2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.9%

Time: 8.3s

Alternatives: 20

Speedup: 76.4×

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: 62.5% 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: 99.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   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-2neg N/A

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

   9. sin-neg-rev N/A

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

   10. mul-1-neg N/A

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

   11. frac-sub N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 62.5%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 99.9

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

6. Applied rewrites 99.9%

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

   1. lift-+.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. cos-sum N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lower-sin.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. negate-sub N/A

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

    4. *-commutative N/A

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

    5. metadata-eval N/A

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

    6. lower-fma.f64 N/A

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

    7. +-commutative N/A

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

    8. lower-fma.f64 N/A

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

    9. pow2 N/A

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

    10. lift-*.f64 N/A

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

    11. pow2 N/A

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

    12. lift-*.f64 99.7

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

11. Applied rewrites 99.7%

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

Alternative 2: 99.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   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-2neg N/A

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

   9. sin-neg-rev N/A

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

   10. mul-1-neg N/A

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

   11. frac-sub N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 62.5%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 99.9

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

6. Applied rewrites 99.9%

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

   1. lift-+.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. cos-sum N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lower-sin.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in eps around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. negate-sub N/A

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

    5. *-commutative N/A

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

    6. metadata-eval N/A

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

    7. lower-fma.f64 N/A

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

    8. pow2 N/A

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

    9. lift-*.f64 N/A

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

    10. pow2 N/A

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

    11. lift-*.f64 99.7

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

11. Applied rewrites 99.7%

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

Alternative 3: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   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-2neg N/A

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

   9. sin-neg-rev N/A

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

   10. mul-1-neg N/A

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

   11. frac-sub N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 62.5%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 99.9

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

6. Applied rewrites 99.9%

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

   1. lift-+.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. cos-sum N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lower-sin.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. pow2 N/A

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

    7. lift-*.f64 99.7

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

11. Applied rewrites 99.7%

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

Alternative 4: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   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-2neg N/A

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

   9. sin-neg-rev N/A

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

   10. mul-1-neg N/A

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

   11. frac-sub N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 62.5%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 99.9

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

6. Applied rewrites 99.9%

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

   1. lift-cos.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. cos-+PI-rev N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-PI.f64 99.9

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

8. Applied rewrites 99.9%

   \[\leadsto \frac{-1 \cdot \sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \color{blue}{\cos \left(x + \pi\right)}} \]
9. Add Preprocessing

Alternative 5: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   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-2neg N/A

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

   9. sin-neg-rev N/A

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

   10. mul-1-neg N/A

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

   11. frac-sub N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 62.5%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 99.9

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

6. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

8. Applied rewrites 99.9%

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

Alternative 6: 99.7% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (*
   (fma
    (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(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(fma(-0.008333333333333333, Float64(eps * eps), 0.16666666666666666), Float64(eps * eps), -1.0) * eps) / Float64(cos(Float64(eps + x)) * Float64(-cos(x))))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(-0.008333333333333333 * N[(eps * eps), $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 62.5%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   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-2neg N/A

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

   9. sin-neg-rev N/A

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

   10. mul-1-neg N/A

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

   11. frac-sub N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 62.5%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. negate-sub N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 99.7

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

9. Applied rewrites 99.7%

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

Alternative 7: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   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-2neg N/A

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

   9. sin-neg-rev N/A

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

   10. mul-1-neg N/A

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

   11. frac-sub N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 62.5%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. negate-sub N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 99.6

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

9. Applied rewrites 99.6%

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

Alternative 8: 99.3% accurate, 1.0× speedup

Math

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

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(Float64(-eps) / Float64(cos(Float64(eps + x)) * Float64(-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 62.5%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   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-2neg N/A

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

   9. sin-neg-rev N/A

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

   10. mul-1-neg N/A

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

   11. frac-sub N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 62.5%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in eps around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.3

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

9. Applied rewrites 99.3%

   \[\leadsto \frac{-\varepsilon}{\cos \left(\varepsilon + x\right) \cdot \left(-\cos x\right)} \]
10. Add Preprocessing

Alternative 9: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (- (fma (- x (* -0.3333333333333333 eps)) eps 1.0) (- (pow (tan x) 2.0)))
  eps))

C

double code(double x, double eps) {
	return (fma((x - (-0.3333333333333333 * eps)), eps, 1.0) - -pow(tan(x), 2.0)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(Float64(x - Float64(-0.3333333333333333 * eps)), eps, 1.0) - Float64(-(tan(x) ^ 2.0))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(x - N[(-0.3333333333333333 * eps), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] - (-N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 99.0

      \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

7. Applied rewrites 99.0%

   \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]
8. Add Preprocessing

Alternative 10: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. quot-tan N/A

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

   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-2neg N/A

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

   9. sin-neg-rev N/A

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

   10. mul-1-neg N/A

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

   11. frac-sub N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 62.5%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in x around 0

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

   1. negate-sub N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 98.5

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

9. Applied rewrites 98.5%

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

10. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. negate-sub N/A

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

    4. *-commutative N/A

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

    5. metadata-eval N/A

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

    6. lower-fma.f64 N/A

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

    7. pow2 N/A

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

    8. lift-*.f64 98.5

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

12. Applied rewrites 98.5%

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

Alternative 11: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.3333333333333333, x, 1.3333333333333333 \cdot \varepsilon\right), x, 1\right), x, 0.3333333333333333 \cdot \varepsilon\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(0.37777777777777777, x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (fma
    (fma
     (fma (fma 1.3333333333333333 x (* 1.3333333333333333 eps)) x 1.0)
     x
     (* 0.3333333333333333 eps))
    eps
    1.0)
   (-
    (*
     (fma (fma 0.37777777777777777 (* x x) 0.6666666666666666) (* x x) 1.0)
     (* x x))))
  eps))

C

double code(double x, double eps) {
	return (fma(fma(fma(fma(1.3333333333333333, x, (1.3333333333333333 * eps)), x, 1.0), x, (0.3333333333333333 * eps)), eps, 1.0) - -(fma(fma(0.37777777777777777, (x * x), 0.6666666666666666), (x * x), 1.0) * (x * x))) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(fma(fma(1.3333333333333333, x, Float64(1.3333333333333333 * eps)), x, 1.0), x, Float64(0.3333333333333333 * eps)), eps, 1.0) - Float64(-Float64(fma(fma(0.37777777777777777, Float64(x * x), 0.6666666666666666), Float64(x * x), 1.0) * Float64(x * x)))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(1.3333333333333333 * x + N[(1.3333333333333333 * eps), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + N[(0.3333333333333333 * eps), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] - (-N[(N[(N[(0.37777777777777777 * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 99.0

      \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

7. Applied rewrites 99.0%

   \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   3. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right) + 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   4. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\left(\left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}, {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   6. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\frac{17}{45} \cdot {x}^{2} + \frac{2}{3}, {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   7. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, {x}^{2}, \frac{2}{3}\right), {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   8. pow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   10. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   11. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   12. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

   13. lift-*.f64 98.5

       \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(0.37777777777777777, x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

10. Applied rewrites 98.5%

    \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(0.37777777777777777, x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

11. Taylor expanded in x around 0

    \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + x \cdot \left(\frac{4}{3} \cdot x - \frac{-4}{3} \cdot \varepsilon\right)\right) - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]
12. Step-by-step derivation

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

       \[\leadsto \left(\mathsf{fma}\left(x \cdot \left(1 + x \cdot \left(\frac{4}{3} \cdot x - \frac{-4}{3} \cdot \varepsilon\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right) \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

    2. *-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(1 + x \cdot \left(\frac{4}{3} \cdot x - \frac{-4}{3} \cdot \varepsilon\right)\right) \cdot x + \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right) \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

    3. metadata-eval N/A

       \[\leadsto \left(\mathsf{fma}\left(\left(1 + x \cdot \left(\frac{4}{3} \cdot x - \frac{-4}{3} \cdot \varepsilon\right)\right) \cdot x + \frac{1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

    4. lower-fma.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(1 + x \cdot \left(\frac{4}{3} \cdot x - \frac{-4}{3} \cdot \varepsilon\right), x, \frac{1}{3} \cdot \varepsilon\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

13. Applied rewrites 98.5%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.3333333333333333, x, 1.3333333333333333 \cdot \varepsilon\right), x, 1\right), x, 0.3333333333333333 \cdot \varepsilon\right), \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(0.37777777777777777, x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]
14. Add Preprocessing

Alternative 12: 98.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.19682539682539682, x \cdot x, -0.37777777777777777\right), x \cdot x, -0.6666666666666666\right), x \cdot x, -1\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (fma (- x (* -0.3333333333333333 eps)) eps 1.0)
   (*
    (fma
     (fma
      (fma -0.19682539682539682 (* x x) -0.37777777777777777)
      (* x x)
      -0.6666666666666666)
     (* x x)
     -1.0)
    (* x x)))
  eps))

C

double code(double x, double eps) {
	return (fma((x - (-0.3333333333333333 * eps)), eps, 1.0) - (fma(fma(fma(-0.19682539682539682, (x * x), -0.37777777777777777), (x * x), -0.6666666666666666), (x * x), -1.0) * (x * x))) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(Float64(x - Float64(-0.3333333333333333 * eps)), eps, 1.0) - Float64(fma(fma(fma(-0.19682539682539682, Float64(x * x), -0.37777777777777777), Float64(x * x), -0.6666666666666666), Float64(x * x), -1.0) * Float64(x * x))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(x - N[(-0.3333333333333333 * eps), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] - N[(N[(N[(N[(-0.19682539682539682 * N[(x * x), $MachinePrecision] + -0.37777777777777777), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 99.0

      \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

7. Applied rewrites 99.0%

   \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   3. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right) + 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   4. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\left(\left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}, {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   6. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\frac{17}{45} \cdot {x}^{2} + \frac{2}{3}, {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   7. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, {x}^{2}, \frac{2}{3}\right), {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   8. pow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   10. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   11. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   12. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

   13. lift-*.f64 98.5

       \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(0.37777777777777777, x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

10. Applied rewrites 98.5%

    \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(0.37777777777777777, x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

11. Taylor expanded in x around 0

    \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - {x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right)\right) \cdot \varepsilon \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right) \cdot {x}^{2}\right) \cdot \varepsilon \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-62}{315} \cdot {x}^{2} - \frac{17}{45}\right) - \frac{2}{3}\right) - 1\right) \cdot {x}^{2}\right) \cdot \varepsilon \]

13. Applied rewrites 98.5%

    \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.19682539682539682, x \cdot x, -0.37777777777777777\right), x \cdot x, -0.6666666666666666\right), x \cdot x, -1\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon \]
14. Add Preprocessing

Alternative 13: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(0.37777777777777777, x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (fma (- x (* -0.3333333333333333 eps)) eps 1.0)
   (-
    (*
     (fma (fma 0.37777777777777777 (* x x) 0.6666666666666666) (* x x) 1.0)
     (* x x))))
  eps))

C

double code(double x, double eps) {
	return (fma((x - (-0.3333333333333333 * eps)), eps, 1.0) - -(fma(fma(0.37777777777777777, (x * x), 0.6666666666666666), (x * x), 1.0) * (x * x))) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(Float64(x - Float64(-0.3333333333333333 * eps)), eps, 1.0) - Float64(-Float64(fma(fma(0.37777777777777777, Float64(x * x), 0.6666666666666666), Float64(x * x), 1.0) * Float64(x * x)))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(x - N[(-0.3333333333333333 * eps), $MachinePrecision]), $MachinePrecision] * eps + 1.0), $MachinePrecision] - (-N[(N[(N[(0.37777777777777777 * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 99.0

      \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

7. Applied rewrites 99.0%

   \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   3. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right) + 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   4. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\left(\left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}, {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   6. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\frac{17}{45} \cdot {x}^{2} + \frac{2}{3}, {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   7. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, {x}^{2}, \frac{2}{3}\right), {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   8. pow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   10. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   11. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   12. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

   13. lift-*.f64 98.5

       \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(0.37777777777777777, x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

10. Applied rewrites 98.5%

    \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(0.37777777777777777, x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]
11. Add Preprocessing

Alternative 14: 98.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 99.0

      \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

7. Applied rewrites 99.0%

   \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

   \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   3. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right) + 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   4. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\left(\left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right) \cdot {x}^{2} + 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}, {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   6. +-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\frac{17}{45} \cdot {x}^{2} + \frac{2}{3}, {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   7. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, {x}^{2}, \frac{2}{3}\right), {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   8. pow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), {x}^{2}, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   10. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   11. lift-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 1\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

   12. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(x - \frac{-1}{3} \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{17}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

   13. lift-*.f64 98.5

       \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(0.37777777777777777, x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

10. Applied rewrites 98.5%

    \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-\mathsf{fma}\left(\mathsf{fma}\left(0.37777777777777777, x \cdot x, 0.6666666666666666\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

11. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. negate-sub N/A

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

    4. metadata-eval N/A

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

    5. lower-fma.f64 N/A

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

    6. pow2 N/A

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

    7. lift-*.f64 N/A

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

    8. pow2 N/A

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

    9. lift-*.f64 98.4

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

13. Applied rewrites 98.4%

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

Alternative 15: 98.3% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

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

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

   3. lower--.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 98.3

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

10. Applied rewrites 98.3%

    \[\leadsto \mathsf{fma}\left(\varepsilon, 1 - -0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right), x \cdot \left(\left(x + \varepsilon\right) \cdot \varepsilon\right)\right) \]
11. Step-by-step derivation

    1. lift-fma.f64 N/A

       \[\leadsto \varepsilon \cdot \left(1 - \frac{-1}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + x \cdot \color{blue}{\left(\left(x + \varepsilon\right) \cdot \varepsilon\right)} \]

    2. lift--.f64 N/A

       \[\leadsto \varepsilon \cdot \left(1 - \frac{-1}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + x \cdot \left(\left(x + \color{blue}{\varepsilon}\right) \cdot \varepsilon\right) \]

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. *-commutative N/A

       \[\leadsto \left(1 - \frac{-1}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon + x \cdot \left(\color{blue}{\left(x + \varepsilon\right)} \cdot \varepsilon\right) \]

    6. pow2 N/A

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

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

       \[\leadsto \left(1 + \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon + x \cdot \left(\left(\color{blue}{x} + \varepsilon\right) \cdot \varepsilon\right) \]

    8. metadata-eval N/A

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

    9. lower-fma.f64 N/A

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

    10. +-commutative N/A

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

    11. *-commutative N/A

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

    12. lower-fma.f64 N/A

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

    13. pow2 N/A

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

    14. lift-*.f64 98.3

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

12. Applied rewrites 98.3%

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

Alternative 16: 98.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 99.0

      \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

7. Applied rewrites 99.0%

   \[\leadsto \left(\mathsf{fma}\left(x - -0.3333333333333333 \cdot \varepsilon, \varepsilon, 1\right) - \left(-{\tan x}^{2}\right)\right) \cdot \varepsilon \]

8. Taylor expanded in x around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 98.3

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

10. Applied rewrites 98.3%

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

Alternative 17: 98.2% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

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

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

   3. lower--.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 98.3

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

10. Applied rewrites 98.3%

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

11. Taylor expanded in eps around 0

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

13. Applied rewrites 98.2%

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

Alternative 18: 98.2% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

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

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

   3. lower--.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. pow2 N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 98.2

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

10. Applied rewrites 98.2%

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

Alternative 19: 98.2% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
3. 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

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 97.8%

   \[\leadsto \varepsilon \]

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 98.2

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

10. Applied rewrites 98.2%

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

Alternative 20: 97.8% accurate, 76.4× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 62.5%

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

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
3. 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

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 97.8%

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

Developer Target 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 62.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (- (/ (+ (tan x) (tan eps)) (- 1.0 (* (tan x) (tan eps)))) (tan x)))

C

double code(double x, double eps) {
	return ((tan(x) + tan(eps)) / (1.0 - (tan(x) * tan(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) + tan(eps)) / (1.0d0 - (tan(x) * tan(eps)))) - tan(x)
end function

Java

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

Python

def code(x, eps):
	return ((math.tan(x) + math.tan(eps)) / (1.0 - (math.tan(x) * math.tan(eps)))) - math.tan(x)

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 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 2025119 
(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 c (/ (sin eps) (* (cos x) (cos (+ x eps)))))

  :alt
  (! :herbie-platform c (- (/ (+ (tan x) (tan eps)) (- 1 (* (tan x) (tan eps)))) (tan x)))

  :alt
  (! :herbie-platform c (+ eps (* eps (tan x) (tan x))))

  (- (tan (+ x eps)) (tan x)))