2tan (problem 3.3.2)

Percentage Accurate: 62.2% → 99.7%

Time: 21.3s

Alternatives: 8

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    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: 62.2% 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

real(8) function code(x, eps)
    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: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin x}^{2}\\ t_1 := {\cos x}^{-2}\\ \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(\left(0.3333333333333333 - t\_1 \cdot \left(-0.3333333333333333 \cdot t\_0\right)\right) + \mathsf{fma}\left({\sin x}^{4}, {\cos x}^{-4}, t\_1 \cdot t\_0\right)\right) + \left(\tan x + {\tan x}^{3}\right)\right)\right) + \frac{t\_0}{{\cos x}^{2}}\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0)) (t_1 (pow (cos x) -2.0)))
   (*
    eps
    (+
     (+
      1.0
      (*
       eps
       (+
        (*
         eps
         (+
          (- 0.3333333333333333 (* t_1 (* -0.3333333333333333 t_0)))
          (fma (pow (sin x) 4.0) (pow (cos x) -4.0) (* t_1 t_0))))
        (+ (tan x) (pow (tan x) 3.0)))))
     (/ t_0 (pow (cos x) 2.0))))))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = pow(cos(x), -2.0);
	return eps * ((1.0 + (eps * ((eps * ((0.3333333333333333 - (t_1 * (-0.3333333333333333 * t_0))) + fma(pow(sin(x), 4.0), pow(cos(x), -4.0), (t_1 * t_0)))) + (tan(x) + pow(tan(x), 3.0))))) + (t_0 / pow(cos(x), 2.0)));
}

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = cos(x) ^ -2.0
	return Float64(eps * Float64(Float64(1.0 + Float64(eps * Float64(Float64(eps * Float64(Float64(0.3333333333333333 - Float64(t_1 * Float64(-0.3333333333333333 * t_0))) + fma((sin(x) ^ 4.0), (cos(x) ^ -4.0), Float64(t_1 * t_0)))) + Float64(tan(x) + (tan(x) ^ 3.0))))) + Float64(t_0 / (cos(x) ^ 2.0))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Cos[x], $MachinePrecision], -2.0], $MachinePrecision]}, N[(eps * N[(N[(1.0 + N[(eps * N[(N[(eps * N[(N[(0.3333333333333333 - N[(t$95$1 * N[(-0.3333333333333333 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] * N[Power[N[Cos[x], $MachinePrecision], -4.0], $MachinePrecision] + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 62.1%

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

   1. tan-sum 62.3%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   2. div-inv 62.3%

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

   3. fma-neg 62.3%

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

4. Applied egg-rr 62.3%

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

5. Taylor expanded in eps around 0 99.4%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

7. Applied egg-rr 99.4%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{{\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333 - \mathsf{fma}\left(-1, {\sin x}^{2} \cdot {\cos x}^{-2}, \mathsf{fma}\left({\sin x}^{2} \cdot {\cos x}^{-2}, -0.3333333333333333, -{\sin x}^{4} \cdot {\cos x}^{-4}\right)\right), -\mathsf{fma}\left(-1, {\tan x}^{3}, -\tan x\right)\right)\right)}^{1}}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
8. Step-by-step derivation

   1. unpow1 99.4%

      \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333 - \mathsf{fma}\left(-1, {\sin x}^{2} \cdot {\cos x}^{-2}, \mathsf{fma}\left({\sin x}^{2} \cdot {\cos x}^{-2}, -0.3333333333333333, -{\sin x}^{4} \cdot {\cos x}^{-4}\right)\right), -\mathsf{fma}\left(-1, {\tan x}^{3}, -\tan x\right)\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

9. Simplified 99.4%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333 - \left(\left(-0.3333333333333333 \cdot \left({\sin x}^{2} \cdot {\cos x}^{-2}\right) - {\sin x}^{4} \cdot {\cos x}^{-4}\right) - {\sin x}^{2} \cdot {\cos x}^{-2}\right), -\left(\left(-{\tan x}^{3}\right) - \tan x\right)\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
10. Step-by-step derivation

    1. pow1 99.4%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{{\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333 - \left(\left(-0.3333333333333333 \cdot \left({\sin x}^{2} \cdot {\cos x}^{-2}\right) - {\sin x}^{4} \cdot {\cos x}^{-4}\right) - {\sin x}^{2} \cdot {\cos x}^{-2}\right), -\left(\left(-{\tan x}^{3}\right) - \tan x\right)\right)\right)}^{1}}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

11. Applied egg-rr 99.4%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{{\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333 - \left(-0.3333333333333333 \cdot \left({\sin x}^{2} \cdot {\cos x}^{-2}\right) - \mathsf{fma}\left({\sin x}^{4}, {\cos x}^{-4}, {\sin x}^{2} \cdot {\cos x}^{-2}\right)\right), -\left(\left(-{\tan x}^{3}\right) - \tan x\right)\right)\right)}^{1}}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
12. Step-by-step derivation

    1. unpow1 99.4%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.3333333333333333 - \left(-0.3333333333333333 \cdot \left({\sin x}^{2} \cdot {\cos x}^{-2}\right) - \mathsf{fma}\left({\sin x}^{4}, {\cos x}^{-4}, {\sin x}^{2} \cdot {\cos x}^{-2}\right)\right), -\left(\left(-{\tan x}^{3}\right) - \tan x\right)\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

    2. fma-undefine 99.4%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(0.3333333333333333 - \left(-0.3333333333333333 \cdot \left({\sin x}^{2} \cdot {\cos x}^{-2}\right) - \mathsf{fma}\left({\sin x}^{4}, {\cos x}^{-4}, {\sin x}^{2} \cdot {\cos x}^{-2}\right)\right)\right) + \left(-\left(\left(-{\tan x}^{3}\right) - \tan x\right)\right)\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

    3. unsub-neg 99.4%

       \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(0.3333333333333333 - \left(-0.3333333333333333 \cdot \left({\sin x}^{2} \cdot {\cos x}^{-2}\right) - \mathsf{fma}\left({\sin x}^{4}, {\cos x}^{-4}, {\sin x}^{2} \cdot {\cos x}^{-2}\right)\right)\right) - \left(\left(-{\tan x}^{3}\right) - \tan x\right)\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

13. Simplified 99.4%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\left(0.3333333333333333 - {\cos x}^{-2} \cdot \left(-0.3333333333333333 \cdot {\sin x}^{2}\right)\right) + \mathsf{fma}\left({\sin x}^{4}, {\cos x}^{-4}, {\sin x}^{2} \cdot {\cos x}^{-2}\right)\right) - \left(\left(-{\tan x}^{3}\right) - \tan x\right)\right)}\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

14. Final simplification 99.4%

    \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(\left(0.3333333333333333 - {\cos x}^{-2} \cdot \left(-0.3333333333333333 \cdot {\sin x}^{2}\right)\right) + \mathsf{fma}\left({\sin x}^{4}, {\cos x}^{-4}, {\cos x}^{-2} \cdot {\sin x}^{2}\right)\right) + \left(\tan x + {\tan x}^{3}\right)\right)\right) + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]
15. Add Preprocessing

Alternative 2: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(1 + \varepsilon \cdot \left(\varepsilon \cdot 0.3333333333333333 + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (/ (pow (sin x) 2.0) (pow (cos x) 2.0))
   (+
    1.0
    (*
     eps
     (+
      (* eps 0.3333333333333333)
      (+ (/ (sin x) (cos x)) (/ (pow (sin x) 3.0) (pow (cos x) 3.0)))))))))

C

double code(double x, double eps) {
	return eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + (1.0 + (eps * ((eps * 0.3333333333333333) + ((sin(x) / cos(x)) + (pow(sin(x), 3.0) / pow(cos(x), 3.0)))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + (1.0d0 + (eps * ((eps * 0.3333333333333333d0) + ((sin(x) / cos(x)) + ((sin(x) ** 3.0d0) / (cos(x) ** 3.0d0)))))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) + (1.0 + (eps * ((eps * 0.3333333333333333) + ((Math.sin(x) / Math.cos(x)) + (Math.pow(Math.sin(x), 3.0) / Math.pow(Math.cos(x), 3.0)))))));
}

Python

def code(x, eps):
	return eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + (1.0 + (eps * ((eps * 0.3333333333333333) + ((math.sin(x) / math.cos(x)) + (math.pow(math.sin(x), 3.0) / math.pow(math.cos(x), 3.0)))))))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + Float64(1.0 + Float64(eps * Float64(Float64(eps * 0.3333333333333333) + Float64(Float64(sin(x) / cos(x)) + Float64((sin(x) ^ 3.0) / (cos(x) ^ 3.0))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + (1.0 + (eps * ((eps * 0.3333333333333333) + ((sin(x) / cos(x)) + ((sin(x) ^ 3.0) / (cos(x) ^ 3.0)))))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(eps * N[(N[(eps * 0.3333333333333333), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.1%

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

   1. tan-sum 62.3%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   2. div-inv 62.3%

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

   3. fma-neg 62.3%

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

4. Applied egg-rr 62.3%

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

5. Taylor expanded in eps around 0 99.4%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

6. Taylor expanded in x around 0 99.4%

   \[\leadsto \varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\color{blue}{0.3333333333333333 \cdot \varepsilon} - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \]

7. Final simplification 99.4%

   \[\leadsto \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(1 + \varepsilon \cdot \left(\varepsilon \cdot 0.3333333333333333 + \left(\frac{\sin x}{\cos x} + \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 3: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ \varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, t\_0 + {t\_0}^{3}, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin x) (cos x))))
   (*
    eps
    (+
     1.0
     (fma
      eps
      (+ t_0 (pow t_0 3.0))
      (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))))))

C

double code(double x, double eps) {
	double t_0 = sin(x) / cos(x);
	return eps * (1.0 + fma(eps, (t_0 + pow(t_0, 3.0)), (pow(sin(x), 2.0) / pow(cos(x), 2.0))));
}

Julia

function code(x, eps)
	t_0 = Float64(sin(x) / cos(x))
	return Float64(eps * Float64(1.0 + fma(eps, Float64(t_0 + (t_0 ^ 3.0)), Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(1.0 + N[(eps * N[(t$95$0 + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.1%

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

   1. tan-sum 62.3%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   2. div-inv 62.3%

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

   3. fma-neg 62.3%

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

4. Applied egg-rr 62.3%

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

   1. fma-neg 62.3%

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

   2. *-commutative 62.3%

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

   3. associate-*l/ 62.3%

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

   4. *-lft-identity 62.3%

      \[\leadsto \frac{\color{blue}{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \]

6. Simplified 62.3%

   \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \tan x} \]

7. Taylor expanded in eps around 0 99.3%

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

9. Simplified 99.3%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \mathsf{fma}\left(\varepsilon, \frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)} \]
10. Add Preprocessing

Alternative 4: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x}{\cos x}\\ \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(1 + \varepsilon \cdot \left(t\_0 + {t\_0}^{3}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (sin x) (cos x))))
   (*
    eps
    (+
     (/ (pow (sin x) 2.0) (pow (cos x) 2.0))
     (+ 1.0 (* eps (+ t_0 (pow t_0 3.0))))))))

C

double code(double x, double eps) {
	double t_0 = sin(x) / cos(x);
	return eps * ((pow(sin(x), 2.0) / pow(cos(x), 2.0)) + (1.0 + (eps * (t_0 + pow(t_0, 3.0)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = sin(x) / cos(x)
    code = eps * (((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)) + (1.0d0 + (eps * (t_0 + (t_0 ** 3.0d0)))))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin(x) / Math.cos(x);
	return eps * ((Math.pow(Math.sin(x), 2.0) / Math.pow(Math.cos(x), 2.0)) + (1.0 + (eps * (t_0 + Math.pow(t_0, 3.0)))));
}

Python

def code(x, eps):
	t_0 = math.sin(x) / math.cos(x)
	return eps * ((math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)) + (1.0 + (eps * (t_0 + math.pow(t_0, 3.0)))))

Julia

function code(x, eps)
	t_0 = Float64(sin(x) / cos(x))
	return Float64(eps * Float64(Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + Float64(1.0 + Float64(eps * Float64(t_0 + (t_0 ^ 3.0))))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = sin(x) / cos(x);
	tmp = eps * (((sin(x) ^ 2.0) / (cos(x) ^ 2.0)) + (1.0 + (eps * (t_0 + (t_0 ^ 3.0)))));
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(eps * N[(N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(eps * N[(t$95$0 + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.1%

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

   1. tan-sum 62.3%

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   2. div-inv 62.3%

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

   3. fma-neg 62.3%

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

4. Applied egg-rr 62.3%

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

5. Taylor expanded in eps around 0 99.4%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(1 + \varepsilon \cdot \left(\varepsilon \cdot \left(0.3333333333333333 - \left(-1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(-1 \cdot \frac{{\sin x}^{4}}{{\cos x}^{4}} + -0.3333333333333333 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)\right)\right) - \left(-1 \cdot \frac{\sin x}{\cos x} + -1 \cdot \frac{{\sin x}^{3}}{{\cos x}^{3}}\right)\right)\right) - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]

6. Taylor expanded in eps around 0 99.3%

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

   1. associate-*r* 99.3%

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

   2. distribute-lft-out 99.3%

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

   3. associate-*r* 99.3%

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

   4. mul-1-neg 99.3%

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

   5. distribute-lft-neg-in 99.3%

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

   6. *-commutative 99.3%

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

   7. mul-1-neg 99.3%

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

   8. remove-double-neg 99.3%

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

   9. cube-mult 99.3%

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

   10. unpow2 99.3%

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

   11. cube-mult 99.3%

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

   12. unpow2 99.3%

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

   13. times-frac 99.3%

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

   14. unpow2 99.3%

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

   15. unpow2 99.3%

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

8. Simplified 99.3%

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

9. Final simplification 99.3%

   \[\leadsto \varepsilon \cdot \left(\frac{{\sin x}^{2}}{{\cos x}^{2}} + \left(1 + \varepsilon \cdot \left(\frac{\sin x}{\cos x} + {\left(\frac{\sin x}{\cos x}\right)}^{3}\right)\right)\right) \]
10. Add Preprocessing

Alternative 5: 99.1% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ 1.0 (/ (pow (sin x) 2.0) (pow (cos x) 2.0)))))

C

double code(double x, double eps) {
	return eps * (1.0 + (pow(sin(x), 2.0) / pow(cos(x), 2.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + ((sin(x) ** 2.0d0) / (cos(x) ** 2.0d0)))
end function

Java

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

Python

def code(x, eps):
	return eps * (1.0 + (math.pow(math.sin(x), 2.0) / math.pow(math.cos(x), 2.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64((sin(x) ^ 2.0) / (cos(x) ^ 2.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + ((sin(x) ^ 2.0) / (cos(x) ^ 2.0)));
end

Wolfram

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

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in eps around 0 98.9%

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

   1. sub-neg 98.9%

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

   2. mul-1-neg 98.9%

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

   3. remove-double-neg 98.9%

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

5. Simplified 98.9%

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

Alternative 6: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\cos \varepsilon} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in x around 0 97.6%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
4. Add Preprocessing

Alternative 7: 60.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

   1. *-un-lft-identity 62.1%

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

   2. *-commutative 62.1%

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

   3. tan-quot 62.1%

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

   4. div-inv 62.1%

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

   5. prod-diff 62.1%

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

4. Applied egg-rr 62.1%

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

   1. fma-undefine 62.1%

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

   2. *-rgt-identity 62.1%

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

   3. +-commutative 62.1%

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

   4. fma-undefine 62.1%

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

   5. distribute-lft-neg-in 62.1%

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

   6. neg-mul-1 62.1%

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

   7. distribute-lft1-in 62.1%

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

   8. metadata-eval 62.1%

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

   9. associate-*l/ 62.1%

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

   10. *-lft-identity 62.1%

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

   11. mul0-lft 62.1%

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

   12. associate-+l+ 62.1%

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

6. Simplified 62.1%

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

7. Taylor expanded in x around 0 60.5%

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

Alternative 8: 7.8% accurate, 102.5× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return -x

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := (-x)

Derivation

1. Initial program 62.1%

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

   1. *-un-lft-identity 62.1%

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

   2. *-commutative 62.1%

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

   3. tan-quot 62.1%

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

   4. div-inv 62.1%

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

   5. prod-diff 62.1%

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

4. Applied egg-rr 62.1%

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

   1. fma-undefine 62.1%

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

   2. *-rgt-identity 62.1%

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

   3. +-commutative 62.1%

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

   4. fma-undefine 62.1%

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

   5. distribute-lft-neg-in 62.1%

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

   6. neg-mul-1 62.1%

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

   7. distribute-lft1-in 62.1%

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

   8. metadata-eval 62.1%

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

   9. associate-*l/ 62.1%

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

   10. *-lft-identity 62.1%

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

   11. mul0-lft 62.1%

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

   12. associate-+l+ 62.1%

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

6. Simplified 62.1%

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

7. Taylor expanded in x around 0 60.5%

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

8. Taylor expanded in x around inf 7.6%

   \[\leadsto \color{blue}{-1 \cdot x} \]
9. Step-by-step derivation

   1. neg-mul-1 7.6%

      \[\leadsto \color{blue}{-x} \]

10. Simplified 7.6%

    \[\leadsto \color{blue}{-x} \]
11. Add Preprocessing

Developer target: 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

real(8) function code(x, eps)
    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]

Reproduce

herbie shell --seed 2024106 
(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
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

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