2tan (problem 3.3.2)

Percentage Accurate: 62.3% → 98.6%

Time: 14.1s

Alternatives: 7

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: 62.3% 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: 98.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \varepsilon, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.37777777777777777, 0.6666666666666666\right), 1\right)\right)\right), \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  eps
  (fma
   x
   eps
   (*
    x
    (*
     x
     (fma (* x x) (fma (* x x) 0.37777777777777777 0.6666666666666666) 1.0))))
  eps))

C

double code(double x, double eps) {
	return fma(eps, fma(x, eps, (x * (x * fma((x * x), fma((x * x), 0.37777777777777777, 0.6666666666666666), 1.0)))), eps);
}

Julia

function code(x, eps)
	return fma(eps, fma(x, eps, Float64(x * Float64(x * fma(Float64(x * x), fma(Float64(x * x), 0.37777777777777777, 0.6666666666666666), 1.0)))), eps)
end

Wolfram

code[x_, eps_] := N[(eps * N[(x * eps + N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.37777777777777777 + 0.6666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]

Derivation

1. Initial program 60.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. accelerator-lowering-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. Simplified 100.0%

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

6. Taylor expanded in x around 0

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

8. Simplified 100.0%

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

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \frac{\varepsilon}{\cos x}, \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)}\right), \varepsilon\right) \]
10. Step-by-step derivation

    1. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \frac{\varepsilon}{\cos x}, \color{blue}{\left(x \cdot x\right)} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right), \varepsilon\right) \]

    2. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \frac{\varepsilon}{\cos x}, \color{blue}{x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)}\right), \varepsilon\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \frac{\varepsilon}{\cos x}, \color{blue}{x \cdot \left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)}\right), \varepsilon\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \frac{\varepsilon}{\cos x}, x \cdot \color{blue}{\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right)\right)\right)}\right), \varepsilon\right) \]

    5. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \frac{\varepsilon}{\cos x}, x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{3} + \frac{17}{45} \cdot {x}^{2}\right) + 1\right)}\right)\right), \varepsilon\right) \]

    6. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \frac{\varepsilon}{\cos x}, x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{3} + \frac{17}{45} \cdot {x}^{2}, 1\right)}\right)\right), \varepsilon\right) \]

    7. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \frac{\varepsilon}{\cos x}, x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} + \frac{17}{45} \cdot {x}^{2}, 1\right)\right)\right), \varepsilon\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \frac{\varepsilon}{\cos x}, x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{3} + \frac{17}{45} \cdot {x}^{2}, 1\right)\right)\right), \varepsilon\right) \]

    9. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \frac{\varepsilon}{\cos x}, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{17}{45} \cdot {x}^{2} + \frac{2}{3}}, 1\right)\right)\right), \varepsilon\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \frac{\varepsilon}{\cos x}, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{17}{45}} + \frac{2}{3}, 1\right)\right)\right), \varepsilon\right) \]

    11. accelerator-lowering-fma.f64 N/A

        \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \frac{\varepsilon}{\cos x}, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{17}{45}, \frac{2}{3}\right)}, 1\right)\right)\right), \varepsilon\right) \]

    12. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \frac{\varepsilon}{\cos x}, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{17}{45}, \frac{2}{3}\right), 1\right)\right)\right), \varepsilon\right) \]

    13. *-lowering-*.f64 100.0

        \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \frac{\varepsilon}{\cos x}, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.37777777777777777, 0.6666666666666666\right), 1\right)\right)\right), \varepsilon\right) \]

11. Simplified 100.0%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \frac{\varepsilon}{\cos x}, \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.37777777777777777, 0.6666666666666666\right), 1\right)\right)}\right), \varepsilon\right) \]

12. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{\varepsilon}, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{17}{45}, \frac{2}{3}\right), 1\right)\right)\right), \varepsilon\right) \]
13. Step-by-step derivation

14. Simplified 100.0%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{\varepsilon}, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.37777777777777777, 0.6666666666666666\right), 1\right)\right)\right), \varepsilon\right) \]
15. Add Preprocessing

Alternative 2: 98.5% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. accelerator-lowering-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. Simplified 100.0%

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

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. *-lowering-*.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. accelerator-lowering-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. accelerator-lowering-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

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

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

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{4}{3}}, \frac{2}{3} \cdot x\right), 1\right), \varepsilon\right), \varepsilon\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \frac{4}{3}, \color{blue}{x \cdot \frac{2}{3}}\right), 1\right), \varepsilon\right), \varepsilon\right) \]

   12. *-lowering-*.f64 99.9

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

8. Simplified 99.9%

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

9. Taylor expanded in eps around 0

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

    1. +-commutative N/A

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

    2. unpow2 N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. accelerator-lowering-fma.f64 N/A

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

    6. *-commutative N/A

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

    7. *-lowering-*.f64 99.9

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

11. Simplified 99.9%

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

Alternative 3: 98.4% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. accelerator-lowering-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. Simplified 100.0%

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

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. *-lowering-*.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. accelerator-lowering-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. accelerator-lowering-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

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

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

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{4}{3}}, \frac{2}{3} \cdot x\right), 1\right), \varepsilon\right), \varepsilon\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \frac{4}{3}, \color{blue}{x \cdot \frac{2}{3}}\right), 1\right), \varepsilon\right), \varepsilon\right) \]

   12. *-lowering-*.f64 99.9

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

8. Simplified 99.9%

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

9. Taylor expanded in eps around 0

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

    1. +-commutative N/A

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

    2. distribute-rgt-in N/A

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

    3. associate-*l* N/A

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

    4. *-lft-identity N/A

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

    5. accelerator-lowering-fma.f64 N/A

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

    6. unpow2 N/A

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

    7. *-lowering-*.f64 N/A

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

    8. *-lowering-*.f64 N/A

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

    9. +-commutative N/A

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

    10. unpow2 N/A

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

    11. associate-*r* N/A

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

    12. *-commutative N/A

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

    13. accelerator-lowering-fma.f64 N/A

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

    14. *-commutative N/A

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

    15. *-lowering-*.f64 99.9

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

11. Simplified 99.9%

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

12. Final simplification 99.9%

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

Alternative 4: 98.4% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. accelerator-lowering-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. Simplified 100.0%

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

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. accelerator-lowering-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. *-lowering-*.f64 N/A

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

   7. +-lowering-+.f64 99.6

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

8. Simplified 99.6%

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

Alternative 5: 98.3% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. accelerator-lowering-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. Simplified 100.0%

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

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. accelerator-lowering-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. *-lowering-*.f64 N/A

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

   7. +-lowering-+.f64 99.6

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

8. Simplified 99.6%

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

9. Taylor expanded in eps around 0

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

    1. *-lowering-*.f64 99.6

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

11. Simplified 99.6%

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

Alternative 6: 97.9% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. accelerator-lowering-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. Simplified 100.0%

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

6. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 98.8

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

8. Simplified 98.8%

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

Alternative 7: 97.9% 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 60.4%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. accelerator-lowering-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. Simplified 100.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\varepsilon} \]
7. Step-by-step derivation

8. Simplified 98.8%

   \[\leadsto \color{blue}{\varepsilon} \]
9. 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: 62.4% 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 2024205 
(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)))