2cos (problem 3.3.5)

Percentage Accurate: 53.2% → 99.8%

Time: 6.6s

Alternatives: 5

Speedup: 1.9×

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.2%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. mult-flip N/A

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

   18. metadata-eval N/A

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

   19. lower-*.f64 N/A

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

3. Applied rewrites 80.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lower-+.f64 N/A

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

   5. lower--.f64 99.6

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

5. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.6

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lift--.f64 N/A

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

   8. +-inverses N/A

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

   9. +-rgt-identity N/A

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

   10. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-+l+ N/A

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

   7. count-2-rev N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. sin-sum N/A

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

   11. lower-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. count-2-rev N/A

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

   18. lower-+.f64 N/A

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

   19. lower-*.f64 N/A

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

   20. lower-cos.f64 N/A

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

   21. *-commutative N/A

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

   22. lift-*.f64 N/A

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

9. Applied rewrites 99.8%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.2%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. mult-flip N/A

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

   18. metadata-eval N/A

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

   19. lower-*.f64 N/A

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

3. Applied rewrites 80.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lower-+.f64 N/A

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

   5. lower--.f64 99.6

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

5. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.6

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lift--.f64 N/A

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

   8. +-inverses N/A

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

   9. +-rgt-identity N/A

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

   10. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.2%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. mult-flip N/A

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

   18. metadata-eval N/A

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

   19. lower-*.f64 N/A

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

3. Applied rewrites 80.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lower-+.f64 N/A

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

   5. lower--.f64 99.6

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

5. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.6

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lift--.f64 N/A

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

   8. +-inverses N/A

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

   9. +-rgt-identity N/A

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

   10. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-flip N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 99.5

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

10. Applied rewrites 99.5%

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

Alternative 4: 99.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.2%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. mult-flip N/A

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

   18. metadata-eval N/A

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

   19. lower-*.f64 N/A

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

3. Applied rewrites 80.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lower-+.f64 N/A

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

   5. lower--.f64 99.6

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

5. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.6

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lift--.f64 N/A

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

   8. +-inverses N/A

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

   9. +-rgt-identity N/A

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

   10. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-flip N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 99.4

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

10. Applied rewrites 99.4%

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

Alternative 5: 79.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.2%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. mult-flip N/A

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

   18. metadata-eval N/A

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

   19. lower-*.f64 N/A

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

3. Applied rewrites 80.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lower-+.f64 N/A

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

   5. lower--.f64 99.6

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

5. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.6

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. lift--.f64 N/A

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

   8. +-inverses N/A

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

   9. +-rgt-identity N/A

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

   10. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right)} \]
9. Step-by-step derivation

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\varepsilon\right)\right) \cdot \sin \color{blue}{x} \]

   4. lower-neg.f64 N/A

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

   5. lower-sin.f64 79.9

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

10. Applied rewrites 79.9%

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

Developer Target 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (-2.0 * sin((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) * sin((x + (eps / 2.0d0)))) * sin((eps / 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{\left(-2 \cdot \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (pow (cbrt (* (* -2.0 (sin (* 0.5 (fma 2.0 x eps)))) (sin (* 0.5 eps)))) 3.0))

C

double code(double x, double eps) {
	return pow(cbrt(((-2.0 * sin((0.5 * fma(2.0, x, eps)))) * sin((0.5 * eps)))), 3.0);
}

Julia

function code(x, eps)
	return cbrt(Float64(Float64(-2.0 * sin(Float64(0.5 * fma(2.0, x, eps)))) * sin(Float64(0.5 * eps)))) ^ 3.0
end

Wolfram

code[x_, eps_] := N[Power[N[Power[N[(N[(-2.0 * N[Sin[N[(0.5 * N[(2.0 * x + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]

Reproduce

herbie shell --seed 2025136 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :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 (sin (+ x (/ eps 2))) (sin (/ eps 2))))

  :alt
  (! :herbie-platform c (pow (cbrt (* -2 (sin (* 1/2 (fma 2 x eps))) (sin (* 1/2 eps)))) 3))

  (- (cos (+ x eps)) (cos x)))