2cos (problem 3.3.5)

Percentage Accurate: 52.6% → 99.6%

Time: 9.3s

Alternatives: 14

Speedup: 25.9×

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: 52.6% 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: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ t_1 := \left(2 \cdot x\right) \cdot 0.5\\ \left(\mathsf{fma}\left(t\_0, \cos t\_1, \mathsf{fma}\left(\mathsf{fma}\left(-2.170138888888889 \cdot 10^{-5}, \varepsilon \cdot \varepsilon, 0.0026041666666666665\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.125, \varepsilon \cdot \varepsilon, 1\right) \cdot \sin t\_1\right) \cdot t\_0\right) \cdot -2 \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 eps))) (t_1 (* (* 2.0 x) 0.5)))
   (*
    (*
     (fma
      t_0
      (cos t_1)
      (*
       (fma
        (-
         (*
          (fma -2.170138888888889e-5 (* eps eps) 0.0026041666666666665)
          (* eps eps))
         0.125)
        (* eps eps)
        1.0)
       (sin t_1)))
     t_0)
    -2.0)))

C

double code(double x, double eps) {
	double t_0 = sin((0.5 * eps));
	double t_1 = (2.0 * x) * 0.5;
	return (fma(t_0, cos(t_1), (fma(((fma(-2.170138888888889e-5, (eps * eps), 0.0026041666666666665) * (eps * eps)) - 0.125), (eps * eps), 1.0) * sin(t_1))) * t_0) * -2.0;
}

Julia

function code(x, eps)
	t_0 = sin(Float64(0.5 * eps))
	t_1 = Float64(Float64(2.0 * x) * 0.5)
	return Float64(Float64(fma(t_0, cos(t_1), Float64(fma(Float64(Float64(fma(-2.170138888888889e-5, Float64(eps * eps), 0.0026041666666666665) * Float64(eps * eps)) - 0.125), Float64(eps * eps), 1.0) * sin(t_1))) * t_0) * -2.0)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * x), $MachinePrecision] * 0.5), $MachinePrecision]}, N[(N[(N[(t$95$0 * N[Cos[t$95$1], $MachinePrecision] + N[(N[(N[(N[(N[(-2.170138888888889e-5 * N[(eps * eps), $MachinePrecision] + 0.0026041666666666665), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 0.125), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Initial program 52.6%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. diff-cos N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 80.8%

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

4. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

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

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. metadata-eval N/A

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

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

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

   8. count-2-rev N/A

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

   9. associate-+l+ N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. associate-+l+ N/A

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

   13. count-2-rev N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-sin.f64 N/A

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

6. Applied rewrites 99.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

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

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

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

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

   10. distribute-rgt-in N/A

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

   11. *-commutative N/A

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

   12. sin-sum N/A

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

   13. lower-fma.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 N/A

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in eps around 0

   \[\leadsto \left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right), \left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {\varepsilon}^{2}\right) - \frac{1}{8}\right)\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2 \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right), \left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {\varepsilon}^{2}\right) - \frac{1}{8}\right) + 1\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2 \]

    2. *-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right), \left(\left({\varepsilon}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {\varepsilon}^{2}\right) - \frac{1}{8}\right) \cdot {\varepsilon}^{2} + 1\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2 \]

    3. lower-fma.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right), \mathsf{fma}\left({\varepsilon}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {\varepsilon}^{2}\right) - \frac{1}{8}, {\varepsilon}^{2}, 1\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2 \]

    4. lower--.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right), \mathsf{fma}\left({\varepsilon}^{2} \cdot \left(\frac{1}{384} + \frac{-1}{46080} \cdot {\varepsilon}^{2}\right) - \frac{1}{8}, {\varepsilon}^{2}, 1\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2 \]

    5. *-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right), \mathsf{fma}\left(\left(\frac{1}{384} + \frac{-1}{46080} \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2} - \frac{1}{8}, {\varepsilon}^{2}, 1\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2 \]

    6. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right), \mathsf{fma}\left(\left(\frac{1}{384} + \frac{-1}{46080} \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2} - \frac{1}{8}, {\varepsilon}^{2}, 1\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2 \]

    7. +-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right), \mathsf{fma}\left(\left(\frac{-1}{46080} \cdot {\varepsilon}^{2} + \frac{1}{384}\right) \cdot {\varepsilon}^{2} - \frac{1}{8}, {\varepsilon}^{2}, 1\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2 \]

    8. lower-fma.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{46080}, {\varepsilon}^{2}, \frac{1}{384}\right) \cdot {\varepsilon}^{2} - \frac{1}{8}, {\varepsilon}^{2}, 1\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2 \]

    9. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{46080}, \varepsilon \cdot \varepsilon, \frac{1}{384}\right) \cdot {\varepsilon}^{2} - \frac{1}{8}, {\varepsilon}^{2}, 1\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2 \]

    10. lift-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{46080}, \varepsilon \cdot \varepsilon, \frac{1}{384}\right) \cdot {\varepsilon}^{2} - \frac{1}{8}, {\varepsilon}^{2}, 1\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2 \]

    11. pow2 N/A

        \[\leadsto \left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{46080}, \varepsilon \cdot \varepsilon, \frac{1}{384}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{8}, {\varepsilon}^{2}, 1\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2 \]

    12. lift-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{46080}, \varepsilon \cdot \varepsilon, \frac{1}{384}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{8}, {\varepsilon}^{2}, 1\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2 \]

    13. pow2 N/A

        \[\leadsto \left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right), \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right), \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{46080}, \varepsilon \cdot \varepsilon, \frac{1}{384}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{8}, \varepsilon \cdot \varepsilon, 1\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot -2 \]

    14. lift-*.f64 99.6

        \[\leadsto \left(\mathsf{fma}\left(\sin \left(0.5 \cdot \varepsilon\right), \cos \left(\left(2 \cdot x\right) \cdot 0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-2.170138888888889 \cdot 10^{-5}, \varepsilon \cdot \varepsilon, 0.0026041666666666665\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.125, \varepsilon \cdot \varepsilon, 1\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot -2 \]

11. Applied rewrites 99.6%

    \[\leadsto \left(\mathsf{fma}\left(\sin \left(0.5 \cdot \varepsilon\right), \cos \left(\left(2 \cdot x\right) \cdot 0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(-2.170138888888889 \cdot 10^{-5}, \varepsilon \cdot \varepsilon, 0.0026041666666666665\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.125, \varepsilon \cdot \varepsilon, 1\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot -2 \]
12. Add Preprocessing

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot x\right) \cdot 0.5\\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.00026041666666666666 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.020833333333333332, \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon, \cos t\_0, \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin t\_0\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot -2 \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* 2.0 x) 0.5)))
   (*
    (*
     (fma
      (*
       (fma
        (- (* 0.00026041666666666666 (* eps eps)) 0.020833333333333332)
        (* eps eps)
        0.5)
       eps)
      (cos t_0)
      (* (cos (* 0.5 eps)) (sin t_0)))
     (sin (* 0.5 eps)))
    -2.0)))

C

double code(double x, double eps) {
	double t_0 = (2.0 * x) * 0.5;
	return (fma((fma(((0.00026041666666666666 * (eps * eps)) - 0.020833333333333332), (eps * eps), 0.5) * eps), cos(t_0), (cos((0.5 * eps)) * sin(t_0))) * sin((0.5 * eps))) * -2.0;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(2.0 * x) * 0.5)
	return Float64(Float64(fma(Float64(fma(Float64(Float64(0.00026041666666666666 * Float64(eps * eps)) - 0.020833333333333332), Float64(eps * eps), 0.5) * eps), cos(t_0), Float64(cos(Float64(0.5 * eps)) * sin(t_0))) * sin(Float64(0.5 * eps))) * -2.0)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(2.0 * x), $MachinePrecision] * 0.5), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(N[(0.00026041666666666666 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 0.020833333333333332), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 0.5), $MachinePrecision] * eps), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision] + N[(N[Cos[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]

Derivation

1. Initial program 52.6%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. diff-cos N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 80.8%

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

4. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

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

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. metadata-eval N/A

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

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

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

   8. count-2-rev N/A

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

   9. associate-+l+ N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. associate-+l+ N/A

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

   13. count-2-rev N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-sin.f64 N/A

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

6. Applied rewrites 99.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. +-commutative N/A

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

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

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

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

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

   10. distribute-rgt-in N/A

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

   11. *-commutative N/A

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

   12. sin-sum N/A

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

   13. lower-fma.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 N/A

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. lower--.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. pow2 N/A

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

    9. lift-*.f64 N/A

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

    10. pow2 N/A

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

    11. lift-*.f64 99.6

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

11. Applied rewrites 99.6%

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

Alternative 3: 99.6% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (*
    (fma
     (fma (sin x) 0.16666666666666666 (* (* (cos x) eps) 0.041666666666666664))
     eps
     (* -0.5 (cos x)))
    eps)
   (sin x))
  eps))

C

double code(double x, double eps) {
	return ((fma(fma(sin(x), 0.16666666666666666, ((cos(x) * eps) * 0.041666666666666664)), eps, (-0.5 * cos(x))) * eps) - sin(x)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(fma(fma(sin(x), 0.16666666666666666, Float64(Float64(cos(x) * eps) * 0.041666666666666664)), eps, Float64(-0.5 * cos(x))) * eps) - sin(x)) * eps)
end

Wolfram

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

Derivation

1. Initial program 52.6%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

Alternative 4: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.6%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. diff-cos N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 80.8%

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

4. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

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

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. metadata-eval N/A

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

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

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

   8. count-2-rev N/A

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

   9. associate-+l+ N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. associate-+l+ N/A

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

   13. count-2-rev N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-sin.f64 N/A

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

9. Applied rewrites 99.6%

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

Alternative 5: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.6%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. diff-cos N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 80.8%

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

4. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

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

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. metadata-eval N/A

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

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

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

   8. count-2-rev N/A

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

   9. associate-+l+ N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. associate-+l+ N/A

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

   13. count-2-rev N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-sin.f64 N/A

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 99.6

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

9. Applied rewrites 99.6%

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

Alternative 6: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.6%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. diff-cos N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 80.8%

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

4. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

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

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. metadata-eval N/A

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

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

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

   8. count-2-rev N/A

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

   9. associate-+l+ N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. associate-+l+ N/A

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

   13. count-2-rev N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-sin.f64 N/A

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 99.5

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

9. Applied rewrites 99.5%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 99.5

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

12. Applied rewrites 99.5%

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

Alternative 7: 99.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.6%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. diff-cos N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 80.8%

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

4. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

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

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. metadata-eval N/A

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

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

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

   8. count-2-rev N/A

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

   9. associate-+l+ N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. associate-+l+ N/A

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

   13. count-2-rev N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-sin.f64 N/A

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in eps around 0

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

   1. lift-*.f64 99.3

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

9. Applied rewrites 99.3%

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

Alternative 8: 98.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (((((eps * eps) * 0.041666666666666664d0) - 0.5d0) * eps) - sin(x)) * eps
end function

Java

public static double code(double x, double eps) {
	return (((((eps * eps) * 0.041666666666666664) - 0.5) * eps) - Math.sin(x)) * eps;
}

Python

def code(x, eps):
	return (((((eps * eps) * 0.041666666666666664) - 0.5) * eps) - math.sin(x)) * eps

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = (((((eps * eps) * 0.041666666666666664) - 0.5) * eps) - sin(x)) * eps;
end

Wolfram

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

Derivation

1. Initial program 52.6%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 98.9

      \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

7. Applied rewrites 98.9%

   \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]
8. Add Preprocessing

Alternative 9: 98.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.6%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. diff-cos N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 80.8%

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

4. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. diff-cos N/A

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

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

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 98.9%

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

Alternative 10: 98.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right) \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (* (- (* (* eps eps) 0.041666666666666664) 0.5) eps)
   (*
    (fma
     (-
      (* (fma -0.0001984126984126984 (* x x) 0.008333333333333333) (* x x))
      0.16666666666666666)
     (* x x)
     1.0)
    x))
  eps))

C

double code(double x, double eps) {
	return (((((eps * eps) * 0.041666666666666664) - 0.5) * eps) - (fma(((fma(-0.0001984126984126984, (x * x), 0.008333333333333333) * (x * x)) - 0.16666666666666666), (x * x), 1.0) * x)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(Float64(Float64(eps * eps) * 0.041666666666666664) - 0.5) * eps) - Float64(fma(Float64(Float64(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333) * Float64(x * x)) - 0.16666666666666666), Float64(x * x), 1.0) * x)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(eps * eps), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * eps), $MachinePrecision] - N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 52.6%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 98.9

      \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

7. Applied rewrites 98.9%

   \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

   \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{24} - \frac{1}{2}\right) \cdot \varepsilon - x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{24} - \frac{1}{2}\right) \cdot \varepsilon - \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot x\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{24} - \frac{1}{2}\right) \cdot \varepsilon - \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot x\right) \cdot \varepsilon \]

10. Applied rewrites 98.4%

    \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.041666666666666664 - 0.5\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right) \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]
11. Add Preprocessing

Alternative 11: 98.0% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.6%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. diff-cos N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 80.8%

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

4. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. diff-cos N/A

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

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

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 98.0

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

9. Applied rewrites 98.0%

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

Alternative 12: 97.8% accurate, 14.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.6%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. diff-cos N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 80.8%

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

4. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. diff-cos N/A

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

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

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-neg.f64 97.8

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

9. Applied rewrites 97.8%

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

Alternative 13: 78.9% accurate, 25.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.6%

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

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right)} \]
3. 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 79.7

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

4. Applied rewrites 79.7%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 78.9%

   \[\leadsto \left(-\varepsilon\right) \cdot x \]
8. Add Preprocessing

Alternative 14: 51.4% accurate, 51.8× speedup

Math

\[\begin{array}{l} \\ 1 - 1 \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.6%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\cos \varepsilon - 1} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \cos \varepsilon - \color{blue}{1} \]

   2. lower-cos.f64 51.5

      \[\leadsto \cos \varepsilon - 1 \]

4. Applied rewrites 51.5%

   \[\leadsto \color{blue}{\cos \varepsilon - 1} \]

5. Taylor expanded in eps around 0

   \[\leadsto 1 - 1 \]
6. Step-by-step derivation

7. Applied rewrites 51.4%

   \[\leadsto 1 - 1 \]
8. 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, 0.5× 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 2025102 
(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 default (* -2 (sin (+ x (/ eps 2))) (sin (/ eps 2))))

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

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