2sin (example 3.3)

Percentage Accurate: 62.4% → 99.3%

Time: 8.3s

Alternatives: 21

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: 62.4% 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: 99.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)} \]
4. 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 100.0

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

5. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. +-commutative N/A

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 100.0

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

10. Applied rewrites 100.0%

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

Alternative 2: 99.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)} \]
4. 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 100.0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 100.0

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

8. Applied rewrites 100.0%

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

Alternative 3: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)} \]
4. 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 100.0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 100.0%

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

Alternative 4: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos x, \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) \cdot \varepsilon, -0.16666666666666666, \left(\mathsf{fma}\left(9.92063492063492 \cdot 10^{-5} \cdot \left(x \cdot x\right) - 0.004166666666666667, x \cdot x, 0.08333333333333333\right) \cdot \left(x \cdot x\right) - 0.5\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (cos x)
  eps
  (*
   (*
    (fma
     (* (fma (* x x) -0.5 1.0) eps)
     -0.16666666666666666
     (*
      (-
       (*
        (fma
         (- (* 9.92063492063492e-5 (* x x)) 0.004166666666666667)
         (* x x)
         0.08333333333333333)
        (* x x))
       0.5)
      x))
    eps)
   eps)))

C

double code(double x, double eps) {
	return fma(cos(x), eps, ((fma((fma((x * x), -0.5, 1.0) * eps), -0.16666666666666666, (((fma(((9.92063492063492e-5 * (x * x)) - 0.004166666666666667), (x * x), 0.08333333333333333) * (x * x)) - 0.5) * x)) * eps) * eps));
}

Julia

function code(x, eps)
	return fma(cos(x), eps, Float64(Float64(fma(Float64(fma(Float64(x * x), -0.5, 1.0) * eps), -0.16666666666666666, Float64(Float64(Float64(fma(Float64(Float64(9.92063492063492e-5 * Float64(x * x)) - 0.004166666666666667), Float64(x * x), 0.08333333333333333) * Float64(x * x)) - 0.5) * x)) * eps) * eps))
end

Wolfram

code[x_, eps_] := N[(N[Cos[x], $MachinePrecision] * eps + N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * eps), $MachinePrecision] * -0.16666666666666666 + N[(N[(N[(N[(N[(N[(9.92063492063492e-5 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.004166666666666667), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)} \]
4. 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 100.0

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

5. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. +-commutative N/A

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 100.0

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

10. Applied rewrites 100.0%

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

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(\cos x, \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) \cdot \varepsilon, \frac{-1}{6}, x \cdot \left({x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{10080} \cdot {x}^{2} - \frac{1}{240}\right)\right) - \frac{1}{2}\right)\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\cos x, \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) \cdot \varepsilon, \frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{10080} \cdot {x}^{2} - \frac{1}{240}\right)\right) - \frac{1}{2}\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \]

    2. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\cos x, \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) \cdot \varepsilon, \frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{10080} \cdot {x}^{2} - \frac{1}{240}\right)\right) - \frac{1}{2}\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \]

13. Applied rewrites 99.9%

    \[\leadsto \mathsf{fma}\left(\cos x, \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) \cdot \varepsilon, -0.16666666666666666, \left(\mathsf{fma}\left(9.92063492063492 \cdot 10^{-5} \cdot \left(x \cdot x\right) - 0.004166666666666667, x \cdot x, 0.08333333333333333\right) \cdot \left(x \cdot x\right) - 0.5\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \]
14. Add Preprocessing

Alternative 5: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos x, \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) \cdot \varepsilon, -0.16666666666666666, \left(\mathsf{fma}\left(-0.004166666666666667, x \cdot x, 0.08333333333333333\right) \cdot \left(x \cdot x\right) - 0.5\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (cos x)
  eps
  (*
   (*
    (fma
     (* (fma (* x x) -0.5 1.0) eps)
     -0.16666666666666666
     (*
      (-
       (* (fma -0.004166666666666667 (* x x) 0.08333333333333333) (* x x))
       0.5)
      x))
    eps)
   eps)))

C

double code(double x, double eps) {
	return fma(cos(x), eps, ((fma((fma((x * x), -0.5, 1.0) * eps), -0.16666666666666666, (((fma(-0.004166666666666667, (x * x), 0.08333333333333333) * (x * x)) - 0.5) * x)) * eps) * eps));
}

Julia

function code(x, eps)
	return fma(cos(x), eps, Float64(Float64(fma(Float64(fma(Float64(x * x), -0.5, 1.0) * eps), -0.16666666666666666, Float64(Float64(Float64(fma(-0.004166666666666667, Float64(x * x), 0.08333333333333333) * Float64(x * x)) - 0.5) * x)) * eps) * eps))
end

Wolfram

code[x_, eps_] := N[(N[Cos[x], $MachinePrecision] * eps + N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * eps), $MachinePrecision] * -0.16666666666666666 + N[(N[(N[(N[(-0.004166666666666667 * N[(x * x), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)} \]
4. 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 100.0

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

5. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. +-commutative N/A

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 100.0

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

10. Applied rewrites 100.0%

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

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(\cos x, \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) \cdot \varepsilon, \frac{-1}{6}, x \cdot \left({x}^{2} \cdot \left(\frac{1}{12} + \frac{-1}{240} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\cos x, \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) \cdot \varepsilon, \frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{1}{12} + \frac{-1}{240} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \]

    2. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\cos x, \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) \cdot \varepsilon, \frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{1}{12} + \frac{-1}{240} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \]

    3. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\cos x, \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) \cdot \varepsilon, \frac{-1}{6}, \left({x}^{2} \cdot \left(\frac{1}{12} + \frac{-1}{240} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\cos x, \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) \cdot \varepsilon, \frac{-1}{6}, \left(\left(\frac{1}{12} + \frac{-1}{240} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \]

    5. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\cos x, \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) \cdot \varepsilon, \frac{-1}{6}, \left(\left(\frac{1}{12} + \frac{-1}{240} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \]

    6. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\cos x, \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) \cdot \varepsilon, \frac{-1}{6}, \left(\left(\frac{-1}{240} \cdot {x}^{2} + \frac{1}{12}\right) \cdot {x}^{2} - \frac{1}{2}\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \]

    7. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\cos x, \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) \cdot \varepsilon, \frac{-1}{6}, \left(\mathsf{fma}\left(\frac{-1}{240}, {x}^{2}, \frac{1}{12}\right) \cdot {x}^{2} - \frac{1}{2}\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \]

    8. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\cos x, \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) \cdot \varepsilon, \frac{-1}{6}, \left(\mathsf{fma}\left(\frac{-1}{240}, x \cdot x, \frac{1}{12}\right) \cdot {x}^{2} - \frac{1}{2}\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \]

    9. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\cos x, \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) \cdot \varepsilon, \frac{-1}{6}, \left(\mathsf{fma}\left(\frac{-1}{240}, x \cdot x, \frac{1}{12}\right) \cdot {x}^{2} - \frac{1}{2}\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \]

    10. pow2 N/A

        \[\leadsto \mathsf{fma}\left(\cos x, \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{2}, 1\right) \cdot \varepsilon, \frac{-1}{6}, \left(\mathsf{fma}\left(\frac{-1}{240}, x \cdot x, \frac{1}{12}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \]

    11. lift-*.f64 99.9

        \[\leadsto \mathsf{fma}\left(\cos x, \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) \cdot \varepsilon, -0.16666666666666666, \left(\mathsf{fma}\left(-0.004166666666666667, x \cdot x, 0.08333333333333333\right) \cdot \left(x \cdot x\right) - 0.5\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \]

13. Applied rewrites 99.9%

    \[\leadsto \mathsf{fma}\left(\cos x, \varepsilon, \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.5, 1\right) \cdot \varepsilon, -0.16666666666666666, \left(\mathsf{fma}\left(-0.004166666666666667, x \cdot x, 0.08333333333333333\right) \cdot \left(x \cdot x\right) - 0.5\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \]
14. Add Preprocessing

Alternative 6: 98.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   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 99.9

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\left(x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]
7. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

8. Applied rewrites 99.8%

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

Alternative 7: 98.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   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 99.9

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]

   3. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(\left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{2} + 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]

   6. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]

   11. lower-*.f64 99.8

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, -0.5, \cos x\right) \cdot \varepsilon \]

8. Applied rewrites 99.8%

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

Alternative 8: 98.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   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 99.9

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 9: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)} \]
4. 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 100.0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot x + \frac{-1}{6} \cdot \varepsilon, \varepsilon, \cos x\right) \cdot \varepsilon \]
7. 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 99.6

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

8. Applied rewrites 99.6%

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

Alternative 10: 98.9% 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 65.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   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 99.9

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.6%

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

Alternative 11: 98.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, -0.5, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (fma
   (*
    (*
     (fma (- (* 0.008333333333333333 (* x x)) 0.16666666666666666) (* x x) 1.0)
     x)
    eps)
   -0.5
   (fma
    (-
     (* (fma (* x x) -0.001388888888888889 0.041666666666666664) (* x x))
     0.5)
    (* x x)
    1.0))
  eps))

C

double code(double x, double eps) {
	return fma(((fma(((0.008333333333333333 * (x * x)) - 0.16666666666666666), (x * x), 1.0) * x) * eps), -0.5, fma(((fma((x * x), -0.001388888888888889, 0.041666666666666664) * (x * x)) - 0.5), (x * x), 1.0)) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(Float64(Float64(fma(Float64(Float64(0.008333333333333333 * Float64(x * x)) - 0.16666666666666666), Float64(x * x), 1.0) * x) * eps), -0.5, fma(Float64(Float64(fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664) * Float64(x * x)) - 0.5), Float64(x * x), 1.0)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision] * -0.5 + N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 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 65.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   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 99.9

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]

   3. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(\left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(\left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{2} + 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]

   6. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \cos x\right) \cdot \varepsilon \]

   11. lower-*.f64 99.8

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, -0.5, \cos x\right) \cdot \varepsilon \]

8. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, -0.5, \cos x\right) \cdot \varepsilon \]

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, 1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right) \cdot \varepsilon \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right) \cdot \varepsilon \]

    2. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2} + 1\right) \cdot \varepsilon \]

    3. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

    4. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

    5. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

    6. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

    7. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

    8. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \mathsf{fma}\left(\left({x}^{2} \cdot \frac{-1}{720} + \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

    9. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

    10. pow2 N/A

        \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

    11. lift-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

    12. pow2 N/A

        \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

    13. lift-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right)\right) \cdot \varepsilon \]

    14. pow2 N/A

        \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{1}{6}, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, \frac{-1}{2}, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right)\right) \cdot \varepsilon \]

    15. lift-*.f64 99.5

        \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, -0.5, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) \cdot \varepsilon \]

11. Applied rewrites 99.5%

    \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon, -0.5, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)\right) \cdot \varepsilon \]
12. Add Preprocessing

Alternative 12: 98.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision] * -0.5 + N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 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 65.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   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 99.9

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.8

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. lower--.f64 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 N/A

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

    7. *-commutative N/A

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

    8. pow2 N/A

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

    9. lift-*.f64 N/A

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

    10. +-commutative N/A

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

    11. lift-fma.f64 N/A

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

    12. pow2 N/A

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

    13. lift-*.f64 N/A

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

    14. pow2 N/A

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

    15. lift-*.f64 99.5

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

11. Applied rewrites 99.5%

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

Alternative 13: 98.3% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon + \color{blue}{{x}^{2} \cdot \left(\frac{-1}{2} \cdot \varepsilon + {x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

8. Applied rewrites 99.3%

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

10. Applied rewrites 99.3%

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

Alternative 14: 98.3% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon + \color{blue}{{x}^{2} \cdot \left(\frac{-1}{2} \cdot \varepsilon + {x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

8. Applied rewrites 99.3%

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

9. Taylor expanded in eps around -inf

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

    1. associate-*r* N/A

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

    2. mul-1-neg N/A

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

    3. lower-*.f64 N/A

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

    4. lower-neg.f64 N/A

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

    5. +-commutative N/A

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

    6. *-commutative N/A

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

    7. lower-fma.f64 N/A

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

    8. lower--.f64 N/A

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

    9. lower-*.f64 N/A

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

    10. pow2 N/A

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

    11. lift-*.f64 N/A

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

    12. pow2 N/A

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

    13. lift-*.f64 99.3

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

11. Applied rewrites 99.3%

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

Alternative 15: 98.3% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (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(-0.001388888888888889, (x * x), 0.041666666666666664) * (x * x)) - 0.5), (x * x), 1.0) * eps;
}

Julia

function code(x, eps)
	return Float64(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[(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] * eps), $MachinePrecision]

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.3

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

8. Applied rewrites 99.3%

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

Alternative 16: 98.2% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon + \color{blue}{{x}^{2} \cdot \left(\frac{-1}{2} \cdot \varepsilon + {x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

8. Applied rewrites 99.3%

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

9. Taylor expanded in x around 0

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

    1. lift-*.f64 99.2

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

11. Applied rewrites 99.2%

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

Alternative 17: 98.2% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 99.2

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

8. Applied rewrites 99.2%

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

Alternative 18: 98.2% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 99.2

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

8. Applied rewrites 99.2%

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

Alternative 19: 98.1% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   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 99.9

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in x around 0

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

   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 99.0

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

8. Applied rewrites 99.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. distribute-lft-out N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-+.f64 99.0

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

10. Applied rewrites 99.0%

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

Alternative 20: 98.0% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon + \color{blue}{{x}^{2} \cdot \left(\frac{-1}{2} \cdot \varepsilon + {x}^{2} \cdot \left(\frac{-1}{720} \cdot \left(\varepsilon \cdot {x}^{2}\right) + \frac{1}{24} \cdot \varepsilon\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

8. Applied rewrites 99.3%

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

9. Taylor expanded in x around 0

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

    1. lift-*.f64 99.0

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

11. Applied rewrites 99.0%

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

Alternative 21: 97.6% 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 65.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   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 99.9

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.2%

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

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025084 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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