2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.7%

Time: 15.1s

Alternatives: 9

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: 62.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ t_1 := \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \tan x\\ t_2 := \mathsf{fma}\left(t\_0, 0.3333333333333333, 0.3333333333333333\right) + {\tan x}^{4}\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333, t\_1, \mathsf{fma}\left(t\_2, \tan x, {\tan x}^{3}\right)\right), \mathsf{fma}\left(\tan x, \tan x, t\_2\right)\right), \varepsilon, t\_1\right), \varepsilon, t\_0\right), \varepsilon, \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))
        (t_1 (* (fma (tan x) (tan x) 1.0) (tan x)))
        (t_2
         (+
          (fma t_0 0.3333333333333333 0.3333333333333333)
          (pow (tan x) 4.0))))
   (fma
    (fma
     (fma
      (fma
       eps
       (fma 0.3333333333333333 t_1 (fma t_2 (tan x) (pow (tan x) 3.0)))
       (fma (tan x) (tan x) t_2))
      eps
      t_1)
     eps
     t_0)
    eps
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	double t_1 = fma(tan(x), tan(x), 1.0) * tan(x);
	double t_2 = fma(t_0, 0.3333333333333333, 0.3333333333333333) + pow(tan(x), 4.0);
	return fma(fma(fma(fma(eps, fma(0.3333333333333333, t_1, fma(t_2, tan(x), pow(tan(x), 3.0))), fma(tan(x), tan(x), t_2)), eps, t_1), eps, t_0), eps, eps);
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	t_1 = Float64(fma(tan(x), tan(x), 1.0) * tan(x))
	t_2 = Float64(fma(t_0, 0.3333333333333333, 0.3333333333333333) + (tan(x) ^ 4.0))
	return fma(fma(fma(fma(eps, fma(0.3333333333333333, t_1, fma(t_2, tan(x), (tan(x) ^ 3.0))), fma(tan(x), tan(x), t_2)), eps, t_1), eps, t_0), eps, eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * 0.3333333333333333 + 0.3333333333333333), $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(eps * N[(0.3333333333333333 * t$95$1 + N[(t$95$2 * N[Tan[x], $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] * eps + t$95$1), $MachinePrecision] * eps + t$95$0), $MachinePrecision] * eps + eps), $MachinePrecision]]]]

Derivation

1. Initial program 62.8%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

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

9. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(0.3333333333333333 - -0.3333333333333333 \cdot {\tan x}^{2}\right) + {\tan x}^{4}\right) + {\tan x}^{2}, \tan x, \left(\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x\right) \cdot \frac{0.3333333333333333}{\cos x}\right), \varepsilon, \left(\left(0.3333333333333333 - -0.3333333333333333 \cdot {\tan x}^{2}\right) + {\tan x}^{4}\right) + {\tan x}^{2}\right), \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right)} \]

11. Applied rewrites 99.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333, \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \tan x, \mathsf{fma}\left({\tan x}^{4} + \mathsf{fma}\left({\tan x}^{2}, 0.3333333333333333, 0.3333333333333333\right), \tan x, {\tan x}^{3}\right)\right), \mathsf{fma}\left(\tan x, \tan x, {\tan x}^{4} + \mathsf{fma}\left({\tan x}^{2}, 0.3333333333333333, 0.3333333333333333\right)\right)\right), \varepsilon, \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \tan x\right), \varepsilon, {\tan x}^{2}\right), \color{blue}{\varepsilon}, \varepsilon\right) \]

12. Final simplification 99.6%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333, \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \tan x, \mathsf{fma}\left(\mathsf{fma}\left({\tan x}^{2}, 0.3333333333333333, 0.3333333333333333\right) + {\tan x}^{4}, \tan x, {\tan x}^{3}\right)\right), \mathsf{fma}\left(\tan x, \tan x, \mathsf{fma}\left({\tan x}^{2}, 0.3333333333333333, 0.3333333333333333\right) + {\tan x}^{4}\right)\right), \varepsilon, \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \tan x\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
13. Add Preprocessing

Alternative 2: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, \frac{\sin x \cdot \mathsf{fma}\left(\tan x, \tan x, 1\right)}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   (fma
    0.3333333333333333
    eps
    (/ (* (sin x) (fma (tan x) (tan x) 1.0)) (cos x)))
   eps
   (pow (tan x) 2.0))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(fma(fma(0.3333333333333333, eps, ((sin(x) * fma(tan(x), tan(x), 1.0)) / cos(x))), eps, pow(tan(x), 2.0)), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(fma(0.3333333333333333, eps, Float64(Float64(sin(x) * fma(tan(x), tan(x), 1.0)) / cos(x))), eps, (tan(x) ^ 2.0)), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(0.3333333333333333 * eps + N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 62.8%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

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

9. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(0.3333333333333333 - -0.3333333333333333 \cdot {\tan x}^{2}\right) + {\tan x}^{4}\right) + {\tan x}^{2}, \tan x, \left(\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x\right) \cdot \frac{0.3333333333333333}{\cos x}\right), \varepsilon, \left(\left(0.3333333333333333 - -0.3333333333333333 \cdot {\tan x}^{2}\right) + {\tan x}^{4}\right) + {\tan x}^{2}\right), \varepsilon, \frac{\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \sin x}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right)} \]

10. Taylor expanded in x around 0

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

12. Applied rewrites 99.2%

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

13. Final simplification 99.2%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, \varepsilon, \frac{\sin x \cdot \mathsf{fma}\left(\tan x, \tan x, 1\right)}{\cos x}\right), \varepsilon, {\tan x}^{2}\right), \varepsilon, \varepsilon\right) \]
14. Add Preprocessing

Alternative 3: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.6

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

5. Applied rewrites 98.6%

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

7. Applied rewrites 98.6%

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

Alternative 4: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.6

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

5. Applied rewrites 98.6%

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

Alternative 5: 99.1% 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 62.8%

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

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.6

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

5. Applied rewrites 98.6%

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

7. Applied rewrites 98.6%

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

Alternative 6: 98.5% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.9%

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

Alternative 7: 98.4% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.8%

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

Alternative 8: 98.4% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.6%

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

Alternative 9: 98.1% 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 62.8%

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

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. remove-double-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 97.4%

   \[\leadsto \varepsilon \]
9. Add Preprocessing

Developer Target 1: 99.1% 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 2024272 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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