2sin (example 3.3)

Percentage Accurate: 62.5% → 99.9%

Time: 6.6s

Alternatives: 16

Speedup: 207.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. diff-sin N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 62.5%

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

4. Taylor expanded in x around inf

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

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

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

   2. metadata-eval N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. metadata-eval N/A

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

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

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

   7. count-2-rev N/A

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

   8. associate-+l+ N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. associate-+l+ N/A

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

   12. count-2-rev N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.9

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

9. Applied rewrites 99.9%

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

Alternative 2: 99.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. diff-sin N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 62.5%

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

4. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.7

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

9. Applied rewrites 99.7%

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

10. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

12. Applied rewrites 99.7%

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

Alternative 3: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. diff-sin N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 62.5%

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

4. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.7

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

9. Applied rewrites 99.7%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-fma.f64 N/A

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

    4. lift--.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. +-commutative N/A

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

    8. pow2 N/A

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

    9. pow2 N/A

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

    10. *-commutative N/A

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

    11. *-commutative N/A

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

    12. distribute-rgt-in N/A

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

    13. *-commutative N/A

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

    14. lower-fma.f64 N/A

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

    15. lower-*.f64 N/A

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

11. Applied rewrites 99.7%

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

Alternative 4: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. diff-sin N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 62.5%

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

4. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.7

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

9. Applied rewrites 99.7%

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

Alternative 5: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. diff-sin N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 62.5%

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

4. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.7

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

9. Applied rewrites 99.7%

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

10. Taylor expanded in eps around 0

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

12. Applied rewrites 99.6%

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

Alternative 6: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. diff-sin N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 62.5%

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

4. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.7

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

9. Applied rewrites 99.7%

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

10. Taylor expanded in eps around 0

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

12. Applied rewrites 99.4%

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

Alternative 7: 99.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lower-cos.f64 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.0%

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

Alternative 8: 99.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 99.0

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

4. Applied rewrites 99.0%

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

Alternative 9: 98.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(-0.5, x, -0.16666666666666666 \cdot \varepsilon\right), \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (fma
   (fma -0.5 x (* -0.16666666666666666 eps))
   eps
   (fma
    (-
     (* (fma -0.001388888888888889 (* x x) 0.041666666666666664) (* x x))
     0.5)
    (* x x)
    1.0))
  eps))

C

double code(double x, double eps) {
	return fma(fma(-0.5, x, (-0.16666666666666666 * eps)), eps, fma(((fma(-0.001388888888888889, (x * x), 0.041666666666666664) * (x * x)) - 0.5), (x * x), 1.0)) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(fma(-0.5, x, Float64(-0.16666666666666666 * eps)), eps, fma(Float64(Float64(fma(-0.001388888888888889, Float64(x * x), 0.041666666666666664) * Float64(x * x)) - 0.5), Float64(x * x), 1.0)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(-0.5 * x + N[(-0.16666666666666666 * eps), $MachinePrecision]), $MachinePrecision] * eps + N[(N[(N[(N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lower-cos.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 99.0

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

7. Applied rewrites 99.0%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, x, \frac{-1}{6} \cdot \varepsilon\right), \varepsilon, 1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, x, \frac{-1}{6} \cdot \varepsilon\right), \varepsilon, {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right) \cdot \varepsilon \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, x, \frac{-1}{6} \cdot \varepsilon\right), \varepsilon, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2} + 1\right) \cdot \varepsilon \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, x, \frac{-1}{6} \cdot \varepsilon\right), \varepsilon, \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

   4. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, x, \frac{-1}{6} \cdot \varepsilon\right), \varepsilon, \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, x, \frac{-1}{6} \cdot \varepsilon\right), \varepsilon, \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, x, \frac{-1}{6} \cdot \varepsilon\right), \varepsilon, \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

   7. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, x, \frac{-1}{6} \cdot \varepsilon\right), \varepsilon, \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, x, \frac{-1}{6} \cdot \varepsilon\right), \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {x}^{2}, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

   9. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, x, \frac{-1}{6} \cdot \varepsilon\right), \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, x, \frac{-1}{6} \cdot \varepsilon\right), \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, x, \frac{-1}{6} \cdot \varepsilon\right), \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, x, \frac{-1}{6} \cdot \varepsilon\right), \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{2}, x, \frac{-1}{6} \cdot \varepsilon\right), \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right)\right) \cdot \varepsilon \]

   14. lower-*.f64 98.6

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.5, x, -0.16666666666666666 \cdot \varepsilon\right), \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) \cdot \varepsilon \]

10. Applied rewrites 98.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.5, x, -0.16666666666666666 \cdot \varepsilon\right), \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) \cdot \varepsilon \]
11. Add Preprocessing

Alternative 10: 98.5% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lower-cos.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 99.0

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

7. Applied rewrites 99.0%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 98.5

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

10. Applied rewrites 98.5%

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

Alternative 11: 98.4% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lower-cos.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

7. Applied rewrites 98.4%

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

9. Applied rewrites 98.4%

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

Alternative 12: 98.4% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lower-cos.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in eps around 0

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

   1. distribute-lft-out N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 98.4

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

10. Applied rewrites 98.4%

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

Alternative 13: 98.4% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lower-cos.f64 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lower-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 98.4

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

7. Applied rewrites 98.4%

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

Alternative 14: 98.4% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lower-cos.f64 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lower-*.f64 N/A

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

   6. lift-+.f64 98.4

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

7. Applied rewrites 98.4%

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

Alternative 15: 98.3% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lower-cos.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 98.3

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

10. Applied rewrites 98.3%

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

Alternative 16: 97.8% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lower-cos.f64 99.4

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 97.8%

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

Reproduce

herbie shell --seed 2025101 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))
  (- (sin (+ x eps)) (sin x)))