2tan (problem 3.3.2)

Percentage Accurate: 62.6% → 99.3%

Time: 16.1s

Alternatives: 10

Speedup: 207.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.6% 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: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 98.9%

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

7. Applied rewrites 98.9%

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

8. Final simplification 98.9%

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

Alternative 2: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 98.9%

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

7. Applied rewrites 98.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-tan.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. associate-+l+ N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

9. Applied rewrites 98.9%

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

    1. lift-cos.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-tan.f64 N/A

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

    4. lift-pow.f64 N/A

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

    5. lift-+.f64 N/A

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

    6. lift-sin.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. lift-tan.f64 N/A

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

    9. lift-pow.f64 N/A

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

    10. lift-+.f64 N/A

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

    11. lift-fma.f64 N/A

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

    12. *-commutative N/A

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

11. Applied rewrites 98.9%

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

Alternative 3: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2} + 1\\ \varepsilon \cdot \mathsf{fma}\left(\frac{\varepsilon}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)}, \sin x \cdot t\_0, t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (pow (tan x) 2.0) 1.0)))
   (*
    eps
    (fma
     (/
      eps
      (fma
       (* x x)
       (fma
        (* x x)
        (fma (* x x) -0.001388888888888889 0.041666666666666664)
        -0.5)
       1.0))
     (* (sin x) t_0)
     t_0))))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0) + 1.0;
	return eps * fma((eps / fma((x * x), fma((x * x), fma((x * x), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0)), (sin(x) * t_0), t_0);
}

Julia

function code(x, eps)
	t_0 = Float64((tan(x) ^ 2.0) + 1.0)
	return Float64(eps * fma(Float64(eps / fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0)), Float64(sin(x) * t_0), t_0))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]}, N[(eps * N[(N[(eps / N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 98.9%

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

7. Applied rewrites 98.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-tan.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. associate-+l+ N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

9. Applied rewrites 98.9%

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

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(\frac{\varepsilon}{\color{blue}{1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}}, \left({\tan x}^{2} + 1\right) \cdot \sin x, {\tan x}^{2} + 1\right) \cdot \varepsilon \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{\varepsilon}{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1}}, \left({\tan x}^{2} + 1\right) \cdot \sin x, {\tan x}^{2} + 1\right) \cdot \varepsilon \]

    2. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\varepsilon}{\color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)}}, \left({\tan x}^{2} + 1\right) \cdot \sin x, {\tan x}^{2} + 1\right) \cdot \varepsilon \]

    3. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\varepsilon}{\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)}, \left({\tan x}^{2} + 1\right) \cdot \sin x, {\tan x}^{2} + 1\right) \cdot \varepsilon \]

    4. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\varepsilon}{\mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)}, \left({\tan x}^{2} + 1\right) \cdot \sin x, {\tan x}^{2} + 1\right) \cdot \varepsilon \]

    5. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{\varepsilon}{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right)}, \left({\tan x}^{2} + 1\right) \cdot \sin x, {\tan x}^{2} + 1\right) \cdot \varepsilon \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\varepsilon}{\mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1\right)}, \left({\tan x}^{2} + 1\right) \cdot \sin x, {\tan x}^{2} + 1\right) \cdot \varepsilon \]

    7. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\varepsilon}{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right)}, 1\right)}, \left({\tan x}^{2} + 1\right) \cdot \sin x, {\tan x}^{2} + 1\right) \cdot \varepsilon \]

    8. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\varepsilon}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right)}, \left({\tan x}^{2} + 1\right) \cdot \sin x, {\tan x}^{2} + 1\right) \cdot \varepsilon \]

    9. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\varepsilon}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, \frac{-1}{2}\right), 1\right)}, \left({\tan x}^{2} + 1\right) \cdot \sin x, {\tan x}^{2} + 1\right) \cdot \varepsilon \]

    10. +-commutative N/A

        \[\leadsto \mathsf{fma}\left(\frac{\varepsilon}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right)}, \left({\tan x}^{2} + 1\right) \cdot \sin x, {\tan x}^{2} + 1\right) \cdot \varepsilon \]

    11. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(\frac{\varepsilon}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right)}, \left({\tan x}^{2} + 1\right) \cdot \sin x, {\tan x}^{2} + 1\right) \cdot \varepsilon \]

    12. lower-fma.f64 N/A

        \[\leadsto \mathsf{fma}\left(\frac{\varepsilon}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right)}, \left({\tan x}^{2} + 1\right) \cdot \sin x, {\tan x}^{2} + 1\right) \cdot \varepsilon \]

    13. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(\frac{\varepsilon}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)}, \left({\tan x}^{2} + 1\right) \cdot \sin x, {\tan x}^{2} + 1\right) \cdot \varepsilon \]

    14. lower-*.f64 98.8

        \[\leadsto \mathsf{fma}\left(\frac{\varepsilon}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)}, \left({\tan x}^{2} + 1\right) \cdot \sin x, {\tan x}^{2} + 1\right) \cdot \varepsilon \]

12. Applied rewrites 98.8%

    \[\leadsto \mathsf{fma}\left(\frac{\varepsilon}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)}}, \left({\tan x}^{2} + 1\right) \cdot \sin x, {\tan x}^{2} + 1\right) \cdot \varepsilon \]

13. Final simplification 98.8%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\frac{\varepsilon}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)}, \sin x \cdot \left({\tan x}^{2} + 1\right), {\tan x}^{2} + 1\right) \]
14. Add Preprocessing

Alternative 4: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2} + 1\\ \varepsilon \cdot \mathsf{fma}\left(\frac{\varepsilon}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)}, \sin x \cdot t\_0, t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (pow (tan x) 2.0) 1.0)))
   (*
    eps
    (fma
     (/ eps (fma (* x x) (fma (* x x) 0.041666666666666664 -0.5) 1.0))
     (* (sin x) t_0)
     t_0))))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0) + 1.0;
	return eps * fma((eps / fma((x * x), fma((x * x), 0.041666666666666664, -0.5), 1.0)), (sin(x) * t_0), t_0);
}

Julia

function code(x, eps)
	t_0 = Float64((tan(x) ^ 2.0) + 1.0)
	return Float64(eps * fma(Float64(eps / fma(Float64(x * x), fma(Float64(x * x), 0.041666666666666664, -0.5), 1.0)), Float64(sin(x) * t_0), t_0))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]}, N[(eps * N[(N[(eps / N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 98.9%

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

7. Applied rewrites 98.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-tan.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. associate-+l+ N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

9. Applied rewrites 98.9%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 N/A

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

    5. sub-neg N/A

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

    6. *-commutative N/A

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

    7. metadata-eval N/A

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

    8. lower-fma.f64 N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 98.7

        \[\leadsto \mathsf{fma}\left(\frac{\varepsilon}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, -0.5\right), 1\right)}, \left({\tan x}^{2} + 1\right) \cdot \sin x, {\tan x}^{2} + 1\right) \cdot \varepsilon \]

12. Applied rewrites 98.7%

    \[\leadsto \mathsf{fma}\left(\frac{\varepsilon}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)}}, \left({\tan x}^{2} + 1\right) \cdot \sin x, {\tan x}^{2} + 1\right) \cdot \varepsilon \]

13. Final simplification 98.7%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\frac{\varepsilon}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.041666666666666664, -0.5\right), 1\right)}, \sin x \cdot \left({\tan x}^{2} + 1\right), {\tan x}^{2} + 1\right) \]
14. Add Preprocessing

Alternative 5: 99.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 98.9%

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

7. Applied rewrites 98.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-tan.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. associate-+l+ N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

9. Applied rewrites 98.9%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 98.6%

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

13. Final simplification 98.6%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \sin x \cdot \left({\tan x}^{2} + 1\right), {\tan x}^{2} + 1\right) \]
14. Add Preprocessing

Alternative 6: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.6

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

5. Applied rewrites 98.6%

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

   1. lift-sin.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-fma.f64 98.6

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

7. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right)} \]
8. Add Preprocessing

Alternative 7: 98.4% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(\left(\frac{2}{3} \cdot x + \frac{5}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon\right)\right)\right)}, \varepsilon\right) \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(\varepsilon + x \cdot \left(1 + x \cdot \left(\left(\frac{2}{3} \cdot x + \frac{5}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon\right)\right)\right)}, \varepsilon\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \color{blue}{\left(x \cdot \left(1 + x \cdot \left(\left(\frac{2}{3} \cdot x + \frac{5}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon\right)\right) + \varepsilon\right)}, \varepsilon\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \color{blue}{\mathsf{fma}\left(x, 1 + x \cdot \left(\left(\frac{2}{3} \cdot x + \frac{5}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon\right), \varepsilon\right)}, \varepsilon\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\left(\frac{2}{3} \cdot x + \frac{5}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon\right) + 1}, \varepsilon\right), \varepsilon\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{2}{3} \cdot x + \frac{5}{6} \cdot \varepsilon\right) - \frac{-1}{2} \cdot \varepsilon, 1\right)}, \varepsilon\right), \varepsilon\right) \]

   6. associate--l+ N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{2}{3} \cdot x + \left(\frac{5}{6} \cdot \varepsilon - \frac{-1}{2} \cdot \varepsilon\right)}, 1\right), \varepsilon\right), \varepsilon\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{3}} + \left(\frac{5}{6} \cdot \varepsilon - \frac{-1}{2} \cdot \varepsilon\right), 1\right), \varepsilon\right), \varepsilon\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{2}{3}, \frac{5}{6} \cdot \varepsilon - \frac{-1}{2} \cdot \varepsilon\right)}, 1\right), \varepsilon\right), \varepsilon\right) \]

   9. distribute-rgt-out-- N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{2}{3}, \color{blue}{\varepsilon \cdot \left(\frac{5}{6} - \frac{-1}{2}\right)}\right), 1\right), \varepsilon\right), \varepsilon\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{2}{3}, \color{blue}{\varepsilon \cdot \left(\frac{5}{6} - \frac{-1}{2}\right)}\right), 1\right), \varepsilon\right), \varepsilon\right) \]

   11. metadata-eval 97.6

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

8. Applied rewrites 97.6%

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

Alternative 8: 98.3% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. distribute-lft-out N/A

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

   5. +-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-+.f64 97.6

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

8. Applied rewrites 97.6%

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

9. Final simplification 97.6%

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

Alternative 9: 98.2% accurate, 17.3× speedup

Math

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

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(x, Float64(x * eps), eps)
end

Wolfram

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

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in eps around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 97.6

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

8. Applied rewrites 97.6%

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

Alternative 10: 97.8% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 62.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-lft-in N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 98.9%

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

7. Applied rewrites 98.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.5%

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

    1. *-lft-identity 97.5

       \[\leadsto \color{blue}{\varepsilon} \]

12. Applied rewrites 97.5%

    \[\leadsto \color{blue}{\varepsilon} \]
13. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 62.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 2024219 
(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 (/ (sin eps) (* (cos x) (cos (+ x eps)))))

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

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

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