sin x, -0.16666666666666666 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

Alternative 5: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left(0.0005208333333333333 \cdot {\varepsilon}^{2} - 0.041666666666666664\right)\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (*
   eps
   (+
    1.0
    (*
     (pow eps 2.0)
     (- (* 0.0005208333333333333 (pow eps 2.0)) 0.041666666666666664))))
  (cos (* (fma 2.0 x eps) -0.5))))

C

double code(double x, double eps) {
	return (eps * (1.0 + (pow(eps, 2.0) * ((0.0005208333333333333 * pow(eps, 2.0)) - 0.041666666666666664)))) * cos((fma(2.0, x, eps) * -0.5));
}

Julia

function code(x, eps)
	return Float64(Float64(eps * Float64(1.0 + Float64((eps ^ 2.0) * Float64(Float64(0.0005208333333333333 * (eps ^ 2.0)) - 0.041666666666666664)))) * cos(Float64(fma(2.0, x, eps) * -0.5)))
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(1.0 + N[(N[Power[eps, 2.0], $MachinePrecision] * N[(N[(0.0005208333333333333 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * x + eps), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

   1. lift--.f64 N/A

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

   2. lift-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. mult-flip N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. add-flip N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 N/A

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

   20. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - \color{blue}{0}\right)\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right)} \]

4. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-pow.f64 99.7

      \[\leadsto \left(\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left(0.0005208333333333333 \cdot {\varepsilon}^{2} - 0.041666666666666664\right)\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right) \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left(0.0005208333333333333 \cdot {\varepsilon}^{2} - 0.041666666666666664\right)\right)\right)} \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right) \]
7. Add Preprocessing

Alternative 6: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. lift--.f64 N/A

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

   2. lift-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. mult-flip N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. add-flip N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 N/A

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

   20. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - \color{blue}{0}\right)\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right)} \]
4. Step-by-step derivation

   1. lift-cos.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{-1}{2}\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{-1}{2}\right)} \]

   3. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\frac{-1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)} \]

   4. lift-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\frac{-1}{2} \cdot \color{blue}{\left(2 \cdot x + \varepsilon\right)}\right) \]

   5. distribute-rgt-in N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \color{blue}{\left(\left(2 \cdot x\right) \cdot \frac{-1}{2} + \varepsilon \cdot \frac{-1}{2}\right)} \]

   6. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\left(2 \cdot x\right) \cdot \frac{-1}{2} + \color{blue}{\frac{-1}{2} \cdot \varepsilon}\right) \]

   7. cos-sum N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right)} \]

   8. lower--.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\cos \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right)} - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   10. lower-cos.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\cos \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right)} \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \color{blue}{\left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right)} \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   12. count-2-rev N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\color{blue}{\left(x + x\right)} \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   13. lower-+.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\color{blue}{\left(x + x\right)} \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   14. lower-cos.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \varepsilon\right)} - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon\right)} - \sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \color{blue}{\sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)}\right) \]

   17. lower-sin.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \color{blue}{\sin \left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right)} \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   18. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \color{blue}{\left(\left(2 \cdot x\right) \cdot \frac{-1}{2}\right)} \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   19. count-2-rev N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\color{blue}{\left(x + x\right)} \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   20. lower-+.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\color{blue}{\left(x + x\right)} \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   21. lower-sin.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \color{blue}{\sin \left(\frac{-1}{2} \cdot \varepsilon\right)}\right) \]

5. Applied rewrites 100.0%

   \[\leadsto \left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(\left(x + x\right) \cdot -0.5\right) \cdot \cos \left(-0.5 \cdot \varepsilon\right) - \sin \left(\left(x + x\right) \cdot -0.5\right) \cdot \sin \left(-0.5 \cdot \varepsilon\right)\right)} \]
6. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) - \color{blue}{\sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)}\right) \]

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

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right) + \left(\mathsf{neg}\left(\sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \varepsilon\right)} + \left(\mathsf{neg}\left(\sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\color{blue}{\cos \left(\frac{-1}{2} \cdot \varepsilon\right) \cdot \cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right)} + \left(\mathsf{neg}\left(\sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \varepsilon\right), \cos \left(\left(x + x\right) \cdot \frac{-1}{2}\right), \left(\mathsf{neg}\left(\sin \left(\left(x + x\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right)} \]

7. Applied rewrites 100.0%

   \[\leadsto \left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\varepsilon \cdot 0.5\right), \cos \left(-1 \cdot x\right), \sin \left(--1 \cdot x\right) \cdot \sin \left(-0.5 \cdot \varepsilon\right)\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right)} \cdot 2\right) \cdot \mathsf{fma}\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(-1 \cdot x\right), \sin \left(--1 \cdot x\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   2. lift--.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \color{blue}{\left(\varepsilon - 0\right)}\right) \cdot 2\right) \cdot \mathsf{fma}\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(-1 \cdot x\right), \sin \left(--1 \cdot x\right) \cdot \sin \left(\frac{-1}{2} \cdot \varepsilon\right)\right) \]

   3. --rgt-identity N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 100.0

      \[\leadsto \left(\sin \color{blue}{\left(\varepsilon \cdot 0.5\right)} \cdot 2\right) \cdot \mathsf{fma}\left(\cos \left(\varepsilon \cdot 0.5\right), \cos \left(-1 \cdot x\right), \sin \left(--1 \cdot x\right) \cdot \sin \left(-0.5 \cdot \varepsilon\right)\right) \]

   6. lift-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-cos.f64 N/A

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

   9. cos-neg-rev N/A

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

   10. lift-neg.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. cos-neg-rev N/A

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

   13. lift-*.f64 N/A

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

   14. distribute-rgt-neg-out N/A

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

   15. metadata-eval N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

9. Applied rewrites 99.9%

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

Alternative 7: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\varepsilon \cdot \left(1 + -0.041666666666666664 \cdot {\varepsilon}^{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (* eps (+ 1.0 (* -0.041666666666666664 (pow eps 2.0))))
  (cos (* (fma 2.0 x eps) -0.5))))

C

double code(double x, double eps) {
	return (eps * (1.0 + (-0.041666666666666664 * pow(eps, 2.0)))) * cos((fma(2.0, x, eps) * -0.5));
}

Julia

function code(x, eps)
	return Float64(Float64(eps * Float64(1.0 + Float64(-0.041666666666666664 * (eps ^ 2.0)))) * cos(Float64(fma(2.0, x, eps) * -0.5)))
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(1.0 + N[(-0.041666666666666664 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * x + eps), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

   1. lift--.f64 N/A

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

   2. lift-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. mult-flip N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. add-flip N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 N/A

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

   20. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - \color{blue}{0}\right)\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right)} \]

4. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 99.7

      \[\leadsto \left(\varepsilon \cdot \left(1 + -0.041666666666666664 \cdot {\varepsilon}^{\color{blue}{2}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right) \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(1 + -0.041666666666666664 \cdot {\varepsilon}^{2}\right)\right)} \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right) \]
7. Add Preprocessing

Alternative 8: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. lift--.f64 N/A

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

   2. lift-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. mult-flip N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. add-flip N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 N/A

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

   20. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - \color{blue}{0}\right)\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right)} \]

4. Taylor expanded in eps around 0

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

   1. lower-*.f64 99.4

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

6. Applied rewrites 99.4%

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

Alternative 9: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 99.0

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

4. Applied rewrites 99.0%

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

Alternative 10: 98.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 99.5

      \[\leadsto \varepsilon \cdot \left(\cos x + -0.5 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 98.2

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

7. Applied rewrites 98.2%

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

Alternative 11: 97.7% accurate, 69.8× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 99.5

      \[\leadsto \varepsilon \cdot \left(\cos x + -0.5 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 97.7

      \[\leadsto \varepsilon \cdot \left(1 + -0.5 \cdot \left(\varepsilon \cdot x\right)\right) \]

7. Applied rewrites 97.7%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{-0.5 \cdot \left(\varepsilon \cdot x\right)}\right) \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.7%

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

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025154 
(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 c (* 2 (cos (+ x (/ eps 2))) (sin (/ eps 2))))

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

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

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