2tan (problem 3.3.2)

Percentage Accurate: 62.8% → 99.7%

Time: 15.1s

Alternatives: 11

Speedup: 17.3×

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: 62.8% 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, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ t_1 := \frac{{\sin x}^{2}}{t\_0}\\ t_2 := -0.16666666666666666 - \left(\mathsf{fma}\left(t\_1, 0.16666666666666666, \mathsf{fma}\left(t\_1, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{t\_0}\right)}{t\_0}\right)\\ t_3 := \frac{\frac{{\sin x}^{3}}{t\_0} + \sin x}{\cos x}\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_2, \frac{\sin x}{\cos x}, t\_3 \cdot 0.3333333333333333\right), \varepsilon, t\_2\right), \varepsilon, t\_3\right), \varepsilon, t\_1\right), \varepsilon, \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0))
        (t_1 (/ (pow (sin x) 2.0) t_0))
        (t_2
         (-
          -0.16666666666666666
          (-
           (fma t_1 0.16666666666666666 (fma t_1 -0.5 -0.5))
           (/ (fma (sin x) (sin x) (/ (pow (sin x) 4.0) t_0)) t_0))))
        (t_3 (/ (+ (/ (pow (sin x) 3.0) t_0) (sin x)) (cos x))))
   (fma
    (fma
     (fma
      (fma (fma t_2 (/ (sin x) (cos x)) (* t_3 0.3333333333333333)) eps t_2)
      eps
      t_3)
     eps
     t_1)
    eps
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	double t_1 = pow(sin(x), 2.0) / t_0;
	double t_2 = -0.16666666666666666 - (fma(t_1, 0.16666666666666666, fma(t_1, -0.5, -0.5)) - (fma(sin(x), sin(x), (pow(sin(x), 4.0) / t_0)) / t_0));
	double t_3 = ((pow(sin(x), 3.0) / t_0) + sin(x)) / cos(x);
	return fma(fma(fma(fma(fma(t_2, (sin(x) / cos(x)), (t_3 * 0.3333333333333333)), eps, t_2), eps, t_3), eps, t_1), eps, eps);
}

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	t_1 = Float64((sin(x) ^ 2.0) / t_0)
	t_2 = Float64(-0.16666666666666666 - Float64(fma(t_1, 0.16666666666666666, fma(t_1, -0.5, -0.5)) - Float64(fma(sin(x), sin(x), Float64((sin(x) ^ 4.0) / t_0)) / t_0)))
	t_3 = Float64(Float64(Float64((sin(x) ^ 3.0) / t_0) + sin(x)) / cos(x))
	return fma(fma(fma(fma(fma(t_2, Float64(sin(x) / cos(x)), Float64(t_3 * 0.3333333333333333)), eps, t_2), eps, t_3), eps, t_1), eps, eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.16666666666666666 - N[(N[(t$95$1 * 0.16666666666666666 + N[(t$95$1 * -0.5 + -0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[x], $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / t$95$0), $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(t$95$2 * N[(N[Sin[x], $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * eps + t$95$2), $MachinePrecision] * eps + t$95$3), $MachinePrecision] * eps + t$95$1), $MachinePrecision] * eps + eps), $MachinePrecision]]]]]

Derivation

1. Initial program 65.3%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right), \frac{\sin x}{\cos x}, 0.3333333333333333 \cdot \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, -0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)\right), \varepsilon, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right)} \]

6. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right), \frac{\sin x}{\cos x}, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x} \cdot 0.3333333333333333\right), \varepsilon, -0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)\right), \varepsilon, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right) \]
7. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(2 \cdot x\right)\\ t_1 := \cos \left(2 \cdot x\right)\\ \frac{\sin \varepsilon}{1 + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_1, -0.5, \mathsf{fma}\left(0.16666666666666666 \cdot \varepsilon, t\_0, -0.5\right)\right), \varepsilon, -t\_0\right), \varepsilon, t\_1\right)} \cdot 2 \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* 2.0 x))) (t_1 (cos (* 2.0 x))))
   (*
    (/
     (sin eps)
     (+
      1.0
      (fma
       (fma
        (fma t_1 -0.5 (fma (* 0.16666666666666666 eps) t_0 -0.5))
        eps
        (- t_0))
       eps
       t_1)))
    2.0)))

C

double code(double x, double eps) {
	double t_0 = sin((2.0 * x));
	double t_1 = cos((2.0 * x));
	return (sin(eps) / (1.0 + fma(fma(fma(t_1, -0.5, fma((0.16666666666666666 * eps), t_0, -0.5)), eps, -t_0), eps, t_1))) * 2.0;
}

Julia

function code(x, eps)
	t_0 = sin(Float64(2.0 * x))
	t_1 = cos(Float64(2.0 * x))
	return Float64(Float64(sin(eps) / Float64(1.0 + fma(fma(fma(t_1, -0.5, fma(Float64(0.16666666666666666 * eps), t_0, -0.5)), eps, Float64(-t_0)), eps, t_1))) * 2.0)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Sin[eps], $MachinePrecision] / N[(1.0 + N[(N[(N[(t$95$1 * -0.5 + N[(N[(0.16666666666666666 * eps), $MachinePrecision] * t$95$0 + -0.5), $MachinePrecision]), $MachinePrecision] * eps + (-t$95$0)), $MachinePrecision] * eps + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]]

Derivation

1. Initial program 65.3%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sin-diff N/A

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

   17. lower-sin.f64 N/A

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

   18. lower--.f64 65.3

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 65.3

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

4. Applied rewrites 65.3%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. cos-mult N/A

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

   8. associate-/r/ N/A

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

   9. lower-*.f64 N/A

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in eps around 0

   \[\leadsto \frac{\sin \left(0 + \varepsilon\right)}{\color{blue}{1 + \left(\cos \left(2 \cdot x\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{-1}{2} \cdot \cos \left(2 \cdot x\right) + \frac{1}{6} \cdot \left(\varepsilon \cdot \sin \left(2 \cdot x\right)\right)\right) - \frac{1}{2}\right) - \sin \left(2 \cdot x\right)\right)\right)}} \cdot 2 \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\sin \left(0 + \varepsilon\right)}{\color{blue}{\left(\cos \left(2 \cdot x\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{-1}{2} \cdot \cos \left(2 \cdot x\right) + \frac{1}{6} \cdot \left(\varepsilon \cdot \sin \left(2 \cdot x\right)\right)\right) - \frac{1}{2}\right) - \sin \left(2 \cdot x\right)\right)\right) + 1}} \cdot 2 \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\sin \left(0 + \varepsilon\right)}{\color{blue}{\left(\cos \left(2 \cdot x\right) + \varepsilon \cdot \left(\varepsilon \cdot \left(\left(\frac{-1}{2} \cdot \cos \left(2 \cdot x\right) + \frac{1}{6} \cdot \left(\varepsilon \cdot \sin \left(2 \cdot x\right)\right)\right) - \frac{1}{2}\right) - \sin \left(2 \cdot x\right)\right)\right) + 1}} \cdot 2 \]

9. Applied rewrites 100.0%

   \[\leadsto \frac{\sin \left(0 + \varepsilon\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(2 \cdot x\right), -0.5, \mathsf{fma}\left(0.16666666666666666 \cdot \varepsilon, \sin \left(2 \cdot x\right), -0.5\right)\right), \varepsilon, -\sin \left(2 \cdot x\right)\right), \varepsilon, \cos \left(2 \cdot x\right)\right) + 1}} \cdot 2 \]

10. Final simplification 100.0%

    \[\leadsto \frac{\sin \varepsilon}{1 + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(2 \cdot x\right), -0.5, \mathsf{fma}\left(0.16666666666666666 \cdot \varepsilon, \sin \left(2 \cdot x\right), -0.5\right)\right), \varepsilon, -\sin \left(2 \cdot x\right)\right), \varepsilon, \cos \left(2 \cdot x\right)\right)} \cdot 2 \]
11. Add Preprocessing

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ \frac{\sin \varepsilon}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t\_0, -0.5, -0.5\right), \varepsilon, -\sin \left(2 \cdot x\right)\right), \varepsilon, t\_0\right) + 1} \cdot 2 \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (cos (* 2.0 x))))
   (*
    (/
     (sin eps)
     (+ (fma (fma (fma t_0 -0.5 -0.5) eps (- (sin (* 2.0 x)))) eps t_0) 1.0))
    2.0)))

C

double code(double x, double eps) {
	double t_0 = cos((2.0 * x));
	return (sin(eps) / (fma(fma(fma(t_0, -0.5, -0.5), eps, -sin((2.0 * x))), eps, t_0) + 1.0)) * 2.0;
}

Julia

function code(x, eps)
	t_0 = cos(Float64(2.0 * x))
	return Float64(Float64(sin(eps) / Float64(fma(fma(fma(t_0, -0.5, -0.5), eps, Float64(-sin(Float64(2.0 * x)))), eps, t_0) + 1.0)) * 2.0)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Sin[eps], $MachinePrecision] / N[(N[(N[(N[(t$95$0 * -0.5 + -0.5), $MachinePrecision] * eps + (-N[Sin[N[(2.0 * x), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * eps + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Initial program 65.3%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sin-diff N/A

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

   17. lower-sin.f64 N/A

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

   18. lower--.f64 65.3

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 65.3

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

4. Applied rewrites 65.3%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. cos-mult N/A

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

   8. associate-/r/ N/A

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

   9. lower-*.f64 N/A

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

9. Applied rewrites 99.9%

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

10. Final simplification 99.9%

    \[\leadsto \frac{\sin \varepsilon}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(2 \cdot x\right), -0.5, -0.5\right), \varepsilon, -\sin \left(2 \cdot x\right)\right), \varepsilon, \cos \left(2 \cdot x\right)\right) + 1} \cdot 2 \]
11. Add Preprocessing

Alternative 4: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sin-diff N/A

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

   17. lower-sin.f64 N/A

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

   18. lower--.f64 65.3

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 65.3

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

4. Applied rewrites 65.3%

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

5. Taylor expanded in eps around inf

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

   1. lower-/.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-cos.f64 99.9

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

7. Applied rewrites 99.9%

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

Alternative 5: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sin-diff N/A

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

   17. lower-sin.f64 N/A

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

   18. lower--.f64 65.3

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 65.3

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

4. Applied rewrites 65.3%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. cos-mult N/A

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

   8. associate-/r/ N/A

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

   9. lower-*.f64 N/A

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)}}{\cos \left(0 + \varepsilon\right) + \cos \left(\left(\varepsilon + x\right) + x\right)} \cdot 2 \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon}}{\cos \left(0 + \varepsilon\right) + \cos \left(\left(\varepsilon + x\right) + x\right)} \cdot 2 \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon}}{\cos \left(0 + \varepsilon\right) + \cos \left(\left(\varepsilon + x\right) + x\right)} \cdot 2 \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot {\varepsilon}^{2} + 1\right)} \cdot \varepsilon}{\cos \left(0 + \varepsilon\right) + \cos \left(\left(\varepsilon + x\right) + x\right)} \cdot 2 \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{{\varepsilon}^{2} \cdot \frac{-1}{6}} + 1\right) \cdot \varepsilon}{\cos \left(0 + \varepsilon\right) + \cos \left(\left(\varepsilon + x\right) + x\right)} \cdot 2 \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{-1}{6}, 1\right)} \cdot \varepsilon}{\cos \left(0 + \varepsilon\right) + \cos \left(\left(\varepsilon + x\right) + x\right)} \cdot 2 \]

   6. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{-1}{6}, 1\right) \cdot \varepsilon}{\cos \left(0 + \varepsilon\right) + \cos \left(\left(\varepsilon + x\right) + x\right)} \cdot 2 \]

   7. lower-*.f64 99.9

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, -0.16666666666666666, 1\right) \cdot \varepsilon}{\cos \left(0 + \varepsilon\right) + \cos \left(\left(\varepsilon + x\right) + x\right)} \cdot 2 \]

9. Applied rewrites 99.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.16666666666666666, 1\right) \cdot \varepsilon}}{\cos \left(0 + \varepsilon\right) + \cos \left(\left(\varepsilon + x\right) + x\right)} \cdot 2 \]

10. Final simplification 99.9%

    \[\leadsto \frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.16666666666666666, 1\right) \cdot \varepsilon}{\cos \left(\left(\varepsilon + x\right) + x\right) + \cos \varepsilon} \cdot 2 \]
11. Add Preprocessing

Alternative 6: 99.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. sin-diff N/A

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

   17. lower-sin.f64 N/A

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

   18. lower--.f64 65.3

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 65.3

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

4. Applied rewrites 65.3%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. cos-mult N/A

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

   8. associate-/r/ N/A

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

   9. lower-*.f64 N/A

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-*.f64 99.0

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

9. Applied rewrites 99.0%

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

10. Final simplification 99.0%

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

Alternative 7: 98.5% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right), \frac{\sin x}{\cos x}, 0.3333333333333333 \cdot \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, -0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)\right), \varepsilon, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \left(\frac{4}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right), \varepsilon, \varepsilon\right) \]
7. Step-by-step derivation

8. Applied rewrites 98.2%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.8888888888888888, \varepsilon \cdot \varepsilon, 1.3333333333333333\right) \cdot x, \varepsilon, \mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, 1\right)\right), x, \mathsf{fma}\left({\varepsilon}^{3}, 0.6666666666666666, \varepsilon\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right), \varepsilon, \varepsilon\right) \]

9. Taylor expanded in eps around 0

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

11. Applied rewrites 98.2%

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

Alternative 8: 98.5% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right), \frac{\sin x}{\cos x}, 0.3333333333333333 \cdot \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, -0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)\right), \varepsilon, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right)} \]

6. Taylor expanded in eps around 0

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

8. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \sin x}{\cos x}, -0.3333333333333333, \sin x \cdot \frac{\mathsf{fma}\left(-{\sin x}^{2}, \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}, \mathsf{fma}\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666\right)}{\cos x}\right), \varepsilon, \mathsf{fma}\left(-{\sin x}^{2}, \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}, \mathsf{fma}\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666\right)\right), \varepsilon, \frac{\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon} \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.2%

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

12. Taylor expanded in eps around 0

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

14. Applied rewrites 98.2%

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

15. Final simplification 98.2%

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

Alternative 9: 98.4% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right), \frac{\sin x}{\cos x}, 0.3333333333333333 \cdot \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, -0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)\right), \varepsilon, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right)} \]

6. Taylor expanded in eps around 0

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

8. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \sin x}{\cos x}, -0.3333333333333333, \sin x \cdot \frac{\mathsf{fma}\left(-{\sin x}^{2}, \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}, \mathsf{fma}\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666\right)}{\cos x}\right), \varepsilon, \mathsf{fma}\left(-{\sin x}^{2}, \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}, \mathsf{fma}\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666\right)\right), \varepsilon, \frac{\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon} \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.2%

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

12. Taylor expanded in eps around 0

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

14. Applied rewrites 98.2%

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

15. Final simplification 98.2%

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

Alternative 10: 98.3% accurate, 17.3× 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 65.3%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right), \frac{\sin x}{\cos x}, 0.3333333333333333 \cdot \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, -0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)\right), \varepsilon, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \left(\frac{4}{3} \cdot {\varepsilon}^{2} + \varepsilon \cdot \left(x \cdot \left(\frac{4}{3} + \frac{17}{9} \cdot {\varepsilon}^{2}\right)\right)\right)\right)\right), \varepsilon, \varepsilon\right) \]
7. Step-by-step derivation

8. Applied rewrites 98.2%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.8888888888888888, \varepsilon \cdot \varepsilon, 1.3333333333333333\right) \cdot x, \varepsilon, \mathsf{fma}\left(1.3333333333333333, \varepsilon \cdot \varepsilon, 1\right)\right), x, \mathsf{fma}\left({\varepsilon}^{3}, 0.6666666666666666, \varepsilon\right)\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right), \varepsilon, \varepsilon\right) \]

9. Taylor expanded in eps around 0

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

11. Applied rewrites 98.1%

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

Alternative 11: 98.3% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.3%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right), \frac{\sin x}{\cos x}, 0.3333333333333333 \cdot \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, -0.16666666666666666 - \left(\mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, 0.16666666666666666, \mathsf{fma}\left(\frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{{\cos x}^{2}}\right)}{{\cos x}^{2}}\right)\right), \varepsilon, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right)} \]

6. Taylor expanded in eps around 0

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

8. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \sin x}{\cos x}, -0.3333333333333333, \sin x \cdot \frac{\mathsf{fma}\left(-{\sin x}^{2}, \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}, \mathsf{fma}\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666\right)}{\cos x}\right), \varepsilon, \mathsf{fma}\left(-{\sin x}^{2}, \frac{1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}}{{\cos x}^{2}}, \mathsf{fma}\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5, \frac{0.16666666666666666 \cdot {\sin x}^{2}}{{\cos x}^{2}}\right) + 0.16666666666666666\right)\right), \varepsilon, \frac{\left(1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \sin x}{\cos x}\right), \varepsilon, 1 + \frac{{\sin x}^{2}}{{\cos x}^{2}}\right) \cdot \varepsilon} \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.2%

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

12. Taylor expanded in eps around 0

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

14. Applied rewrites 98.1%

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

Developer Target 1: 99.0% 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

real(8) function code(x, eps)
    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 2024268 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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