2tan (problem 3.3.2)

Percentage Accurate: 62.0% → 99.7%

Time: 14.5s

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: 62.0% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ t_1 := {\sin x}^{2}\\ t_2 := \mathsf{fma}\left(t\_1, \frac{-1}{t\_0}, -1\right)\\ t_3 := \frac{t\_1}{t\_0}\\ t_4 := \mathsf{fma}\left(t\_3, \sin x, \sin x\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(\left(\varepsilon \cdot \frac{\mathsf{fma}\left(t\_4, -0.3333333333333333, \mathsf{fma}\left(\frac{t\_2}{t\_0} \cdot t\_1, \sin x, t\_4 \cdot -0.3333333333333333\right)\right)}{-\cos x} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{\sin x}{t\_0}, \sin x, 1\right), -0.5, \frac{\mathsf{fma}\left(t\_2, t\_1, t\_1 \cdot 0.16666666666666666\right)}{t\_0} + 0.16666666666666666\right)\right) \cdot \varepsilon, \varepsilon, \frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_3, \varepsilon, \varepsilon\right), \sin x, \frac{t\_1}{\cos x}\right)}{\cos x}\right), \varepsilon, \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0))
        (t_1 (pow (sin x) 2.0))
        (t_2 (fma t_1 (/ -1.0 t_0) -1.0))
        (t_3 (/ t_1 t_0))
        (t_4 (fma t_3 (sin x) (sin x))))
   (fma
    (fma
     (*
      (-
       (*
        eps
        (/
         (fma
          t_4
          -0.3333333333333333
          (fma (* (/ t_2 t_0) t_1) (sin x) (* t_4 -0.3333333333333333)))
         (- (cos x))))
       (fma
        (fma (/ (sin x) t_0) (sin x) 1.0)
        -0.5
        (+
         (/ (fma t_2 t_1 (* t_1 0.16666666666666666)) t_0)
         0.16666666666666666)))
      eps)
     eps
     (/ (fma (fma t_3 eps eps) (sin x) (/ t_1 (cos x))) (cos x)))
    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);
	double t_2 = fma(t_1, (-1.0 / t_0), -1.0);
	double t_3 = t_1 / t_0;
	double t_4 = fma(t_3, sin(x), sin(x));
	return fma(fma((((eps * (fma(t_4, -0.3333333333333333, fma(((t_2 / t_0) * t_1), sin(x), (t_4 * -0.3333333333333333))) / -cos(x))) - fma(fma((sin(x) / t_0), sin(x), 1.0), -0.5, ((fma(t_2, t_1, (t_1 * 0.16666666666666666)) / t_0) + 0.16666666666666666))) * eps), eps, (fma(fma(t_3, eps, eps), sin(x), (t_1 / cos(x))) / cos(x))), eps, eps);
}

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	t_1 = sin(x) ^ 2.0
	t_2 = fma(t_1, Float64(-1.0 / t_0), -1.0)
	t_3 = Float64(t_1 / t_0)
	t_4 = fma(t_3, sin(x), sin(x))
	return fma(fma(Float64(Float64(Float64(eps * Float64(fma(t_4, -0.3333333333333333, fma(Float64(Float64(t_2 / t_0) * t_1), sin(x), Float64(t_4 * -0.3333333333333333))) / Float64(-cos(x)))) - fma(fma(Float64(sin(x) / t_0), sin(x), 1.0), -0.5, Float64(Float64(fma(t_2, t_1, Float64(t_1 * 0.16666666666666666)) / t_0) + 0.16666666666666666))) * eps), eps, Float64(fma(fma(t_3, eps, eps), sin(x), Float64(t_1 / cos(x))) / cos(x))), 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[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(-1.0 / t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Sin[x], $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(eps * N[(N[(t$95$4 * -0.3333333333333333 + N[(N[(N[(t$95$2 / t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(t$95$4 * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[Cos[x], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision] * N[Sin[x], $MachinePrecision] + 1.0), $MachinePrecision] * -0.5 + N[(N[(N[(t$95$2 * t$95$1 + N[(t$95$1 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps + N[(N[(N[(t$95$3 * eps + eps), $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(t$95$1 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]]]]]]

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (*
   (+
    (fma
     (* 0.3333333333333333 (* eps eps))
     (fma (tan x) (sin x) (cos x))
     (cos x))
    (* (tan x) (sin x)))
   eps)
  (* (fma (- (tan x)) (tan eps) 1.0) (cos x))))

C

double code(double x, double eps) {
	return ((fma((0.3333333333333333 * (eps * eps)), fma(tan(x), sin(x), cos(x)), cos(x)) + (tan(x) * sin(x))) * eps) / (fma(-tan(x), tan(eps), 1.0) * cos(x));
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(fma(Float64(0.3333333333333333 * Float64(eps * eps)), fma(tan(x), sin(x), cos(x)), cos(x)) + Float64(tan(x) * sin(x))) * eps) / Float64(fma(Float64(-tan(x)), tan(eps), 1.0) * cos(x)))
end

Wolfram

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

Derivation

1. Initial program 62.5%

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

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

   4. tan-sum N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. frac-sub N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 62.7%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

7. Applied rewrites 99.7%

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

9. Applied rewrites 99.7%

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

11. Applied rewrites 99.7%

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

Alternative 3: 99.7% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma (tan x) (sin x) (cos x))))
   (/
    (* (fma t_0 (* 0.3333333333333333 (* eps eps)) t_0) eps)
    (* (fma (- (tan x)) (tan eps) 1.0) (cos x)))))

C

double code(double x, double eps) {
	double t_0 = fma(tan(x), sin(x), cos(x));
	return (fma(t_0, (0.3333333333333333 * (eps * eps)), t_0) * eps) / (fma(-tan(x), tan(eps), 1.0) * cos(x));
}

Julia

function code(x, eps)
	t_0 = fma(tan(x), sin(x), cos(x))
	return Float64(Float64(fma(t_0, Float64(0.3333333333333333 * Float64(eps * eps)), t_0) * eps) / Float64(fma(Float64(-tan(x)), tan(eps), 1.0) * cos(x)))
end

Wolfram

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

Derivation

1. Initial program 62.5%

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

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

   4. tan-sum N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. frac-sub N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 62.7%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

7. Applied rewrites 99.7%

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

9. Applied rewrites 99.7%

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

11. Applied rewrites 99.7%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \sin x, \cos x\right), 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right), \mathsf{fma}\left(\tan x, \sin x, \cos x\right)\right) \cdot \varepsilon}{\mathsf{fma}\left(-\tan x, \tan \varepsilon, 1\right) \cdot \cos x} \]
12. Add Preprocessing

Alternative 4: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.3333333333333333\right), \cos x\right) + \sin x \cdot \tan x\right) \cdot \varepsilon}{\mathsf{fma}\left(-\tan x, \tan \varepsilon, 1\right) \cdot \cos x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (*
   (+
    (fma
     (* eps eps)
     (fma 0.16666666666666666 (* x x) 0.3333333333333333)
     (cos x))
    (* (sin x) (tan x)))
   eps)
  (* (fma (- (tan x)) (tan eps) 1.0) (cos x))))

C

double code(double x, double eps) {
	return ((fma((eps * eps), fma(0.16666666666666666, (x * x), 0.3333333333333333), cos(x)) + (sin(x) * tan(x))) * eps) / (fma(-tan(x), tan(eps), 1.0) * cos(x));
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(fma(Float64(eps * eps), fma(0.16666666666666666, Float64(x * x), 0.3333333333333333), cos(x)) + Float64(sin(x) * tan(x))) * eps) / Float64(fma(Float64(-tan(x)), tan(eps), 1.0) * cos(x)))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(eps * eps), $MachinePrecision] * N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] / N[(N[((-N[Tan[x], $MachinePrecision]) * N[Tan[eps], $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

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

   4. tan-sum N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. frac-sub N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 62.7%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

7. Applied rewrites 99.7%

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

9. Applied rewrites 99.7%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 99.5%

    \[\leadsto \frac{\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.3333333333333333\right), \cos x\right) - \sin x \cdot \left(-\tan x\right)\right) \cdot \varepsilon}{\mathsf{fma}\left(-\tan x, \tan \varepsilon, 1\right) \cdot \cos x} \]

13. Final simplification 99.5%

    \[\leadsto \frac{\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.3333333333333333\right), \cos x\right) + \sin x \cdot \tan x\right) \cdot \varepsilon}{\mathsf{fma}\left(-\tan x, \tan \varepsilon, 1\right) \cdot \cos x} \]
14. Add Preprocessing

Alternative 5: 99.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (* (+ (fma (* eps eps) 0.3333333333333333 (cos x)) (* (sin x) (tan x))) eps)
  (* (fma (- (tan x)) (tan eps) 1.0) (cos x))))

C

double code(double x, double eps) {
	return ((fma((eps * eps), 0.3333333333333333, cos(x)) + (sin(x) * tan(x))) * eps) / (fma(-tan(x), tan(eps), 1.0) * cos(x));
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(fma(Float64(eps * eps), 0.3333333333333333, cos(x)) + Float64(sin(x) * tan(x))) * eps) / Float64(fma(Float64(-tan(x)), tan(eps), 1.0) * cos(x)))
end

Wolfram

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

Derivation

1. Initial program 62.5%

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

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

   4. tan-sum N/A

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

   5. lift-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. frac-sub N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 62.7%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

7. Applied rewrites 99.7%

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

9. Applied rewrites 99.7%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 99.4%

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

13. Final simplification 99.4%

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

Alternative 6: 99.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (+ (* (* (tan x) (+ (tan x) (fma (pow (tan x) 2.0) eps eps))) eps) eps))

C

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

Julia

function code(x, eps)
	return Float64(Float64(Float64(tan(x) * Float64(tan(x) + fma((tan(x) ^ 2.0), eps, eps))) * eps) + eps)
end

Wolfram

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in eps around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.2%

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

9. Applied rewrites 99.2%

   \[\leadsto \left(\tan x \cdot \left(\tan x + \mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)\right)\right) \cdot \varepsilon + \varepsilon \]
10. Add Preprocessing

Alternative 7: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in eps around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.2%

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

9. Applied rewrites 99.2%

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

Alternative 8: 99.0% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (+ (* (fma (fma (* x x) eps eps) (tan x) (pow (tan x) 2.0)) eps) eps))

C

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

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(Float64(x * x), eps, eps), tan(x), (tan(x) ^ 2.0)) * eps) + eps)
end

Wolfram

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in eps around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.5%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \varepsilon, \varepsilon\right), \tan x, {\tan x}^{2}\right) \cdot \varepsilon + \varepsilon \]
11. Add Preprocessing

Alternative 9: 98.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.8333333333333334, \varepsilon, \mathsf{fma}\left(0.6666666666666666, x, 0.5 \cdot \varepsilon\right)\right), x, 1\right), x, \varepsilon\right) \cdot x, \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (*
   (fma
    (fma
     (fma 0.8333333333333334 eps (fma 0.6666666666666666 x (* 0.5 eps)))
     x
     1.0)
    x
    eps)
   x)
  eps
  eps))

C

double code(double x, double eps) {
	return fma((fma(fma(fma(0.8333333333333334, eps, fma(0.6666666666666666, x, (0.5 * eps))), x, 1.0), x, eps) * x), eps, eps);
}

Julia

function code(x, eps)
	return fma(Float64(fma(fma(fma(0.8333333333333334, eps, fma(0.6666666666666666, x, Float64(0.5 * eps))), x, 1.0), x, eps) * x), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(0.8333333333333334 * eps + N[(0.6666666666666666 * x + N[(0.5 * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + eps), $MachinePrecision] * x), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in eps around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.6%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.8333333333333334, \varepsilon, \mathsf{fma}\left(0.6666666666666666, x, 0.5 \cdot \varepsilon\right)\right), x, 1\right), x, \varepsilon\right) \cdot x, \varepsilon, \varepsilon\right) \]
9. Add Preprocessing

Alternative 10: 98.6% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in eps around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.6%

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

Alternative 11: 98.4% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in eps around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.3%

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

Alternative 12: 98.5% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in eps around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.3%

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

Alternative 13: 98.0% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in eps around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.4%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 96.4%

    \[\leadsto \mathsf{fma}\left(x, \varepsilon, 1\right) \cdot \varepsilon \]
12. 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 2024337 
(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)))