2sin (example 3.3)

Percentage Accurate: 62.5% → 100.0%

Time: 6.2s

Alternatives: 12

Speedup: 17.7×

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: 62.5% 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: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + x\right) \cdot -0.5\\ \left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \left(\cos t\_0 \cdot \cos \left(\varepsilon \cdot -0.5\right) - \sin t\_0 \cdot \sin \left(\varepsilon \cdot -0.5\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (+ x x) -0.5)))
   (*
    (* (sin (* 0.5 (- eps 0.0))) 2.0)
    (- (* (cos t_0) (cos (* eps -0.5))) (* (sin t_0) (sin (* eps -0.5)))))))

C

double code(double x, double eps) {
	double t_0 = (x + x) * -0.5;
	return (sin((0.5 * (eps - 0.0))) * 2.0) * ((cos(t_0) * cos((eps * -0.5))) - (sin(t_0) * sin((eps * -0.5))));
}

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
    real(8) :: t_0
    t_0 = (x + x) * (-0.5d0)
    code = (sin((0.5d0 * (eps - 0.0d0))) * 2.0d0) * ((cos(t_0) * cos((eps * (-0.5d0)))) - (sin(t_0) * sin((eps * (-0.5d0)))))
end function

Java

public static double code(double x, double eps) {
	double t_0 = (x + x) * -0.5;
	return (Math.sin((0.5 * (eps - 0.0))) * 2.0) * ((Math.cos(t_0) * Math.cos((eps * -0.5))) - (Math.sin(t_0) * Math.sin((eps * -0.5))));
}

Python

def code(x, eps):
	t_0 = (x + x) * -0.5
	return (math.sin((0.5 * (eps - 0.0))) * 2.0) * ((math.cos(t_0) * math.cos((eps * -0.5))) - (math.sin(t_0) * math.sin((eps * -0.5))))

Julia

function code(x, eps)
	t_0 = Float64(Float64(x + x) * -0.5)
	return Float64(Float64(sin(Float64(0.5 * Float64(eps - 0.0))) * 2.0) * Float64(Float64(cos(t_0) * cos(Float64(eps * -0.5))) - Float64(sin(t_0) * sin(Float64(eps * -0.5)))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = (x + x) * -0.5;
	tmp = (sin((0.5 * (eps - 0.0))) * 2.0) * ((cos(t_0) * cos((eps * -0.5))) - (sin(t_0) * sin((eps * -0.5))));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x + x), $MachinePrecision] * -0.5), $MachinePrecision]}, N[(N[(N[Sin[N[(0.5 * N[(eps - 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(N[Cos[t$95$0], $MachinePrecision] * N[Cos[N[(eps * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t$95$0], $MachinePrecision] * N[Sin[N[(eps * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-fma.f64 N/A

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

   5. distribute-rgt-in N/A

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

   6. cos-sum N/A

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

   7. lower--.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. count-2-rev N/A

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

   12. lower-+.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. count-2-rev N/A

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

   19. lower-+.f64 N/A

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

   20. lower-sin.f64 N/A

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

   21. lower-*.f64 100.0

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

5. Applied rewrites 100.0%

   \[\leadsto \left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \color{blue}{\left(\cos \left(\left(x + x\right) \cdot -0.5\right) \cdot \cos \left(\varepsilon \cdot -0.5\right) - \sin \left(\left(x + x\right) \cdot -0.5\right) \cdot \sin \left(\varepsilon \cdot -0.5\right)\right)} \]
6. Add Preprocessing

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2, \varepsilon\right), 0.5, \left(0.5 \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (* (sin (* 0.5 (- eps 0.0))) 2.0)
  (sin (fma (fma x 2.0 eps) 0.5 (* (* 0.5 (cbrt (* PI PI))) (cbrt PI))))))

C

double code(double x, double eps) {
	return (sin((0.5 * (eps - 0.0))) * 2.0) * sin(fma(fma(x, 2.0, eps), 0.5, ((0.5 * cbrt((((double) M_PI) * ((double) M_PI)))) * cbrt(((double) M_PI)))));
}

Julia

function code(x, eps)
	return Float64(Float64(sin(Float64(0.5 * Float64(eps - 0.0))) * 2.0) * sin(fma(fma(x, 2.0, eps), 0.5, Float64(Float64(0.5 * cbrt(Float64(pi * pi))) * cbrt(pi)))))
end

Wolfram

code[x_, eps_] := N[(N[(N[Sin[N[(0.5 * N[(eps - 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * N[Sin[N[(N[(x * 2.0 + eps), $MachinePrecision] * 0.5 + N[(N[(0.5 * N[Power[N[(Pi * Pi), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lift-*.f64 N/A

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

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

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

   6. lift-fma.f64 N/A

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

   7. count-2-rev N/A

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

   8. associate-+l+ N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. mult-flip N/A

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

   14. lower-sin.f64 N/A

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

   15. mult-flip N/A

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

   16. +-commutative N/A

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

   17. lift-+.f64 N/A

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

   18. associate-+l+ N/A

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

   19. count-2-rev N/A

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

   20. lift-fma.f64 N/A

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

   21. metadata-eval N/A

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

5. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f64 N/A

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

   4. add-cube-cbrt N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2, \varepsilon\right), \frac{1}{2}, \frac{1}{2} \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2, \varepsilon\right), \frac{1}{2}, \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2, \varepsilon\right), \frac{1}{2}, \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2, \varepsilon\right), \frac{1}{2}, \color{blue}{\left(\frac{1}{2} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \]

   8. lift-PI.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2, \varepsilon\right), \frac{1}{2}, \left(\frac{1}{2} \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \]

   9. lift-PI.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2, \varepsilon\right), \frac{1}{2}, \left(\frac{1}{2} \cdot \left(\sqrt[3]{\pi} \cdot \sqrt[3]{\color{blue}{\pi}}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \]

   10. cbrt-unprod N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2, \varepsilon\right), \frac{1}{2}, \left(\frac{1}{2} \cdot \color{blue}{\sqrt[3]{\pi \cdot \pi}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2, \varepsilon\right), \frac{1}{2}, \left(\frac{1}{2} \cdot \sqrt[3]{\color{blue}{\pi \cdot \pi}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \]

   12. lower-unsound-*.f32 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2, \varepsilon\right), \frac{1}{2}, \left(\frac{1}{2} \cdot \sqrt[3]{\color{blue}{\pi \cdot \pi}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \]

   13. lower-cbrt.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2, \varepsilon\right), \frac{1}{2}, \left(\frac{1}{2} \cdot \color{blue}{\sqrt[3]{\pi \cdot \pi}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \]

   14. lower-unsound-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2, \varepsilon\right), \frac{1}{2}, \left(\frac{1}{2} \cdot \sqrt[3]{\color{blue}{\pi \cdot \pi}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \]

   15. lift-PI.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2, \varepsilon\right), \frac{1}{2}, \left(\frac{1}{2} \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\color{blue}{\pi}}\right)\right) \]

   16. lower-cbrt.f64 99.9

       \[\leadsto \left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2, \varepsilon\right), 0.5, \left(0.5 \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right)\right) \]

7. Applied rewrites 99.9%

   \[\leadsto \left(\sin \left(0.5 \cdot \left(\varepsilon - 0\right)\right) \cdot 2\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(x, 2, \varepsilon\right), 0.5, \color{blue}{\left(0.5 \cdot \sqrt[3]{\pi \cdot \pi}\right) \cdot \sqrt[3]{\pi}}\right)\right) \]
8. Add Preprocessing

Alternative 3: 99.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lift-*.f64 N/A

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

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

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

   6. lift-fma.f64 N/A

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

   7. count-2-rev N/A

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

   8. associate-+l+ N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. mult-flip N/A

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

   14. lower-sin.f64 N/A

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

   15. mult-flip N/A

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

   16. +-commutative N/A

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

   17. lift-+.f64 N/A

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

   18. associate-+l+ N/A

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

   19. count-2-rev N/A

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

   20. lift-fma.f64 N/A

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

   21. metadata-eval N/A

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

5. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. --rgt-identity N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 99.9

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

7. Applied rewrites 99.9%

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

Alternative 4: 99.9% 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 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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. count-2-rev N/A

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

   12. associate-+l+ N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   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

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

   21. lift-+.f64 N/A

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

   22. associate-+l+ N/A

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

   23. count-2-rev N/A

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

   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 5: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	return Float64(Float64(eps * Float64(1.0 + Float64(-0.041666666666666664 * (eps ^ 2.0)))) * sin(fma(fma(x, 2.0, eps), 0.5, Float64(pi * 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[Sin[N[(N[(x * 2.0 + eps), $MachinePrecision] * 0.5 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lift-*.f64 N/A

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

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

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

   6. lift-fma.f64 N/A

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

   7. count-2-rev N/A

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

   8. associate-+l+ N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. mult-flip N/A

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

   14. lower-sin.f64 N/A

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

   15. mult-flip N/A

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

   16. +-commutative N/A

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

   17. lift-+.f64 N/A

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

   18. associate-+l+ N/A

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

   19. count-2-rev N/A

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

   20. lift-fma.f64 N/A

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

   21. metadata-eval N/A

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 99.7

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

8. Applied rewrites 99.7%

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

Alternative 6: 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 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 7: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lift-*.f64 N/A

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

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

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

   6. lift-fma.f64 N/A

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

   7. count-2-rev N/A

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

   8. associate-+l+ N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. mult-flip N/A

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

   14. lower-sin.f64 N/A

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

   15. mult-flip N/A

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

   16. +-commutative N/A

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

   17. lift-+.f64 N/A

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

   18. associate-+l+ N/A

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

   19. count-2-rev N/A

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

   20. lift-fma.f64 N/A

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

   21. metadata-eval N/A

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in eps around 0

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

   1. lower-*.f64 99.5

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

8. Applied rewrites 99.5%

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

Alternative 8: 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 62.5%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 9: 99.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 99.0

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

4. Applied rewrites 99.0%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. mult-flip N/A

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

   6. metadata-eval N/A

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

   7. lift-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-fma.f64 99.0

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

6. Applied rewrites 99.0%

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

Alternative 10: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 99.0

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

4. Applied rewrites 99.0%

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

Alternative 11: 98.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * (1.0 + (-0.5 * pow(x, 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 = eps * (1.0d0 + ((-0.5d0) * (x ** 2.0d0)))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + (-0.5 * Math.pow(x, 2.0)));
}

Python

def code(x, eps):
	return eps * (1.0 + (-0.5 * math.pow(x, 2.0)))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + (-0.5 * (x ^ 2.0)));
end

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 99.0

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 98.3

      \[\leadsto \varepsilon \cdot \left(1 + -0.5 \cdot {x}^{2}\right) \]

7. Applied rewrites 98.3%

   \[\leadsto \varepsilon \cdot \left(1 + \color{blue}{-0.5 \cdot {x}^{2}}\right) \]
8. Add Preprocessing

Alternative 12: 97.9% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot 1 \end{array} \]

FPCore

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

C

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

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 = eps * 1.0d0
end function

Java

public static double code(double x, double eps) {
	return eps * 1.0;
}

Python

def code(x, eps):
	return eps * 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 99.0

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot 1 \]
6. Step-by-step derivation

7. Applied rewrites 97.9%

   \[\leadsto \varepsilon \cdot 1 \]
8. 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 2025159 
(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)))