2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.5%

Time: 13.4s

Alternatives: 13

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: 62.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 99.5% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0)) (t_1 (/ (pow (sin x) 2.0) t_0)))
   (fma
    (fma
     (fma
      (-
       -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)))
      eps
      (/ (+ (/ (pow (sin x) 3.0) t_0) (sin x)) (cos x)))
     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;
	return fma(fma(fma((-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))), eps, (((pow(sin(x), 3.0) / t_0) + sin(x)) / cos(x))), 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)
	return fma(fma(fma(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))), eps, Float64(Float64(Float64((sin(x) ^ 3.0) / t_0) + sin(x)) / cos(x))), 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]}, N[(N[(N[(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] * eps + 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]), $MachinePrecision] * eps + t$95$1), $MachinePrecision] * eps + eps), $MachinePrecision]]]

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

Alternative 2: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	return fma(fma(fma(0.3333333333333333, eps, (((pow(sin(x), 3.0) / t_0) + sin(x)) / cos(x))), eps, (pow(sin(x), 2.0) / t_0)), eps, eps);
}

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	return fma(fma(fma(0.3333333333333333, eps, Float64(Float64(Float64((sin(x) ^ 3.0) / t_0) + sin(x)) / cos(x))), eps, Float64((sin(x) ^ 2.0) / t_0)), eps, eps)
end

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 100.0%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, \frac{\frac{{\sin x}^{3}}{{\cos x}^{2}} + \sin x}{\cos x}\right), \varepsilon, \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon, \varepsilon\right) \]
9. Add Preprocessing

Alternative 3: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 61.0

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

4. Applied rewrites 61.0%

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

5. Taylor expanded in eps around inf

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

   1. lower-sin.f64 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

10. Applied rewrites 100.0%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 100.0%

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

14. Final simplification 100.0%

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

Alternative 4: 99.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 61.0

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

4. Applied rewrites 61.0%

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

5. Taylor expanded in eps around inf

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

   1. lower-sin.f64 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower--.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-cos.f64 100.0

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

10. Applied rewrites 100.0%

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

Alternative 5: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 61.0

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

4. Applied rewrites 61.0%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. associate--l+ N/A

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

   9. +-inverses N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. inv-pow N/A

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

   13. lower-pow.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. distribute-lft-neg-in N/A

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

   17. lower-*.f64 N/A

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

   18. lower-neg.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Final simplification 99.9%

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

Alternative 6: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{0.5 \cdot \left(\cos \left(\left(\varepsilon + x\right) + x\right) + \cos \varepsilon\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/ (sin eps) (* 0.5 (+ (cos (+ (+ eps x) x)) (cos eps)))))

C

double code(double x, double eps) {
	return sin(eps) / (0.5 * (cos(((eps + x) + x)) + cos(eps)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = sin(eps) / (0.5d0 * (cos(((eps + x) + x)) + cos(eps)))
end function

Java

public static double code(double x, double eps) {
	return Math.sin(eps) / (0.5 * (Math.cos(((eps + x) + x)) + Math.cos(eps)));
}

Python

def code(x, eps):
	return math.sin(eps) / (0.5 * (math.cos(((eps + x) + x)) + math.cos(eps)))

Julia

function code(x, eps)
	return Float64(sin(eps) / Float64(0.5 * Float64(cos(Float64(Float64(eps + x) + x)) + cos(eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = sin(eps) / (0.5 * (cos(((eps + x) + x)) + cos(eps)));
end

Wolfram

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

Derivation

1. Initial program 61.0%

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

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 61.0

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

4. Applied rewrites 61.0%

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

5. Taylor expanded in eps around inf

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

   1. lower-sin.f64 99.9

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

7. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-mult N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-+.f64 N/A

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

   10. associate--l+ N/A

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

   11. +-inverses N/A

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

   12. +-rgt-identity N/A

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

   13. lower-+.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-+.f64 99.9

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

9. Applied rewrites 99.9%

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

10. Final simplification 99.9%

    \[\leadsto \frac{\sin \varepsilon}{0.5 \cdot \left(\cos \left(\left(\varepsilon + x\right) + x\right) + \cos \varepsilon\right)} \]
11. Add Preprocessing

Alternative 7: 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 61.0%

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

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 61.0

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

4. Applied rewrites 61.0%

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

5. Taylor expanded in eps around inf

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

   1. lower-sin.f64 99.9

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

7. Applied rewrites 99.9%

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

Alternative 8: 99.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 61.0

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

4. Applied rewrites 61.0%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. associate--l+ N/A

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

   9. +-inverses N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. inv-pow N/A

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

   13. lower-pow.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. distribute-lft-neg-in N/A

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

   17. lower-*.f64 N/A

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

   18. lower-neg.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in eps around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.9

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

9. Applied rewrites 99.9%

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

10. Final simplification 99.9%

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

Alternative 9: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\cos \left(2 \cdot x\right) \cdot 0.5 + 0.5} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/ (sin eps) (+ (* (cos (* 2.0 x)) 0.5) 0.5)))

C

double code(double x, double eps) {
	return sin(eps) / ((cos((2.0 * x)) * 0.5) + 0.5);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = sin(eps) / ((cos((2.0d0 * x)) * 0.5d0) + 0.5d0)
end function

Java

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

Python

def code(x, eps):
	return math.sin(eps) / ((math.cos((2.0 * x)) * 0.5) + 0.5)

Julia

function code(x, eps)
	return Float64(sin(eps) / Float64(Float64(cos(Float64(2.0 * x)) * 0.5) + 0.5))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 61.0

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

4. Applied rewrites 61.0%

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

5. Taylor expanded in eps around inf

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

   1. lower-sin.f64 99.9

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in eps around 0

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

   1. lower-pow.f64 N/A

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

   2. lower-cos.f64 99.6

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

10. Applied rewrites 99.6%

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

12. Applied rewrites 99.6%

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

13. Final simplification 99.6%

    \[\leadsto \frac{\sin \varepsilon}{\cos \left(2 \cdot x\right) \cdot 0.5 + 0.5} \]
14. Add Preprocessing

Alternative 10: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps / pow(cos(x), 2.0);
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 61.0

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

4. Applied rewrites 61.0%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 99.6

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

7. Applied rewrites 99.6%

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

Alternative 11: 98.3% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.0%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 99.0%

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

12. Final simplification 99.0%

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

Alternative 12: 98.3% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.0%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 99.0%

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

12. Final simplification 99.0%

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

Alternative 13: 98.2% 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 61.0%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.0%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 98.9%

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

Developer Target 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 2024249 
(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)))