2sin (example 3.3)

Percentage Accurate: 62.4% → 99.9%

Time: 6.5s

Alternatives: 11

Speedup: 69.8×

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \mathsf{fma}\left(\left(-\sin x\right) \cdot \varepsilon, 0.5, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.125 \cdot \cos x - \left(-0.020833333333333332 \cdot \varepsilon\right) \cdot \sin x, \cos x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (* (sin (* 0.5 (- eps 0.0))) 2.0)
  (fma
   (* (- (sin x)) eps)
   0.5
   (fma
    (* eps eps)
    (- (* -0.125 (cos x)) (* (* -0.020833333333333332 eps) (sin x)))
    (cos x)))))

C

double code(double x, double eps) {
	return (sin((0.5 * (eps - 0.0))) * 2.0) * fma((-sin(x) * eps), 0.5, fma((eps * eps), ((-0.125 * cos(x)) - ((-0.020833333333333332 * eps) * sin(x))), cos(x)));
}

Julia

function code(x, eps)
	return Float64(Float64(sin(Float64(0.5 * Float64(eps - 0.0))) * 2.0) * fma(Float64(Float64(-sin(x)) * eps), 0.5, fma(Float64(eps * eps), Float64(Float64(-0.125 * cos(x)) - Float64(Float64(-0.020833333333333332 * eps) * sin(x))), cos(x))))
end

Wolfram

code[x_, eps_] := N[(N[(N[Sin[N[(0.5 * N[(eps - 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[((-N[Sin[x], $MachinePrecision]) * eps), $MachinePrecision] * 0.5 + N[(N[(eps * eps), $MachinePrecision] * N[(N[(-0.125 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(-0.020833333333333332 * eps), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. mult-flip N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. add-flip N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 N/A

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

   20. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - \color{blue}{0}\right)\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right)} \]

4. Taylor expanded in eps around 0

   \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \left(\frac{1}{2} \cdot \sin \left(-1 \cdot x\right) + \varepsilon \cdot \left(\frac{-1}{8} \cdot \cos \left(-1 \cdot x\right) + \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right)} \]
5. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \color{blue}{\varepsilon \cdot \left(\frac{1}{2} \cdot \sin \left(-1 \cdot x\right) + \varepsilon \cdot \left(\frac{-1}{8} \cdot \cos \left(-1 \cdot x\right) + \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \color{blue}{\varepsilon} \cdot \left(\frac{1}{2} \cdot \sin \left(-1 \cdot x\right) + \varepsilon \cdot \left(\frac{-1}{8} \cdot \cos \left(-1 \cdot x\right) + \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \left(\frac{1}{2} \cdot \sin \left(-1 \cdot x\right) + \varepsilon \cdot \left(\frac{-1}{8} \cdot \cos \left(-1 \cdot x\right) + \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin \left(-1 \cdot x\right) + \varepsilon \cdot \left(\frac{-1}{8} \cdot \cos \left(-1 \cdot x\right) + \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)}\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sin \left(-1 \cdot x\right)}, \varepsilon \cdot \left(\frac{-1}{8} \cdot \cos \left(-1 \cdot x\right) + \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right) \]

   6. lower-sin.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \mathsf{fma}\left(\frac{1}{2}, \sin \left(-1 \cdot x\right), \varepsilon \cdot \left(\frac{-1}{8} \cdot \cos \left(-1 \cdot x\right) + \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \mathsf{fma}\left(\frac{1}{2}, \sin \left(-1 \cdot x\right), \varepsilon \cdot \left(\frac{-1}{8} \cdot \cos \left(-1 \cdot x\right) + \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \mathsf{fma}\left(\frac{1}{2}, \sin \left(-1 \cdot x\right), \varepsilon \cdot \left(\frac{-1}{8} \cdot \cos \left(-1 \cdot x\right) + \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \mathsf{fma}\left(\frac{1}{2}, \sin \left(-1 \cdot x\right), \varepsilon \cdot \mathsf{fma}\left(\frac{-1}{8}, \cos \left(-1 \cdot x\right), \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right) \]

   10. lower-cos.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \mathsf{fma}\left(\frac{1}{2}, \sin \left(-1 \cdot x\right), \varepsilon \cdot \mathsf{fma}\left(\frac{-1}{8}, \cos \left(-1 \cdot x\right), \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \mathsf{fma}\left(\frac{1}{2}, \sin \left(-1 \cdot x\right), \varepsilon \cdot \mathsf{fma}\left(\frac{-1}{8}, \cos \left(-1 \cdot x\right), \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right) \]

6. Applied rewrites 99.8%

   \[\leadsto \left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \mathsf{fma}\left(0.5, \sin \left(-1 \cdot x\right), \varepsilon \cdot \mathsf{fma}\left(-0.125, \cos \left(-1 \cdot x\right), -0.020833333333333332 \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right)} \]

8. Applied rewrites 99.8%

   \[\leadsto \left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \mathsf{fma}\left(\left(-\sin x\right) \cdot \varepsilon, \color{blue}{0.5}, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.125 \cdot \cos x - \left(-0.020833333333333332 \cdot \varepsilon\right) \cdot \sin x, \cos x\right)\right) \]
9. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\varepsilon \cdot \left(1 + -0.041666666666666664 \cdot {\varepsilon}^{2}\right)\right) \cdot \mathsf{fma}\left(\left(-\sin x\right) \cdot \varepsilon, 0.5, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.125 \cdot \cos x - \left(-0.020833333333333332 \cdot \varepsilon\right) \cdot \sin x, \cos x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (* eps (+ 1.0 (* -0.041666666666666664 (pow eps 2.0))))
  (fma
   (* (- (sin x)) eps)
   0.5
   (fma
    (* eps eps)
    (- (* -0.125 (cos x)) (* (* -0.020833333333333332 eps) (sin x)))
    (cos x)))))

C

double code(double x, double eps) {
	return (eps * (1.0 + (-0.041666666666666664 * pow(eps, 2.0)))) * fma((-sin(x) * eps), 0.5, fma((eps * eps), ((-0.125 * cos(x)) - ((-0.020833333333333332 * eps) * sin(x))), cos(x)));
}

Julia

function code(x, eps)
	return Float64(Float64(eps * Float64(1.0 + Float64(-0.041666666666666664 * (eps ^ 2.0)))) * fma(Float64(Float64(-sin(x)) * eps), 0.5, fma(Float64(eps * eps), Float64(Float64(-0.125 * cos(x)) - Float64(Float64(-0.020833333333333332 * eps) * sin(x))), cos(x))))
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(1.0 + N[(-0.041666666666666664 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[((-N[Sin[x], $MachinePrecision]) * eps), $MachinePrecision] * 0.5 + N[(N[(eps * eps), $MachinePrecision] * N[(N[(-0.125 * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[(-0.020833333333333332 * eps), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. mult-flip N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. add-flip N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 N/A

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

   20. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - \color{blue}{0}\right)\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right)} \]

4. Taylor expanded in eps around 0

   \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \left(\frac{1}{2} \cdot \sin \left(-1 \cdot x\right) + \varepsilon \cdot \left(\frac{-1}{8} \cdot \cos \left(-1 \cdot x\right) + \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right)} \]
5. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \color{blue}{\varepsilon \cdot \left(\frac{1}{2} \cdot \sin \left(-1 \cdot x\right) + \varepsilon \cdot \left(\frac{-1}{8} \cdot \cos \left(-1 \cdot x\right) + \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \color{blue}{\varepsilon} \cdot \left(\frac{1}{2} \cdot \sin \left(-1 \cdot x\right) + \varepsilon \cdot \left(\frac{-1}{8} \cdot \cos \left(-1 \cdot x\right) + \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \left(\frac{1}{2} \cdot \sin \left(-1 \cdot x\right) + \varepsilon \cdot \left(\frac{-1}{8} \cdot \cos \left(-1 \cdot x\right) + \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \color{blue}{\left(\frac{1}{2} \cdot \sin \left(-1 \cdot x\right) + \varepsilon \cdot \left(\frac{-1}{8} \cdot \cos \left(-1 \cdot x\right) + \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)}\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\sin \left(-1 \cdot x\right)}, \varepsilon \cdot \left(\frac{-1}{8} \cdot \cos \left(-1 \cdot x\right) + \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right) \]

   6. lower-sin.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \mathsf{fma}\left(\frac{1}{2}, \sin \left(-1 \cdot x\right), \varepsilon \cdot \left(\frac{-1}{8} \cdot \cos \left(-1 \cdot x\right) + \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \mathsf{fma}\left(\frac{1}{2}, \sin \left(-1 \cdot x\right), \varepsilon \cdot \left(\frac{-1}{8} \cdot \cos \left(-1 \cdot x\right) + \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \mathsf{fma}\left(\frac{1}{2}, \sin \left(-1 \cdot x\right), \varepsilon \cdot \left(\frac{-1}{8} \cdot \cos \left(-1 \cdot x\right) + \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \mathsf{fma}\left(\frac{1}{2}, \sin \left(-1 \cdot x\right), \varepsilon \cdot \mathsf{fma}\left(\frac{-1}{8}, \cos \left(-1 \cdot x\right), \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right) \]

   10. lower-cos.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \mathsf{fma}\left(\frac{1}{2}, \sin \left(-1 \cdot x\right), \varepsilon \cdot \mathsf{fma}\left(\frac{-1}{8}, \cos \left(-1 \cdot x\right), \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \mathsf{fma}\left(\frac{1}{2}, \sin \left(-1 \cdot x\right), \varepsilon \cdot \mathsf{fma}\left(\frac{-1}{8}, \cos \left(-1 \cdot x\right), \frac{-1}{48} \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right) \]

6. Applied rewrites 99.8%

   \[\leadsto \left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(-1 \cdot x\right) + \varepsilon \cdot \mathsf{fma}\left(0.5, \sin \left(-1 \cdot x\right), \varepsilon \cdot \mathsf{fma}\left(-0.125, \cos \left(-1 \cdot x\right), -0.020833333333333332 \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)\right)\right)} \]

8. Applied rewrites 99.8%

   \[\leadsto \left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \mathsf{fma}\left(\left(-\sin x\right) \cdot \varepsilon, \color{blue}{0.5}, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.125 \cdot \cos x - \left(-0.020833333333333332 \cdot \varepsilon\right) \cdot \sin x, \cos x\right)\right) \]

9. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(1 + \frac{-1}{24} \cdot {\varepsilon}^{2}\right)\right)} \cdot \mathsf{fma}\left(\left(-\sin x\right) \cdot \varepsilon, \frac{1}{2}, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{8} \cdot \cos x - \left(\frac{-1}{48} \cdot \varepsilon\right) \cdot \sin x, \cos x\right)\right) \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \left(\varepsilon \cdot \color{blue}{\left(1 + \frac{-1}{24} \cdot {\varepsilon}^{2}\right)}\right) \cdot \mathsf{fma}\left(\left(-\sin x\right) \cdot \varepsilon, \frac{1}{2}, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{8} \cdot \cos x - \left(\frac{-1}{48} \cdot \varepsilon\right) \cdot \sin x, \cos x\right)\right) \]

    2. lower-+.f64 N/A

       \[\leadsto \left(\varepsilon \cdot \left(1 + \color{blue}{\frac{-1}{24} \cdot {\varepsilon}^{2}}\right)\right) \cdot \mathsf{fma}\left(\left(-\sin x\right) \cdot \varepsilon, \frac{1}{2}, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{8} \cdot \cos x - \left(\frac{-1}{48} \cdot \varepsilon\right) \cdot \sin x, \cos x\right)\right) \]

    3. lower-*.f64 N/A

       \[\leadsto \left(\varepsilon \cdot \left(1 + \frac{-1}{24} \cdot \color{blue}{{\varepsilon}^{2}}\right)\right) \cdot \mathsf{fma}\left(\left(-\sin x\right) \cdot \varepsilon, \frac{1}{2}, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{-1}{8} \cdot \cos x - \left(\frac{-1}{48} \cdot \varepsilon\right) \cdot \sin x, \cos x\right)\right) \]

    4. lower-pow.f64 99.8

       \[\leadsto \left(\varepsilon \cdot \left(1 + -0.041666666666666664 \cdot {\varepsilon}^{\color{blue}{2}}\right)\right) \cdot \mathsf{fma}\left(\left(-\sin x\right) \cdot \varepsilon, 0.5, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.125 \cdot \cos x - \left(-0.020833333333333332 \cdot \varepsilon\right) \cdot \sin x, \cos x\right)\right) \]

11. Applied rewrites 99.8%

    \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(1 + -0.041666666666666664 \cdot {\varepsilon}^{2}\right)\right)} \cdot \mathsf{fma}\left(\left(-\sin x\right) \cdot \varepsilon, 0.5, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.125 \cdot \cos x - \left(-0.020833333333333332 \cdot \varepsilon\right) \cdot \sin x, \cos x\right)\right) \]
12. Add Preprocessing

Alternative 3: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. mult-flip N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. add-flip N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 N/A

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

   20. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - \color{blue}{0}\right)\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right)} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right)} \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{-1}{2}\right) \]

   2. lift--.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \color{blue}{\left(\varepsilon - 0\right)}\right) \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \frac{-1}{2}\right) \]

   3. --rgt-identity N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 99.9

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

   6. lift-cos.f64 N/A

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

   7. cos-neg-rev N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. lift-fma.f64 N/A

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

   11. +-commutative N/A

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

   12. count-2-rev N/A

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

   13. associate-+l+ N/A

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

   14. +-commutative N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. mult-flip N/A

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

   18. lower-cos.f64 N/A

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

   19. mult-flip N/A

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

   20. +-commutative N/A

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

   21. associate-+l+ N/A

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

   22. count-2-rev N/A

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

   23. +-commutative N/A

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

   24. lift-fma.f64 N/A

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

   25. metadata-eval N/A

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

5. Applied rewrites 99.9%

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

Alternative 4: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\varepsilon \cdot \left(1 + -0.041666666666666664 \cdot {\varepsilon}^{2}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (* eps (+ 1.0 (* -0.041666666666666664 (pow eps 2.0))))
  (cos (* (fma 2.0 x eps) -0.5))))

C

double code(double x, double eps) {
	return (eps * (1.0 + (-0.041666666666666664 * pow(eps, 2.0)))) * cos((fma(2.0, x, eps) * -0.5));
}

Julia

function code(x, eps)
	return Float64(Float64(eps * Float64(1.0 + Float64(-0.041666666666666664 * (eps ^ 2.0)))) * cos(Float64(fma(2.0, x, eps) * -0.5)))
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(1.0 + N[(-0.041666666666666664 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * x + eps), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. mult-flip N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. add-flip N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 N/A

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

   20. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - \color{blue}{0}\right)\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right)} \]

4. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 99.7

      \[\leadsto \left(\varepsilon \cdot \left(1 + -0.041666666666666664 \cdot {\varepsilon}^{\color{blue}{2}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right) \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(1 + -0.041666666666666664 \cdot {\varepsilon}^{2}\right)\right)} \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right) \]
7. Add Preprocessing

Alternative 5: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	return Float64(Float64(Float64(0.5 * eps) * 2.0) * sin(Float64(Float64(fma(x, 2.0, eps) + pi) / 2.0)))
end

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. mult-flip N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. add-flip N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 N/A

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

   20. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - \color{blue}{0}\right)\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right)} \]

4. Taylor expanded in eps around 0

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

   1. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. metadata-eval N/A

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

   8. lift-fma.f64 N/A

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

   9. count-2-rev N/A

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

   10. associate-+l+ N/A

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

   11. +-commutative N/A

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

   12. metadata-eval N/A

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

   13. mult-flip N/A

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

   14. +-commutative N/A

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

   15. associate-+l+ N/A

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

   16. count-2-rev N/A

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

   17. lift-fma.f64 N/A

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

   18. div-add-rev N/A

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

   19. lower-/.f64 N/A

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

   20. lower-+.f64 N/A

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

   21. lift-fma.f64 N/A

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

   22. *-commutative N/A

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

   23. lower-fma.f64 N/A

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

   24. lower-PI.f64 99.5

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

8. Applied rewrites 99.5%

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

Alternative 6: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. mult-flip N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. add-flip N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 N/A

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

   20. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - \color{blue}{0}\right)\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right)} \]

4. Taylor expanded in eps around 0

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

   1. lower-*.f64 99.5

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

6. Applied rewrites 99.5%

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

Alternative 7: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \cos \left(-1 \cdot x\right) \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * cos(((-1.0d0) * x))
end function

Java

public static double code(double x, double eps) {
	return eps * Math.cos((-1.0 * x));
}

Python

def code(x, eps):
	return eps * math.cos((-1.0 * x))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * cos((-1.0 * x));
end

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. mult-flip N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. add-flip N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 N/A

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

   20. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - \color{blue}{0}\right)\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right)} \]

4. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

      \[\leadsto \varepsilon \cdot \cos \left(-1 \cdot x\right) \]

   3. lower-*.f64 99.0

      \[\leadsto \varepsilon \cdot \cos \left(-1 \cdot x\right) \]

6. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\varepsilon \cdot \cos \left(-1 \cdot x\right)} \]
7. Add Preprocessing

Alternative 8: 98.3% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. mult-flip N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. add-flip N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 N/A

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

   20. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - \color{blue}{0}\right)\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right)} \]

4. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-*.f64 99.5

      \[\leadsto \varepsilon \cdot \left(\cos \left(-1 \cdot x\right) + 0.5 \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right) \]

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos \left(-1 \cdot x\right) + 0.5 \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)} \]

7. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 98.3

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

9. Applied rewrites 98.3%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 98.3

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

    6. lift-fma.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. lift-pow.f64 N/A

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

    9. pow2 N/A

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

    10. lift-*.f64 N/A

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

    11. distribute-lft-out N/A

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

    12. *-commutative N/A

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

    13. lower-*.f64 N/A

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

    14. lift-*.f64 N/A

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

    15. lift-*.f64 N/A

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

    16. distribute-lft-out N/A

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

    17. lower-*.f64 N/A

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

    18. lower-+.f64 98.3

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

11. Applied rewrites 98.3%

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

Alternative 9: 98.2% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. mult-flip N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. add-flip N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 N/A

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

   20. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - \color{blue}{0}\right)\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right)} \]

4. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-*.f64 99.5

      \[\leadsto \varepsilon \cdot \left(\cos \left(-1 \cdot x\right) + 0.5 \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right) \]

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos \left(-1 \cdot x\right) + 0.5 \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)} \]

7. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 98.3

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

9. Applied rewrites 98.3%

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

10. Taylor expanded in x around inf

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

    1. lower-*.f64 N/A

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

    2. lower-*.f64 98.2

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

12. Applied rewrites 98.2%

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

Alternative 10: 97.8% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

2. Taylor expanded in x around 0

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

   1. lower-sin.f64 97.8

      \[\leadsto \sin \varepsilon \]

4. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\sin \varepsilon} \]

5. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 97.8

      \[\leadsto \varepsilon \cdot \left(1 + -0.16666666666666666 \cdot {\varepsilon}^{2}\right) \]

7. Applied rewrites 97.8%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {\varepsilon}^{2}\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

   11. cube-unmult N/A

       \[\leadsto {\varepsilon}^{3} \cdot \frac{-1}{6} + \varepsilon \]

   12. lower-pow.f32 N/A

       \[\leadsto {\varepsilon}^{3} \cdot \frac{-1}{6} + \varepsilon \]

   13. lower-unsound-pow.f32 N/A

       \[\leadsto {\varepsilon}^{3} \cdot \frac{-1}{6} + \varepsilon \]

   14. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{3}, \frac{-1}{6}, \varepsilon\right) \]

   15. lower-unsound-pow.f32 97.8

       \[\leadsto \mathsf{fma}\left(\left( {\varepsilon}^{3} \right)_{\text{binary32}}, -0.16666666666666666, \varepsilon\right) \]

   16. lower-pow.f32 N/A

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{3}, \frac{-1}{6}, \varepsilon\right) \]

   17. metadata-eval N/A

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

   18. pow-plus N/A

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

   19. lift-pow.f64 N/A

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

   20. lower-*.f64 97.8

       \[\leadsto \mathsf{fma}\left({\varepsilon}^{2} \cdot \varepsilon, -0.16666666666666666, \varepsilon\right) \]

   21. lift-pow.f64 N/A

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

   22. unpow2 N/A

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

   23. lower-*.f64 97.8

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

9. Applied rewrites 97.8%

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

Alternative 11: 97.8% accurate, 69.8× 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 62.4%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. mult-flip N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. add-flip N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 N/A

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

   20. +-inverses N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - \color{blue}{0}\right)\right) \cdot 2\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \cos \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot -0.5\right)} \]

4. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-*.f64 99.5

      \[\leadsto \varepsilon \cdot \left(\cos \left(-1 \cdot x\right) + 0.5 \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right) \]

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos \left(-1 \cdot x\right) + 0.5 \cdot \left(\varepsilon \cdot \sin \left(-1 \cdot x\right)\right)\right)} \]

7. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 98.3

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

9. Applied rewrites 98.3%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 97.8%

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

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \left(\cos \varepsilon - 1\right) + \cos x \cdot \sin \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+ (* (sin x) (- (cos eps) 1.0)) (* (cos x) (sin eps))))

C

double code(double x, double eps) {
	return (sin(x) * (cos(eps) - 1.0)) + (cos(x) * sin(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 = (sin(x) * (cos(eps) - 1.0d0)) + (cos(x) * sin(eps))
end function

Java

public static double code(double x, double eps) {
	return (Math.sin(x) * (Math.cos(eps) - 1.0)) + (Math.cos(x) * Math.sin(eps));
}

Python

def code(x, eps):
	return (math.sin(x) * (math.cos(eps) - 1.0)) + (math.cos(x) * math.sin(eps))

Julia

function code(x, eps)
	return Float64(Float64(sin(x) * Float64(cos(eps) - 1.0)) + Float64(cos(x) * sin(eps)))
end

MATLAB

function tmp = code(x, eps)
	tmp = (sin(x) * (cos(eps) - 1.0)) + (cos(x) * sin(eps));
end

Wolfram

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

Developer Target 3: 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 2025164 
(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 c (* 2 (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  :alt
  (! :herbie-platform c (+ (* (sin x) (- (cos eps) 1)) (* (cos x) (sin eps))))

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

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