2cos (problem 3.3.5)

Percentage Accurate: 52.1% → 99.4%

Time: 13.6s

Alternatives: 8

Speedup: 25.9×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

Alternative 2: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in eps around 0

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

   1. remove-double-neg N/A

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

   2. distribute-lft-neg-out N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.2%

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

12. Applied rewrites 99.5%

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

Alternative 3: 98.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return -eps * (sin(x) + (0.5 * 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 * (sin(x) + (0.5d0 * eps))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in eps around 0

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

   1. remove-double-neg N/A

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

   2. distribute-lft-neg-out N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.2%

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

12. Applied rewrites 99.3%

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

Alternative 4: 98.1% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in eps around 0

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

   1. remove-double-neg N/A

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

   2. distribute-lft-neg-out N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.3%

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

Alternative 5: 97.9% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in eps around 0

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

   1. remove-double-neg N/A

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

   2. distribute-lft-neg-out N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.1%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, \varepsilon, -0.16666666666666666 \cdot x\right), x, 1\right), x, 0.5 \cdot \varepsilon\right) \cdot \left(-\color{blue}{\varepsilon}\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.1%

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

Alternative 6: 97.6% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in eps around 0

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

   1. remove-double-neg N/A

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

   2. distribute-lft-neg-out N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.1%

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

Alternative 7: 97.4% accurate, 14.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in eps around 0

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

   1. remove-double-neg N/A

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

   2. distribute-lft-neg-out N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.8%

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

Alternative 8: 78.2% accurate, 25.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-sin.f64 79.2

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

5. Applied rewrites 79.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 78.9%

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

Developer Target 1: 98.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2025010 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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