2tan (problem 3.3.2)

Percentage Accurate: 61.9% → 99.8%

Time: 8.0s

Alternatives: 23

Speedup: 207.0×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \frac{\frac{t\_0 \cdot t\_0 - 1}{t\_0 - 1} \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (*
          (-
           (*
            (fma -0.0001984126984126984 (* eps eps) 0.008333333333333333)
            (* eps eps))
           0.16666666666666666)
          (* eps eps))))
   (/
    (* (/ (- (* t_0 t_0) 1.0) (- t_0 1.0)) eps)
    (* (- (* (cos x) (cos eps)) (* (sin x) (sin eps))) (cos x)))))

C

double code(double x, double eps) {
	double t_0 = ((fma(-0.0001984126984126984, (eps * eps), 0.008333333333333333) * (eps * eps)) - 0.16666666666666666) * (eps * eps);
	return ((((t_0 * t_0) - 1.0) / (t_0 - 1.0)) * eps) / (((cos(x) * cos(eps)) - (sin(x) * sin(eps))) * cos(x));
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(fma(-0.0001984126984126984, Float64(eps * eps), 0.008333333333333333) * Float64(eps * eps)) - 0.16666666666666666) * Float64(eps * eps))
	return Float64(Float64(Float64(Float64(Float64(t_0 * t_0) - 1.0) / Float64(t_0 - 1.0)) * eps) / Float64(Float64(Float64(cos(x) * cos(eps)) - Float64(sin(x) * sin(eps))) * cos(x)))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(-0.0001984126984126984 * N[(eps * eps), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - 1.0), $MachinePrecision] / N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. +-commutative N/A

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

   5. quot-tan N/A

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

   6. lift-tan.f64 N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

4. Applied rewrites 62.0%

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

5. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{\varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{\varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

7. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \color{blue}{\left(\varepsilon + x\right)} \cdot \cos x} \]

   2. lift-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\color{blue}{\cos \left(\varepsilon + x\right)} \cdot \cos x} \]

   3. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \color{blue}{\left(x + \varepsilon\right)} \cdot \cos x} \]

   4. cos-sum N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} \cdot \cos x} \]

   5. cos-neg-rev N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\varepsilon\right)\right)} - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   6. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \color{blue}{\left(-1 \cdot \varepsilon\right)} - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   7. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\color{blue}{\left(\cos x \cdot \cos \left(-1 \cdot \varepsilon\right) - \sin x \cdot \sin \varepsilon\right)} \cdot \cos x} \]

   8. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\varepsilon\right)\right)} - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   9. cos-neg-rev N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \color{blue}{\cos \varepsilon} - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\color{blue}{\cos x \cdot \cos \varepsilon} - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   11. lift-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\color{blue}{\cos x} \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \color{blue}{\cos \varepsilon} - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \color{blue}{\sin x \cdot \sin \varepsilon}\right) \cdot \cos x} \]

   14. lower-sin.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \color{blue}{\sin x} \cdot \sin \varepsilon\right) \cdot \cos x} \]

   15. lower-sin.f64 99.8

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \color{blue}{\sin \varepsilon}\right) \cdot \cos x} \]

9. Applied rewrites 99.8%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} \cdot \cos x} \]
10. Step-by-step derivation

    1. lift-fma.f64 N/A

       \[\leadsto \frac{\left(\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

    2. lift--.f64 N/A

       \[\leadsto \frac{\left(\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{\left(\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

    4. lift-fma.f64 N/A

       \[\leadsto \frac{\left(\left(\left(\frac{-1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right) + \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

    5. lift-*.f64 N/A

       \[\leadsto \frac{\left(\left(\left(\frac{-1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right) + \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

    6. lift-*.f64 N/A

       \[\leadsto \frac{\left(\left(\left(\frac{-1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right) + \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

    7. lift-*.f64 N/A

       \[\leadsto \frac{\left(\left(\left(\frac{-1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right) + \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) + 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

    8. flip-+ N/A

       \[\leadsto \frac{\frac{\left(\left(\left(\frac{-1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right) + \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(\left(\frac{-1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right) + \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 1 \cdot 1}{\left(\left(\frac{-1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right) + \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 1} \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

    9. lower-/.f64 N/A

       \[\leadsto \frac{\frac{\left(\left(\left(\frac{-1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right) + \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(\left(\frac{-1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right) + \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 1 \cdot 1}{\left(\left(\frac{-1}{5040} \cdot \left(\varepsilon \cdot \varepsilon\right) + \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 1} \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

11. Applied rewrites 99.8%

    \[\leadsto \frac{\frac{\left(\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 1}{\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 1} \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]
12. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.5\right)\right) - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (*
   (fma
    (-
     (*
      (fma -0.0001984126984126984 (* eps eps) 0.008333333333333333)
      (* eps eps))
     0.16666666666666666)
    (* eps eps)
    1.0)
   eps)
  (*
   (-
    (*
     (cos x)
     (+
      1.0
      (*
       (* eps eps)
       (-
        (*
         (* eps eps)
         (+ 0.041666666666666664 (* -0.001388888888888889 (* eps eps))))
        0.5))))
    (* (sin x) (sin eps)))
   (cos x))))

C

double code(double x, double eps) {
	return (fma(((fma(-0.0001984126984126984, (eps * eps), 0.008333333333333333) * (eps * eps)) - 0.16666666666666666), (eps * eps), 1.0) * eps) / (((cos(x) * (1.0 + ((eps * eps) * (((eps * eps) * (0.041666666666666664 + (-0.001388888888888889 * (eps * eps)))) - 0.5)))) - (sin(x) * sin(eps))) * cos(x));
}

Julia

function code(x, eps)
	return Float64(Float64(fma(Float64(Float64(fma(-0.0001984126984126984, Float64(eps * eps), 0.008333333333333333) * Float64(eps * eps)) - 0.16666666666666666), Float64(eps * eps), 1.0) * eps) / Float64(Float64(Float64(cos(x) * Float64(1.0 + Float64(Float64(eps * eps) * Float64(Float64(Float64(eps * eps) * Float64(0.041666666666666664 + Float64(-0.001388888888888889 * Float64(eps * eps)))) - 0.5)))) - Float64(sin(x) * sin(eps))) * cos(x)))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(eps * eps), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(1.0 + N[(N[(eps * eps), $MachinePrecision] * N[(N[(N[(eps * eps), $MachinePrecision] * N[(0.041666666666666664 + N[(-0.001388888888888889 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. +-commutative N/A

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

   5. quot-tan N/A

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

   6. lift-tan.f64 N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

4. Applied rewrites 62.0%

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

5. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{\varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{\varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

7. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \color{blue}{\left(\varepsilon + x\right)} \cdot \cos x} \]

   2. lift-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\color{blue}{\cos \left(\varepsilon + x\right)} \cdot \cos x} \]

   3. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \color{blue}{\left(x + \varepsilon\right)} \cdot \cos x} \]

   4. cos-sum N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} \cdot \cos x} \]

   5. cos-neg-rev N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\varepsilon\right)\right)} - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   6. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \color{blue}{\left(-1 \cdot \varepsilon\right)} - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   7. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\color{blue}{\left(\cos x \cdot \cos \left(-1 \cdot \varepsilon\right) - \sin x \cdot \sin \varepsilon\right)} \cdot \cos x} \]

   8. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\varepsilon\right)\right)} - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   9. cos-neg-rev N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \color{blue}{\cos \varepsilon} - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\color{blue}{\cos x \cdot \cos \varepsilon} - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   11. lift-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\color{blue}{\cos x} \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \color{blue}{\cos \varepsilon} - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \color{blue}{\sin x \cdot \sin \varepsilon}\right) \cdot \cos x} \]

   14. lower-sin.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \color{blue}{\sin x} \cdot \sin \varepsilon\right) \cdot \cos x} \]

   15. lower-sin.f64 99.8

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \color{blue}{\sin \varepsilon}\right) \cdot \cos x} \]

9. Applied rewrites 99.8%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} \cdot \cos x} \]

10. Taylor expanded in eps around 0

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \color{blue}{\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\varepsilon}^{2}\right) - \frac{1}{2}\right)\right)} - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]
11. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \left(1 + \color{blue}{{\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\varepsilon}^{2}\right) - \frac{1}{2}\right)}\right) - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \left(1 + {\varepsilon}^{2} \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\varepsilon}^{2}\right) - \frac{1}{2}\right)}\right) - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

    3. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\varepsilon}^{2}\right)} - \frac{1}{2}\right)\right) - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

    4. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\varepsilon}^{2}\right)} - \frac{1}{2}\right)\right) - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

    5. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\varepsilon}^{2}\right) - \color{blue}{\frac{1}{2}}\right)\right) - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

    6. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\varepsilon}^{2}\right) - \frac{1}{2}\right)\right) - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

    7. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\varepsilon}^{2}\right) - \frac{1}{2}\right)\right) - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

    8. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\varepsilon}^{2}\right) - \frac{1}{2}\right)\right) - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

    9. lower-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\varepsilon}^{2}\right) - \frac{1}{2}\right)\right) - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

    10. lower-*.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\varepsilon}^{2}\right) - \frac{1}{2}\right)\right) - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

    11. pow2 N/A

        \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - \frac{1}{2}\right)\right) - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

    12. lift-*.f64 99.8

        \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.5\right)\right) - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

12. Applied rewrites 99.8%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \color{blue}{\left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) - 0.5\right)\right)} - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]
13. Add Preprocessing

Alternative 3: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \left(\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666\right)\right)\right)\right) \cdot \cos x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (*
   (fma
    (-
     (*
      (fma -0.0001984126984126984 (* eps eps) 0.008333333333333333)
      (* eps eps))
     0.16666666666666666)
    (* eps eps)
    1.0)
   eps)
  (*
   (-
    (* (cos x) (cos eps))
    (*
     (sin x)
     (*
      eps
      (+
       1.0
       (*
        (* eps eps)
        (- (* 0.008333333333333333 (* eps eps)) 0.16666666666666666))))))
   (cos x))))

C

double code(double x, double eps) {
	return (fma(((fma(-0.0001984126984126984, (eps * eps), 0.008333333333333333) * (eps * eps)) - 0.16666666666666666), (eps * eps), 1.0) * eps) / (((cos(x) * cos(eps)) - (sin(x) * (eps * (1.0 + ((eps * eps) * ((0.008333333333333333 * (eps * eps)) - 0.16666666666666666)))))) * cos(x));
}

Julia

function code(x, eps)
	return Float64(Float64(fma(Float64(Float64(fma(-0.0001984126984126984, Float64(eps * eps), 0.008333333333333333) * Float64(eps * eps)) - 0.16666666666666666), Float64(eps * eps), 1.0) * eps) / Float64(Float64(Float64(cos(x) * cos(eps)) - Float64(sin(x) * Float64(eps * Float64(1.0 + Float64(Float64(eps * eps) * Float64(Float64(0.008333333333333333 * Float64(eps * eps)) - 0.16666666666666666)))))) * cos(x)))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(eps * eps), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision] / N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[(eps * N[(1.0 + N[(N[(eps * eps), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. +-commutative N/A

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

   5. quot-tan N/A

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

   6. lift-tan.f64 N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

4. Applied rewrites 62.0%

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

5. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{\varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{\varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

7. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \color{blue}{\left(\varepsilon + x\right)} \cdot \cos x} \]

   2. lift-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\color{blue}{\cos \left(\varepsilon + x\right)} \cdot \cos x} \]

   3. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \color{blue}{\left(x + \varepsilon\right)} \cdot \cos x} \]

   4. cos-sum N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} \cdot \cos x} \]

   5. cos-neg-rev N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\varepsilon\right)\right)} - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   6. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \color{blue}{\left(-1 \cdot \varepsilon\right)} - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   7. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\color{blue}{\left(\cos x \cdot \cos \left(-1 \cdot \varepsilon\right) - \sin x \cdot \sin \varepsilon\right)} \cdot \cos x} \]

   8. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \color{blue}{\left(\mathsf{neg}\left(\varepsilon\right)\right)} - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   9. cos-neg-rev N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \color{blue}{\cos \varepsilon} - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\color{blue}{\cos x \cdot \cos \varepsilon} - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   11. lift-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\color{blue}{\cos x} \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   12. lower-cos.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \color{blue}{\cos \varepsilon} - \sin x \cdot \sin \varepsilon\right) \cdot \cos x} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \color{blue}{\sin x \cdot \sin \varepsilon}\right) \cdot \cos x} \]

   14. lower-sin.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \color{blue}{\sin x} \cdot \sin \varepsilon\right) \cdot \cos x} \]

   15. lower-sin.f64 99.8

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \color{blue}{\sin \varepsilon}\right) \cdot \cos x} \]

9. Applied rewrites 99.8%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} \cdot \cos x} \]

10. Taylor expanded in eps around 0

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \color{blue}{\left(\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)\right)}\right) \cdot \cos x} \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \left(\varepsilon \cdot \color{blue}{\left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)}\right)\right) \cdot \cos x} \]

    2. lower-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \left(\varepsilon \cdot \left(1 + \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)}\right)\right)\right) \cdot \cos x} \]

    3. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \left(\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)}\right)\right)\right) \cdot \cos x} \]

    4. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \left(\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {\varepsilon}^{2}} - \frac{1}{6}\right)\right)\right)\right) \cdot \cos x} \]

    5. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \left(\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {\varepsilon}^{2}} - \frac{1}{6}\right)\right)\right)\right) \cdot \cos x} \]

    6. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \left(\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \cdot \cos x} \]

    7. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \left(\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)\right)\right) \cdot \cos x} \]

    8. pow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \left(\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\frac{1}{120} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}\right)\right)\right)\right) \cdot \cos x} \]

    9. lift-*.f64 99.8

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \left(\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666\right)\right)\right)\right) \cdot \cos x} \]

12. Applied rewrites 99.8%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \color{blue}{\left(\varepsilon \cdot \left(1 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.008333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666\right)\right)\right)}\right) \cdot \cos x} \]
13. Add Preprocessing

Alternative 4: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\sin \left(\left(x + \varepsilon\right) + \frac{\pi}{2}\right) \cdot \cos x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (*
   (fma
    (-
     (*
      (fma -0.0001984126984126984 (* eps eps) 0.008333333333333333)
      (* eps eps))
     0.16666666666666666)
    (* eps eps)
    1.0)
   eps)
  (* (sin (+ (+ x eps) (/ PI 2.0))) (cos x))))

C

double code(double x, double eps) {
	return (fma(((fma(-0.0001984126984126984, (eps * eps), 0.008333333333333333) * (eps * eps)) - 0.16666666666666666), (eps * eps), 1.0) * eps) / (sin(((x + eps) + (((double) M_PI) / 2.0))) * cos(x));
}

Julia

function code(x, eps)
	return Float64(Float64(fma(Float64(Float64(fma(-0.0001984126984126984, Float64(eps * eps), 0.008333333333333333) * Float64(eps * eps)) - 0.16666666666666666), Float64(eps * eps), 1.0) * eps) / Float64(sin(Float64(Float64(x + eps) + Float64(pi / 2.0))) * cos(x)))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(eps * eps), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision] / N[(N[Sin[N[(N[(x + eps), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. +-commutative N/A

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

   5. quot-tan N/A

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

   6. lift-tan.f64 N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

4. Applied rewrites 62.0%

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

5. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{\varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{\varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

7. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \color{blue}{\left(\varepsilon + x\right)} \cdot \cos x} \]

   2. lift-cos.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\color{blue}{\cos \left(\varepsilon + x\right)} \cdot \cos x} \]

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

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\color{blue}{\sin \left(\left(\varepsilon + x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \cos x} \]

   4. lower-sin.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\color{blue}{\sin \left(\left(\varepsilon + x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \cos x} \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\sin \color{blue}{\left(\left(\varepsilon + x\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \cos x} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\sin \left(\color{blue}{\left(x + \varepsilon\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos x} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\sin \left(\color{blue}{\left(x + \varepsilon\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \cos x} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{5040}, \varepsilon \cdot \varepsilon, \frac{1}{120}\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\sin \left(\left(x + \varepsilon\right) + \color{blue}{\frac{\mathsf{PI}\left(\right)}{2}}\right) \cdot \cos x} \]

   9. lift-PI.f64 99.7

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\sin \left(\left(x + \varepsilon\right) + \frac{\color{blue}{\pi}}{2}\right) \cdot \cos x} \]

9. Applied rewrites 99.7%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\color{blue}{\sin \left(\left(x + \varepsilon\right) + \frac{\pi}{2}\right)} \cdot \cos x} \]
10. Add Preprocessing

Alternative 5: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \cos x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (*
   (fma
    (-
     (*
      (fma -0.0001984126984126984 (* eps eps) 0.008333333333333333)
      (* eps eps))
     0.16666666666666666)
    (* eps eps)
    1.0)
   eps)
  (* (cos (+ eps x)) (cos x))))

C

double code(double x, double eps) {
	return (fma(((fma(-0.0001984126984126984, (eps * eps), 0.008333333333333333) * (eps * eps)) - 0.16666666666666666), (eps * eps), 1.0) * eps) / (cos((eps + x)) * cos(x));
}

Julia

function code(x, eps)
	return Float64(Float64(fma(Float64(Float64(fma(-0.0001984126984126984, Float64(eps * eps), 0.008333333333333333) * Float64(eps * eps)) - 0.16666666666666666), Float64(eps * eps), 1.0) * eps) / Float64(cos(Float64(eps + x)) * cos(x)))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(eps * eps), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision] / N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. +-commutative N/A

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

   5. quot-tan N/A

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

   6. lift-tan.f64 N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

4. Applied rewrites 62.0%

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

5. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{\varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left(1 + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {\varepsilon}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{\varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

7. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, \varepsilon \cdot \varepsilon, 0.008333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
8. Add Preprocessing

Alternative 6: 99.7% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (*
   (fma
    (- (* 0.008333333333333333 (* eps eps)) 0.16666666666666666)
    (* eps eps)
    1.0)
   eps)
  (* (cos (+ eps x)) (cos x))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. +-commutative N/A

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

   5. quot-tan N/A

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

   6. lift-tan.f64 N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

4. Applied rewrites 62.0%

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

5. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{\varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{\varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
8. Add Preprocessing

Alternative 7: 99.6% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (* (fma -0.16666666666666666 (* eps eps) 1.0) eps)
  (* (sin (- (+ eps (* 0.5 PI)) (* -1.0 x))) (cos x))))

C

double code(double x, double eps) {
	return (fma(-0.16666666666666666, (eps * eps), 1.0) * eps) / (sin(((eps + (0.5 * ((double) M_PI))) - (-1.0 * x))) * cos(x));
}

Julia

function code(x, eps)
	return Float64(Float64(fma(-0.16666666666666666, Float64(eps * eps), 1.0) * eps) / Float64(sin(Float64(Float64(eps + Float64(0.5 * pi)) - Float64(-1.0 * x))) * cos(x)))
end

Wolfram

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

Derivation

1. Initial program 61.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. +-commutative N/A

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

   5. quot-tan N/A

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

   6. lift-tan.f64 N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

4. Applied rewrites 62.0%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-cos.f64 N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-PI.f64 99.6

      \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\sin \left(\left(x + \varepsilon\right) + \frac{\color{blue}{\pi}}{2}\right) \cdot \cos x} \]

9. Applied rewrites 99.6%

   \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\color{blue}{\sin \left(\left(x + \varepsilon\right) + \frac{\pi}{2}\right)} \cdot \cos x} \]

10. Taylor expanded in x around -inf

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

    1. sin-+PI/2 N/A

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

    2. cos-sum-rev N/A

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

    3. lower-sin.f64 N/A

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

    4. lower--.f64 N/A

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

    5. lower-+.f64 N/A

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

    6. lower-*.f64 N/A

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

    7. lift-PI.f64 N/A

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

    8. lower-*.f64 99.6

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

12. Applied rewrites 99.6%

    \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\color{blue}{\sin \left(\left(\varepsilon + 0.5 \cdot \pi\right) - -1 \cdot x\right)} \cdot \cos x} \]
13. Add Preprocessing

Alternative 8: 99.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (* (fma -0.16666666666666666 (* eps eps) 1.0) eps)
  (* (cos (+ eps x)) (cos x))))

C

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

Julia

function code(x, eps)
	return Float64(Float64(fma(-0.16666666666666666, Float64(eps * eps), 1.0) * eps) / Float64(cos(Float64(eps + x)) * cos(x)))
end

Wolfram

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

Derivation

1. Initial program 61.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. +-commutative N/A

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

   5. quot-tan N/A

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

   6. lift-tan.f64 N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

4. Applied rewrites 62.0%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

Alternative 9: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. +-commutative N/A

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

   5. quot-tan N/A

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

   6. lift-tan.f64 N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

4. Applied rewrites 62.0%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-cos.f64 N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-PI.f64 99.6

      \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\sin \left(\left(x + \varepsilon\right) + \frac{\color{blue}{\pi}}{2}\right) \cdot \cos x} \]

9. Applied rewrites 99.6%

   \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\color{blue}{\sin \left(\left(x + \varepsilon\right) + \frac{\pi}{2}\right)} \cdot \cos x} \]

10. Taylor expanded in eps around 0

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

12. Applied rewrites 99.4%

    \[\leadsto \frac{\varepsilon}{\sin \left(\left(x + \varepsilon\right) + \frac{\pi}{2}\right) \cdot \cos x} \]
13. Add Preprocessing

Alternative 10: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps / (cos((eps + x)) * 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((eps + x)) * cos(x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. +-commutative N/A

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

   5. quot-tan N/A

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

   6. lift-tan.f64 N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

4. Applied rewrites 62.0%

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

5. Taylor expanded in eps around 0

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

7. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{\varepsilon}}{\cos \left(\varepsilon + x\right) \cdot \cos x} \]
8. Add Preprocessing

Alternative 11: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{{\cos x}^{2}} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ eps (pow (cos x) 2.0)))

C

double code(double x, double eps) {
	return eps / pow(cos(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 / (cos(x) ** 2.0d0)
end function

Java

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

Python

def code(x, eps):
	return eps / math.pow(math.cos(x), 2.0)

Julia

function code(x, eps)
	return Float64(eps / (cos(x) ^ 2.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps / (cos(x) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 61.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. +-commutative N/A

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

   5. quot-tan N/A

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

   6. lift-tan.f64 N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

4. Applied rewrites 62.0%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{\varepsilon}{{\cos x}^{2}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\varepsilon}{\color{blue}{{\cos x}^{2}}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{\varepsilon}{{\cos x}^{\color{blue}{2}}} \]

   3. lift-cos.f64 98.9

      \[\leadsto \frac{\varepsilon}{{\cos x}^{2}} \]

7. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\frac{\varepsilon}{{\cos x}^{2}}} \]
8. Add Preprocessing

Alternative 12: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. mul-1-neg N/A

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

   5. unpow2 N/A

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

   6. unpow2 N/A

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

   7. frac-times N/A

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

   8. tan-quot N/A

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

   9. tan-quot N/A

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

   10. lower-neg.f64 N/A

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

   11. pow2 N/A

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

   12. lower-pow.f64 N/A

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

   13. lift-tan.f64 98.9

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

5. Applied rewrites 98.9%

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

Alternative 13: 98.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (* (fma -0.16666666666666666 (* eps eps) 1.0) eps)
  (*
   (cos (+ eps x))
   (fma
    (-
     (* (fma -0.001388888888888889 (* x x) 0.041666666666666664) (* x x))
     0.5)
    (* x x)
    1.0))))

C

double code(double x, double eps) {
	return (fma(-0.16666666666666666, (eps * eps), 1.0) * eps) / (cos((eps + x)) * fma(((fma(-0.001388888888888889, (x * x), 0.041666666666666664) * (x * x)) - 0.5), (x * x), 1.0));
}

Julia

function code(x, eps)
	return Float64(Float64(fma(-0.16666666666666666, Float64(eps * eps), 1.0) * eps) / Float64(cos(Float64(eps + x)) * fma(Float64(Float64(fma(-0.001388888888888889, Float64(x * x), 0.041666666666666664) * Float64(x * x)) - 0.5), Float64(x * x), 1.0)))
end

Wolfram

code[x_, eps_] := N[(N[(N[(-0.16666666666666666 * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision] / N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. +-commutative N/A

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

   5. quot-tan N/A

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

   6. lift-tan.f64 N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

4. Applied rewrites 62.0%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + \color{blue}{1}\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \left(\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2} + 1\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, \color{blue}{{x}^{2}}, 1\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)} \]

   7. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {x}^{2}, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot \color{blue}{x}, 1\right)} \]

   14. lower-*.f64 98.5

       \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot \color{blue}{x}, 1\right)} \]

10. Applied rewrites 98.5%

    \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}} \]
11. Add Preprocessing

Alternative 14: 98.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. +-commutative N/A

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

   5. quot-tan N/A

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

   6. lift-tan.f64 N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

4. Applied rewrites 62.0%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 98.4

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

10. Applied rewrites 98.4%

    \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \color{blue}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}} \]
11. Add Preprocessing

Alternative 15: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  eps
  (*
   (cos (+ eps x))
   (fma
    (-
     (* (fma -0.001388888888888889 (* x x) 0.041666666666666664) (* x x))
     0.5)
    (* x x)
    1.0))))

C

double code(double x, double eps) {
	return eps / (cos((eps + x)) * fma(((fma(-0.001388888888888889, (x * x), 0.041666666666666664) * (x * x)) - 0.5), (x * x), 1.0));
}

Julia

function code(x, eps)
	return Float64(eps / Float64(cos(Float64(eps + x)) * fma(Float64(Float64(fma(-0.001388888888888889, Float64(x * x), 0.041666666666666664) * Float64(x * x)) - 0.5), Float64(x * x), 1.0)))
end

Wolfram

code[x_, eps_] := N[(eps / N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. +-commutative N/A

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

   5. quot-tan N/A

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

   6. lift-tan.f64 N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

4. Applied rewrites 62.0%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + \color{blue}{1}\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \left(\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2} + 1\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, \color{blue}{{x}^{2}}, 1\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)} \]

   7. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, {x}^{2}, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot \color{blue}{x}, 1\right)} \]

   14. lower-*.f64 98.5

       \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot \color{blue}{x}, 1\right)} \]

10. Applied rewrites 98.5%

    \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}} \]

11. Taylor expanded in eps around 0

    \[\leadsto \frac{\varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right) \cdot \left(x \cdot x\right) - \frac{1}{2}, x \cdot x, 1\right)} \]
12. Step-by-step derivation

13. Applied rewrites 98.5%

    \[\leadsto \frac{\varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)} \]
14. Add Preprocessing

Alternative 16: 98.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. +-commutative N/A

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

   5. quot-tan N/A

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

   6. lift-tan.f64 N/A

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

   7. tan-quot N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. sin-diff-rev N/A

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

   13. lower-sin.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-*.f64 N/A

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

4. Applied rewrites 62.0%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 98.2

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

10. Applied rewrites 98.2%

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

Alternative 17: 98.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, 1\right), x, \mathsf{fma}\left(0.6666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right) + 1\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+
   (fma
    (fma
     (fma 1.3333333333333333 (* eps eps) 1.0)
     x
     (* (fma 0.6666666666666666 (* eps eps) 1.0) eps))
    x
    (* (* eps eps) 0.3333333333333333))
   1.0)
  eps))

C

double code(double x, double eps) {
	return (fma(fma(fma(1.3333333333333333, (eps * eps), 1.0), x, (fma(0.6666666666666666, (eps * eps), 1.0) * eps)), x, ((eps * eps) * 0.3333333333333333)) + 1.0) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(fma(1.3333333333333333, Float64(eps * eps), 1.0), x, Float64(fma(0.6666666666666666, Float64(eps * eps), 1.0) * eps)), x, Float64(Float64(eps * eps) * 0.3333333333333333)) + 1.0) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(1.3333333333333333 * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] * x + N[(N[(0.6666666666666666 * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot {\sin x}^{2}}{{\cos x}^{2}}, -1, \mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \left(\mathsf{fma}\left(\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot {\sin x}^{2}}{{\cos x}^{2}}, -1, \mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right)\right) + 0.16666666666666666\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 98.2%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, 1\right), x, \mathsf{fma}\left(0.6666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right) + 1\right) \cdot \varepsilon \]
9. Add Preprocessing

Alternative 18: 98.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(x + \varepsilon \cdot \left(1 + \mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot x, 1.3333333333333333 \cdot \left(x \cdot x\right)\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right) + 1\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+
   (fma
    (+
     x
     (*
      eps
      (+
       1.0
       (fma 1.3333333333333333 (* eps x) (* 1.3333333333333333 (* x x))))))
    x
    (* (* eps eps) 0.3333333333333333))
   1.0)
  eps))

C

double code(double x, double eps) {
	return (fma((x + (eps * (1.0 + fma(1.3333333333333333, (eps * x), (1.3333333333333333 * (x * x)))))), x, ((eps * eps) * 0.3333333333333333)) + 1.0) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(Float64(x + Float64(eps * Float64(1.0 + fma(1.3333333333333333, Float64(eps * x), Float64(1.3333333333333333 * Float64(x * x)))))), x, Float64(Float64(eps * eps) * 0.3333333333333333)) + 1.0) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(x + N[(eps * N[(1.0 + N[(1.3333333333333333 * N[(eps * x), $MachinePrecision] + N[(1.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot {\sin x}^{2}}{{\cos x}^{2}}, -1, \mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \left(\mathsf{fma}\left(\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot {\sin x}^{2}}{{\cos x}^{2}}, -1, \mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right)\right) + 0.16666666666666666\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\frac{1}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]

8. Applied rewrites 97.7%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot x, \varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right) + 1\right) \cdot \varepsilon \]

9. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \left(\frac{4}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]

11. Applied rewrites 98.2%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.8888888888888888, \varepsilon \cdot \varepsilon, 1.3333333333333333\right) \cdot x, \varepsilon, 1.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + 1, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666, 1\right) \cdot \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right) + 1\right) \cdot \varepsilon \]

12. Taylor expanded in eps around 0

    \[\leadsto \left(\mathsf{fma}\left(x + \varepsilon \cdot \left(1 + \left(\frac{4}{3} \cdot \left(\varepsilon \cdot x\right) + \frac{4}{3} \cdot {x}^{2}\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3}\right) + 1\right) \cdot \varepsilon \]
13. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x + \varepsilon \cdot \left(1 + \left(\frac{4}{3} \cdot \left(\varepsilon \cdot x\right) + \frac{4}{3} \cdot {x}^{2}\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3}\right) + 1\right) \cdot \varepsilon \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x + \varepsilon \cdot \left(1 + \left(\frac{4}{3} \cdot \left(\varepsilon \cdot x\right) + \frac{4}{3} \cdot {x}^{2}\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3}\right) + 1\right) \cdot \varepsilon \]

    3. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x + \varepsilon \cdot \left(1 + \left(\frac{4}{3} \cdot \left(\varepsilon \cdot x\right) + \frac{4}{3} \cdot {x}^{2}\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3}\right) + 1\right) \cdot \varepsilon \]

    4. lower-fma.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x + \varepsilon \cdot \left(1 + \mathsf{fma}\left(\frac{4}{3}, \varepsilon \cdot x, \frac{4}{3} \cdot {x}^{2}\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3}\right) + 1\right) \cdot \varepsilon \]

    5. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x + \varepsilon \cdot \left(1 + \mathsf{fma}\left(\frac{4}{3}, \varepsilon \cdot x, \frac{4}{3} \cdot {x}^{2}\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3}\right) + 1\right) \cdot \varepsilon \]

    6. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x + \varepsilon \cdot \left(1 + \mathsf{fma}\left(\frac{4}{3}, \varepsilon \cdot x, \frac{4}{3} \cdot {x}^{2}\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3}\right) + 1\right) \cdot \varepsilon \]

    7. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(x + \varepsilon \cdot \left(1 + \mathsf{fma}\left(\frac{4}{3}, \varepsilon \cdot x, \frac{4}{3} \cdot \left(x \cdot x\right)\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3}\right) + 1\right) \cdot \varepsilon \]

    8. lift-*.f64 98.2

       \[\leadsto \left(\mathsf{fma}\left(x + \varepsilon \cdot \left(1 + \mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot x, 1.3333333333333333 \cdot \left(x \cdot x\right)\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right) + 1\right) \cdot \varepsilon \]

14. Applied rewrites 98.2%

    \[\leadsto \left(\mathsf{fma}\left(x + \varepsilon \cdot \left(1 + \mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot x, 1.3333333333333333 \cdot \left(x \cdot x\right)\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right) + 1\right) \cdot \varepsilon \]
15. Add Preprocessing

Alternative 19: 98.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(x + \varepsilon \cdot \left(1 + 1.3333333333333333 \cdot \left(x \cdot x\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right) + 1\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (+
   (fma
    (+ x (* eps (+ 1.0 (* 1.3333333333333333 (* x x)))))
    x
    (* (* eps eps) 0.3333333333333333))
   1.0)
  eps))

C

double code(double x, double eps) {
	return (fma((x + (eps * (1.0 + (1.3333333333333333 * (x * x))))), x, ((eps * eps) * 0.3333333333333333)) + 1.0) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(Float64(x + Float64(eps * Float64(1.0 + Float64(1.3333333333333333 * Float64(x * x))))), x, Float64(Float64(eps * eps) * 0.3333333333333333)) + 1.0) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(x + N[(eps * N[(1.0 + N[(1.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot {\sin x}^{2}}{{\cos x}^{2}}, -1, \mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \left(\mathsf{fma}\left(\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot {\sin x}^{2}}{{\cos x}^{2}}, -1, \mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right)\right) + 0.16666666666666666\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\frac{1}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]

8. Applied rewrites 97.7%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot x, \varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right) + 1\right) \cdot \varepsilon \]

9. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \left(\frac{4}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]

11. Applied rewrites 98.2%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.8888888888888888, \varepsilon \cdot \varepsilon, 1.3333333333333333\right) \cdot x, \varepsilon, 1.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + 1, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666, 1\right) \cdot \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right) + 1\right) \cdot \varepsilon \]

12. Taylor expanded in eps around 0

    \[\leadsto \left(\mathsf{fma}\left(x + \varepsilon \cdot \left(1 + \frac{4}{3} \cdot {x}^{2}\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3}\right) + 1\right) \cdot \varepsilon \]
13. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x + \varepsilon \cdot \left(1 + \frac{4}{3} \cdot {x}^{2}\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3}\right) + 1\right) \cdot \varepsilon \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x + \varepsilon \cdot \left(1 + \frac{4}{3} \cdot {x}^{2}\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3}\right) + 1\right) \cdot \varepsilon \]

    3. lower-+.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x + \varepsilon \cdot \left(1 + \frac{4}{3} \cdot {x}^{2}\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3}\right) + 1\right) \cdot \varepsilon \]

    4. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x + \varepsilon \cdot \left(1 + \frac{4}{3} \cdot {x}^{2}\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3}\right) + 1\right) \cdot \varepsilon \]

    5. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(x + \varepsilon \cdot \left(1 + \frac{4}{3} \cdot \left(x \cdot x\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{3}\right) + 1\right) \cdot \varepsilon \]

    6. lift-*.f64 98.2

       \[\leadsto \left(\mathsf{fma}\left(x + \varepsilon \cdot \left(1 + 1.3333333333333333 \cdot \left(x \cdot x\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right) + 1\right) \cdot \varepsilon \]

14. Applied rewrites 98.2%

    \[\leadsto \left(\mathsf{fma}\left(x + \varepsilon \cdot \left(1 + 1.3333333333333333 \cdot \left(x \cdot x\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right) + 1\right) \cdot \varepsilon \]
15. Add Preprocessing

Alternative 20: 98.2% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + x \cdot \left(\varepsilon + x\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lift-+.f64 98.2

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

8. Applied rewrites 98.2%

   \[\leadsto \mathsf{fma}\left(\varepsilon + x, x, 1\right) \cdot \varepsilon \]
9. Add Preprocessing

Alternative 21: 98.1% accurate, 14.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-\varepsilon\right) \cdot \mathsf{fma}\left(\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}, -0.5, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot {\sin x}^{2}}{{\cos x}^{2}}, -1, \mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right)\right) + 0.16666666666666666, \sin x, 0.16666666666666666 \cdot \left(\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x\right)\right)}{\cos x}\right) - \left(\mathsf{fma}\left(\frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot {\sin x}^{2}}{{\cos x}^{2}}, -1, \mathsf{fma}\left(1 - \left(-{\tan x}^{2}\right), -0.5, {\tan x}^{2} \cdot 0.16666666666666666\right)\right) + 0.16666666666666666\right), \varepsilon, 1 \cdot \frac{\left(1 - \left(-{\tan x}^{2}\right)\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(\frac{1}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right)\right)\right) + 1\right) \cdot \varepsilon \]

8. Applied rewrites 97.7%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, \varepsilon \cdot \varepsilon, 1\right) \cdot x, \varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right) + 1\right) \cdot \varepsilon \]

9. Taylor expanded in x around 0

   \[\leadsto \left(1 + \left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \left(\frac{4}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right)\right)\right) \cdot \varepsilon \]

11. Applied rewrites 98.2%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.8888888888888888, \varepsilon \cdot \varepsilon, 1.3333333333333333\right) \cdot x, \varepsilon, 1.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + 1, x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.6666666666666666, 1\right) \cdot \varepsilon\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right) + 1\right) \cdot \varepsilon \]

12. Taylor expanded in eps around 0

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

    1. pow2 N/A

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

    2. lift-*.f64 98.1

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

14. Applied rewrites 98.1%

    \[\leadsto \left(x \cdot x + 1\right) \cdot \varepsilon \]
15. Add Preprocessing

Alternative 22: 97.7% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon + \color{blue}{{\varepsilon}^{2} \cdot x} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto {\varepsilon}^{2} \cdot x + \varepsilon \]

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 97.7

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

8. Applied rewrites 97.7%

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

Alternative 23: 97.7% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 61.9%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.7%

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

Developer Target 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (+ eps (* (* eps (tan x)) (tan x))))

C

double code(double x, double eps) {
	return eps + ((eps * tan(x)) * tan(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 + ((eps * tan(x)) * tan(x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025089 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (+ eps (* eps (tan x) (tan x))))

  (- (tan (+ x eps)) (tan x)))