2sin (example 3.3)

Percentage Accurate: 62.4% → 99.9%

Time: 8.6s

Alternatives: 15

Speedup: 207.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 62.8%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. diff-sin N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 62.8%

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

5. Taylor expanded in x around inf

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

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

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

   2. metadata-eval N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. metadata-eval N/A

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

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

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

   7. count-2-rev N/A

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

   8. associate-+l+ N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. associate-+l+ N/A

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

   12. count-2-rev N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-*.f64 99.9

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

7. Applied rewrites 99.9%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

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

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

   5. lower-sin.f64 N/A

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

   6. +-commutative N/A

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

   7. count-2-rev N/A

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

   8. associate-+l+ N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. associate-+l+ N/A

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

   12. count-2-rev N/A

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

   13. +-commutative N/A

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

   14. lift-fma.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-PI.f64 100.0

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

9. Applied rewrites 100.0%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. distribute-lft-out N/A

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

    3. lower-fma.f64 N/A

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

    4. lower-+.f64 N/A

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

    5. lift-PI.f64 100.0

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

12. Applied rewrites 100.0%

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

Alternative 2: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. diff-sin N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 62.8%

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

5. Taylor expanded in x around inf

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

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

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

   2. metadata-eval N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. metadata-eval N/A

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

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

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

   7. count-2-rev N/A

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

   8. associate-+l+ N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. associate-+l+ N/A

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

   12. count-2-rev N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-*.f64 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.9

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

10. Applied rewrites 99.9%

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

Alternative 3: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. diff-sin N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 62.8%

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

5. Taylor expanded in x around inf

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

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

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

   2. metadata-eval N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. metadata-eval N/A

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

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

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

   7. count-2-rev N/A

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

   8. associate-+l+ N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. associate-+l+ N/A

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

   12. count-2-rev N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-*.f64 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 99.8%

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

Alternative 4: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. diff-sin N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 62.8%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.7

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

10. Applied rewrites 99.7%

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

Alternative 5: 99.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. diff-sin N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 62.8%

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

5. Taylor expanded in x around inf

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

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

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

   2. metadata-eval N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. metadata-eval N/A

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

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

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

   7. count-2-rev N/A

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

   8. associate-+l+ N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. associate-+l+ N/A

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

   12. count-2-rev N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-*.f64 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 99.6

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

10. Applied rewrites 99.6%

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

Alternative 6: 99.4% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (* (* (sin (/ (+ (fma 2.0 x eps) (PI)) 2.0)) (* 0.5 eps)) 2.0))

Derivation

1. Initial program 62.8%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. diff-sin N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 62.8%

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

5. Taylor expanded in eps around 0

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

   1. lower-*.f64 99.4

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(0.5 \cdot \color{blue}{\varepsilon}\right)\right) \cdot 2 \]

7. Applied rewrites 99.4%

   \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \color{blue}{\left(0.5 \cdot \varepsilon\right)}\right) \cdot 2 \]
8. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-+.f64 N/A

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

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

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

   6. lower-sin.f64 N/A

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

   7. div-add-rev N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. associate-+l+ N/A

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

   11. count-2-rev N/A

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

   12. +-commutative N/A

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

   13. lift-fma.f64 N/A

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

   14. lower-PI.f64 99.4

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

9. Applied rewrites 99.4%

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

Alternative 7: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. diff-sin N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 62.8%

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

5. Taylor expanded in eps around 0

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

   1. lower-*.f64 99.4

      \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \left(0.5 \cdot \color{blue}{\varepsilon}\right)\right) \cdot 2 \]

7. Applied rewrites 99.4%

   \[\leadsto \left(\cos \left(\frac{\left(\varepsilon + x\right) + x}{2}\right) \cdot \color{blue}{\left(0.5 \cdot \varepsilon\right)}\right) \cdot 2 \]

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.4

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

10. Applied rewrites 99.4%

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

Alternative 8: 99.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lower-cos.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.1%

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

Alternative 9: 99.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 98.9

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

5. Applied rewrites 98.9%

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

Alternative 10: 98.5% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lower-cos.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 99.1

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

8. Applied rewrites 99.1%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. lower--.f64 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 N/A

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

    7. pow2 N/A

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

    8. lift-*.f64 N/A

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

    9. pow2 N/A

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

    10. lift-*.f64 98.7

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

11. Applied rewrites 98.7%

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

Alternative 11: 98.3% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 98.9

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. +-commutative N/A

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

    5. *-commutative N/A

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

    6. lower-fma.f64 N/A

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

    7. *-commutative N/A

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

    8. pow2 N/A

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

    9. lift-*.f64 N/A

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

    10. lift-*.f64 N/A

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

    11. lift-*.f64 N/A

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

    12. pow2 N/A

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

    13. lift-*.f64 98.5

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

11. Applied rewrites 98.5%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot \varepsilon, 0.041666666666666664, -0.5 \cdot \varepsilon\right), \color{blue}{x \cdot x}, \varepsilon\right) \]
12. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-fma.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-fma.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. associate-*r* N/A

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

    8. lower-fma.f64 N/A

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

    9. lower-*.f64 N/A

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

    10. lift-*.f64 N/A

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

    11. lift-*.f64 N/A

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

    12. lift-fma.f64 N/A

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

    13. lift-*.f64 98.5

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

13. Applied rewrites 98.5%

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

Alternative 12: 98.3% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 98.9

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. +-commutative N/A

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

    5. *-commutative N/A

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

    6. lower-fma.f64 N/A

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

    7. *-commutative N/A

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

    8. pow2 N/A

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

    9. lift-*.f64 N/A

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

    10. lift-*.f64 N/A

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

    11. lift-*.f64 N/A

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

    12. pow2 N/A

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

    13. lift-*.f64 98.5

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

11. Applied rewrites 98.5%

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

12. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower--.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

    6. pow2 N/A

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

    7. lift-*.f64 98.5

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

14. Applied rewrites 98.5%

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

Alternative 13: 98.3% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lower-cos.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lower-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 98.3

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

8. Applied rewrites 98.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. distribute-lft-out N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 98.3

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

10. Applied rewrites 98.3%

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

Alternative 14: 98.2% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 98.9

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in x around 0

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

    1. lift-*.f64 98.2

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

11. Applied rewrites 98.2%

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

Alternative 15: 97.8% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 62.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lower-cos.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.9%

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

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025053 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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