2cos (problem 3.3.5)

Percentage Accurate: 52.2% → 98.8%

Time: 9.3s

Alternatives: 10

Speedup: 14.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (- (* (fma (fma 0.25 x (* 0.16666666666666666 eps)) x -0.5) eps) (sin x))
  eps))

C

double code(double x, double eps) {
	return ((fma(fma(0.25, x, (0.16666666666666666 * eps)), x, -0.5) * eps) - sin(x)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(fma(fma(0.25, x, Float64(0.16666666666666666 * eps)), x, -0.5) * eps) - sin(x)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(0.25 * x + N[(0.16666666666666666 * eps), $MachinePrecision]), $MachinePrecision] * x + -0.5), $MachinePrecision] * eps), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 52.2%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

   \[\leadsto \left(\left(x \cdot \left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x\right) - \frac{1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. sub-flip N/A

      \[\leadsto \left(\left(x \cdot \left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   2. *-commutative N/A

      \[\leadsto \left(\left(\left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x\right) \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x, x, \frac{-1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   5. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{4} \cdot x + \frac{1}{6} \cdot \varepsilon, x, \frac{-1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   7. lower-*.f64 98.8

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

7. Applied rewrites 98.8%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]
8. Add Preprocessing

Alternative 2: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.027777777777777776, \varepsilon \cdot x, 0.25\right), x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (*
    (fma
     (fma
      (fma -0.027777777777777776 (* eps x) 0.25)
      x
      (* 0.16666666666666666 eps))
     x
     -0.5)
    eps)
   (*
    (fma (fma 0.008333333333333333 (* x x) -0.16666666666666666) (* x x) 1.0)
    x))
  eps))

C

double code(double x, double eps) {
	return ((fma(fma(fma(-0.027777777777777776, (eps * x), 0.25), x, (0.16666666666666666 * eps)), x, -0.5) * eps) - (fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * x), 1.0) * x)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(fma(fma(fma(-0.027777777777777776, Float64(eps * x), 0.25), x, Float64(0.16666666666666666 * eps)), x, -0.5) * eps) - Float64(fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * x)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(-0.027777777777777776 * N[(eps * x), $MachinePrecision] + 0.25), $MachinePrecision] * x + N[(0.16666666666666666 * eps), $MachinePrecision]), $MachinePrecision] * x + -0.5), $MachinePrecision] * eps), $MachinePrecision] - N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 52.2%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

   \[\leadsto \left(\left(x \cdot \left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x\right) - \frac{1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. sub-flip N/A

      \[\leadsto \left(\left(x \cdot \left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   2. *-commutative N/A

      \[\leadsto \left(\left(\left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x\right) \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x, x, \frac{-1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   5. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{4} \cdot x + \frac{1}{6} \cdot \varepsilon, x, \frac{-1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   7. lower-*.f64 98.8

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

7. Applied rewrites 98.8%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x\right) \cdot \varepsilon \]

   3. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right) \cdot x\right) \cdot \varepsilon \]

   4. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{2} + 1\right) \cdot x\right) \cdot \varepsilon \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   6. sub-flip N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   7. metadata-eval N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} + \frac{-1}{6}, {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, {x}^{2}, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   9. pow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   11. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

   12. lift-*.f64 98.4

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

10. Applied rewrites 98.4%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

11. Taylor expanded in x around 0

    \[\leadsto \left(\left(x \cdot \left(\frac{1}{6} \cdot \varepsilon + x \cdot \left(\frac{1}{4} + \frac{-1}{36} \cdot \left(\varepsilon \cdot x\right)\right)\right) - \frac{1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]
12. Step-by-step derivation

    1. sub-flip N/A

       \[\leadsto \left(\left(x \cdot \left(\frac{1}{6} \cdot \varepsilon + x \cdot \left(\frac{1}{4} + \frac{-1}{36} \cdot \left(\varepsilon \cdot x\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

    2. *-commutative N/A

       \[\leadsto \left(\left(\left(\frac{1}{6} \cdot \varepsilon + x \cdot \left(\frac{1}{4} + \frac{-1}{36} \cdot \left(\varepsilon \cdot x\right)\right)\right) \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

    3. metadata-eval N/A

       \[\leadsto \left(\left(\left(\frac{1}{6} \cdot \varepsilon + x \cdot \left(\frac{1}{4} + \frac{-1}{36} \cdot \left(\varepsilon \cdot x\right)\right)\right) \cdot x + \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

    4. lower-fma.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{1}{6} \cdot \varepsilon + x \cdot \left(\frac{1}{4} + \frac{-1}{36} \cdot \left(\varepsilon \cdot x\right)\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

    5. +-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(\frac{1}{4} + \frac{-1}{36} \cdot \left(\varepsilon \cdot x\right)\right) + \frac{1}{6} \cdot \varepsilon, x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

    6. *-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{4} + \frac{-1}{36} \cdot \left(\varepsilon \cdot x\right)\right) \cdot x + \frac{1}{6} \cdot \varepsilon, x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

    7. lower-fma.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4} + \frac{-1}{36} \cdot \left(\varepsilon \cdot x\right), x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

    8. +-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{36} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

    9. lower-fma.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{36}, \varepsilon \cdot x, \frac{1}{4}\right), x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

    10. lower-*.f64 N/A

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{36}, \varepsilon \cdot x, \frac{1}{4}\right), x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

    11. lift-*.f64 98.4

        \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.027777777777777776, \varepsilon \cdot x, 0.25\right), x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

13. Applied rewrites 98.4%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.027777777777777776, \varepsilon \cdot x, 0.25\right), x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]
14. Add Preprocessing

Alternative 3: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (* (fma (fma 0.25 x (* 0.16666666666666666 eps)) x -0.5) eps)
   (*
    (fma (fma 0.008333333333333333 (* x x) -0.16666666666666666) (* x x) 1.0)
    x))
  eps))

C

double code(double x, double eps) {
	return ((fma(fma(0.25, x, (0.16666666666666666 * eps)), x, -0.5) * eps) - (fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * x), 1.0) * x)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(fma(fma(0.25, x, Float64(0.16666666666666666 * eps)), x, -0.5) * eps) - Float64(fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * x)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(0.25 * x + N[(0.16666666666666666 * eps), $MachinePrecision]), $MachinePrecision] * x + -0.5), $MachinePrecision] * eps), $MachinePrecision] - N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 52.2%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

   \[\leadsto \left(\left(x \cdot \left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x\right) - \frac{1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. sub-flip N/A

      \[\leadsto \left(\left(x \cdot \left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   2. *-commutative N/A

      \[\leadsto \left(\left(\left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x\right) \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x, x, \frac{-1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   5. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{4} \cdot x + \frac{1}{6} \cdot \varepsilon, x, \frac{-1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   7. lower-*.f64 98.8

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

7. Applied rewrites 98.8%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x\right) \cdot \varepsilon \]

   3. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right) \cdot x\right) \cdot \varepsilon \]

   4. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{2} + 1\right) \cdot x\right) \cdot \varepsilon \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   6. sub-flip N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   7. metadata-eval N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} + \frac{-1}{6}, {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, {x}^{2}, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   9. pow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   11. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

   12. lift-*.f64 98.4

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

10. Applied rewrites 98.4%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]
11. Add Preprocessing

Alternative 4: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(0.25 \cdot x, x, -0.5\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (* (fma (* 0.25 x) x -0.5) eps)
   (*
    (fma (fma 0.008333333333333333 (* x x) -0.16666666666666666) (* x x) 1.0)
    x))
  eps))

C

double code(double x, double eps) {
	return ((fma((0.25 * x), x, -0.5) * eps) - (fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * x), 1.0) * x)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(fma(Float64(0.25 * x), x, -0.5) * eps) - Float64(fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * x)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(0.25 * x), $MachinePrecision] * x + -0.5), $MachinePrecision] * eps), $MachinePrecision] - N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 52.2%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

   \[\leadsto \left(\left(x \cdot \left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x\right) - \frac{1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. sub-flip N/A

      \[\leadsto \left(\left(x \cdot \left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   2. *-commutative N/A

      \[\leadsto \left(\left(\left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x\right) \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x, x, \frac{-1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   5. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{4} \cdot x + \frac{1}{6} \cdot \varepsilon, x, \frac{-1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   7. lower-*.f64 98.8

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

7. Applied rewrites 98.8%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x\right) \cdot \varepsilon \]

   3. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right) \cdot x\right) \cdot \varepsilon \]

   4. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{2} + 1\right) \cdot x\right) \cdot \varepsilon \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   6. sub-flip N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   7. metadata-eval N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} + \frac{-1}{6}, {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, {x}^{2}, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   9. pow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   11. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

   12. lift-*.f64 98.4

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

10. Applied rewrites 98.4%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

11. Taylor expanded in x around inf

    \[\leadsto \left(\mathsf{fma}\left(\frac{1}{4} \cdot x, x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]
12. Step-by-step derivation

    1. lower-*.f64 98.3

       \[\leadsto \left(\mathsf{fma}\left(0.25 \cdot x, x, -0.5\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

13. Applied rewrites 98.3%

    \[\leadsto \left(\mathsf{fma}\left(0.25 \cdot x, x, -0.5\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]
14. Add Preprocessing

Alternative 5: 98.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (* (fma (fma 0.25 x (* 0.16666666666666666 eps)) x -0.5) eps)
   (* (fma (* x x) -0.16666666666666666 1.0) x))
  eps))

C

double code(double x, double eps) {
	return ((fma(fma(0.25, x, (0.16666666666666666 * eps)), x, -0.5) * eps) - (fma((x * x), -0.16666666666666666, 1.0) * x)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(fma(fma(0.25, x, Float64(0.16666666666666666 * eps)), x, -0.5) * eps) - Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(0.25 * x + N[(0.16666666666666666 * eps), $MachinePrecision]), $MachinePrecision] * x + -0.5), $MachinePrecision] * eps), $MachinePrecision] - N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 52.2%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

   \[\leadsto \left(\left(x \cdot \left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x\right) - \frac{1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. sub-flip N/A

      \[\leadsto \left(\left(x \cdot \left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   2. *-commutative N/A

      \[\leadsto \left(\left(\left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x\right) \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x, x, \frac{-1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   5. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{4} \cdot x + \frac{1}{6} \cdot \varepsilon, x, \frac{-1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   7. lower-*.f64 98.8

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

7. Applied rewrites 98.8%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x\right) \cdot \varepsilon \]

   3. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right) \cdot x\right) \cdot \varepsilon \]

   4. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{2} + 1\right) \cdot x\right) \cdot \varepsilon \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   6. sub-flip N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   7. metadata-eval N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} + \frac{-1}{6}, {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, {x}^{2}, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   9. pow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   11. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

   12. lift-*.f64 98.4

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

10. Applied rewrites 98.4%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

11. Taylor expanded in x around 0

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot x\right) \cdot \varepsilon \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left(1 + \frac{-1}{6} \cdot {x}^{2}\right) \cdot x\right) \cdot \varepsilon \]

    3. +-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left(\frac{-1}{6} \cdot {x}^{2} + 1\right) \cdot x\right) \cdot \varepsilon \]

    4. *-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left({x}^{2} \cdot \frac{-1}{6} + 1\right) \cdot x\right) \cdot \varepsilon \]

    5. lower-fma.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left({x}^{2}, \frac{-1}{6}, 1\right) \cdot x\right) \cdot \varepsilon \]

    6. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right) \cdot x\right) \cdot \varepsilon \]

    7. lift-*.f64 98.2

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\right) \cdot \varepsilon \]

13. Applied rewrites 98.2%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\right) \cdot \varepsilon \]
14. Add Preprocessing

Alternative 6: 98.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

   \[\leadsto \left(\left(x \cdot \left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x\right) - \frac{1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]
6. Step-by-step derivation

   1. sub-flip N/A

      \[\leadsto \left(\left(x \cdot \left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   2. *-commutative N/A

      \[\leadsto \left(\left(\left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x\right) \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{6} \cdot \varepsilon + \frac{1}{4} \cdot x, x, \frac{-1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   5. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{4} \cdot x + \frac{1}{6} \cdot \varepsilon, x, \frac{-1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

   7. lower-*.f64 98.8

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

7. Applied rewrites 98.8%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right) \cdot x\right) \cdot \varepsilon \]

   3. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left({x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right) \cdot x\right) \cdot \varepsilon \]

   4. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \left(\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right) \cdot {x}^{2} + 1\right) \cdot x\right) \cdot \varepsilon \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}, {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   6. sub-flip N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   7. metadata-eval N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\frac{1}{120} \cdot {x}^{2} + \frac{-1}{6}, {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, {x}^{2}, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   9. pow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), {x}^{2}, 1\right) \cdot x\right) \cdot \varepsilon \]

   11. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, x, \frac{1}{6} \cdot \varepsilon\right), x, \frac{-1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, x \cdot x, \frac{-1}{6}\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

   12. lift-*.f64 98.4

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

10. Applied rewrites 98.4%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 0.16666666666666666 \cdot \varepsilon\right), x, -0.5\right) \cdot \varepsilon - \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]

11. Taylor expanded in x around 0

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

13. Applied rewrites 98.2%

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

Alternative 7: 97.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. sub-flip N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 97.9

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

7. Applied rewrites 97.9%

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

Alternative 8: 97.9% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. sub-flip N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 97.9

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

7. Applied rewrites 97.9%

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

8. Taylor expanded in eps around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 97.9

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

10. Applied rewrites 97.9%

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

Alternative 9: 97.7% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, eps):
	return ((-0.5 * eps) - x) * eps

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

2. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. sub-flip N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 97.9

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

7. Applied rewrites 97.9%

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

8. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 97.7

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

10. Applied rewrites 97.7%

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

    1. lift-fma.f64 N/A

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

    2. lift-neg.f64 N/A

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

    3. sub-flip-reverse N/A

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

    4. lower--.f64 N/A

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

    5. lift-*.f64 97.7

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

12. Applied rewrites 97.7%

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

Alternative 10: 78.7% accurate, 14.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

2. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-sin.f64 79.6

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

4. Applied rewrites 79.6%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 78.7%

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

Developer Target 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 98.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2025134 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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

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