2tan (problem 3.3.2)

Percentage Accurate: 62.1% → 99.6%

Time: 15.8s

Alternatives: 12

Speedup: 34.5×

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: 62.1% 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: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 100.0%

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

7. Applied rewrites 100.0%

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

9. Applied rewrites 100.0%

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

11. Applied rewrites 100.0%

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

12. Final simplification 100.0%

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

Alternative 2: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 100.0%

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

7. Applied rewrites 100.0%

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

9. Applied rewrites 100.0%

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

11. Applied rewrites 100.0%

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

12. Final simplification 100.0%

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

Alternative 3: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \mathsf{fma}\left(\varepsilon, \tan x, 1\right)\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), x, 0.3333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \frac{\left({t\_0}^{3} + {t\_1}^{3}\right) \cdot \varepsilon}{\mathsf{fma}\left(t\_0, t\_0 - t\_1, {t\_1}^{2}\right)} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (fma (tan x) (tan x) 1.0) (fma eps (tan x) 1.0)))
        (t_1
         (*
          (fma
           (fma 0.6666666666666666 eps (* 1.3333333333333333 x))
           x
           0.3333333333333333)
          (* eps eps))))
   (/
    (* (+ (pow t_0 3.0) (pow t_1 3.0)) eps)
    (fma t_0 (- t_0 t_1) (pow t_1 2.0)))))

C

double code(double x, double eps) {
	double t_0 = fma(tan(x), tan(x), 1.0) * fma(eps, tan(x), 1.0);
	double t_1 = fma(fma(0.6666666666666666, eps, (1.3333333333333333 * x)), x, 0.3333333333333333) * (eps * eps);
	return ((pow(t_0, 3.0) + pow(t_1, 3.0)) * eps) / fma(t_0, (t_0 - t_1), pow(t_1, 2.0));
}

Julia

function code(x, eps)
	t_0 = Float64(fma(tan(x), tan(x), 1.0) * fma(eps, tan(x), 1.0))
	t_1 = Float64(fma(fma(0.6666666666666666, eps, Float64(1.3333333333333333 * x)), x, 0.3333333333333333) * Float64(eps * eps))
	return Float64(Float64(Float64((t_0 ^ 3.0) + (t_1 ^ 3.0)) * eps) / fma(t_0, Float64(t_0 - t_1), (t_1 ^ 2.0)))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision] * N[(eps * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.6666666666666666 * eps + N[(1.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] * x + 0.3333333333333333), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] / N[(t$95$0 * N[(t$95$0 - t$95$1), $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 60.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), x, 0.3333333333333333\right) \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot \left(\frac{\sin x \cdot \varepsilon}{\cos x} + 1\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

10. Applied rewrites 99.8%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), x, 0.3333333333333333\right) \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot \frac{{\left(\varepsilon \cdot \tan x\right)}^{2} - 1}{\mathsf{fma}\left(\varepsilon, \tan x, -1\right)}\right) \cdot \varepsilon \]

12. Applied rewrites 99.8%

    \[\leadsto \frac{\left({\left(\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \mathsf{fma}\left(\varepsilon, \tan x, 1\right)\right)}^{3} + {\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), x, 0.3333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}^{3}\right) \cdot \varepsilon}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \mathsf{fma}\left(\varepsilon, \tan x, 1\right), \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \mathsf{fma}\left(\varepsilon, \tan x, 1\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), x, 0.3333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right), {\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), x, 0.3333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}^{2}\right)}} \]
13. Add Preprocessing

Alternative 4: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({\left(\varepsilon \cdot \tan x\right)}^{2} - 1, {\left(\mathsf{fma}\left(\varepsilon, \tan x, -1\right)\right)}^{-1} \cdot \mathsf{fma}\left(\tan x, \tan x, 1\right), \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), x, 0.3333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (fma
   (- (pow (* eps (tan x)) 2.0) 1.0)
   (* (pow (fma eps (tan x) -1.0) -1.0) (fma (tan x) (tan x) 1.0))
   (*
    (fma
     (fma 0.6666666666666666 eps (* 1.3333333333333333 x))
     x
     0.3333333333333333)
    (* eps eps)))
  eps))

C

double code(double x, double eps) {
	return fma((pow((eps * tan(x)), 2.0) - 1.0), (pow(fma(eps, tan(x), -1.0), -1.0) * fma(tan(x), tan(x), 1.0)), (fma(fma(0.6666666666666666, eps, (1.3333333333333333 * x)), x, 0.3333333333333333) * (eps * eps))) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(Float64((Float64(eps * tan(x)) ^ 2.0) - 1.0), Float64((fma(eps, tan(x), -1.0) ^ -1.0) * fma(tan(x), tan(x), 1.0)), Float64(fma(fma(0.6666666666666666, eps, Float64(1.3333333333333333 * x)), x, 0.3333333333333333) * Float64(eps * eps))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[Power[N[(eps * N[Tan[x], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[Power[N[(eps * N[Tan[x], $MachinePrecision] + -1.0), $MachinePrecision], -1.0], $MachinePrecision] * N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.6666666666666666 * eps + N[(1.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] * x + 0.3333333333333333), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 60.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), x, 0.3333333333333333\right) \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot \left(\frac{\sin x \cdot \varepsilon}{\cos x} + 1\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

10. Applied rewrites 99.8%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), x, 0.3333333333333333\right) \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot \frac{{\left(\varepsilon \cdot \tan x\right)}^{2} - 1}{\mathsf{fma}\left(\varepsilon, \tan x, -1\right)}\right) \cdot \varepsilon \]
11. Step-by-step derivation

12. Applied rewrites 99.8%

    \[\leadsto \mathsf{fma}\left({\left(\varepsilon \cdot \tan x\right)}^{2} - 1, {\left(\mathsf{fma}\left(\varepsilon, \tan x, -1\right)\right)}^{-1} \cdot \mathsf{fma}\left(\tan x, \tan x, 1\right), \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), x, 0.3333333333333333\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]
13. Add Preprocessing

Alternative 5: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), x, 0.3333333333333333\right) \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \mathsf{fma}\left(\varepsilon, \tan x, 1\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (fma
   (*
    (fma
     (fma 0.6666666666666666 eps (* 1.3333333333333333 x))
     x
     0.3333333333333333)
    eps)
   eps
   (* (fma (tan x) (tan x) 1.0) (fma eps (tan x) 1.0)))
  eps))

C

double code(double x, double eps) {
	return fma((fma(fma(0.6666666666666666, eps, (1.3333333333333333 * x)), x, 0.3333333333333333) * eps), eps, (fma(tan(x), tan(x), 1.0) * fma(eps, tan(x), 1.0))) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(Float64(fma(fma(0.6666666666666666, eps, Float64(1.3333333333333333 * x)), x, 0.3333333333333333) * eps), eps, Float64(fma(tan(x), tan(x), 1.0) * fma(eps, tan(x), 1.0))) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(0.6666666666666666 * eps + N[(1.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] * x + 0.3333333333333333), $MachinePrecision] * eps), $MachinePrecision] * eps + N[(N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision] * N[(eps * N[Tan[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 60.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), x, 0.3333333333333333\right) \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot \left(\frac{\sin x \cdot \varepsilon}{\cos x} + 1\right)\right) \cdot \varepsilon \]
9. Step-by-step derivation

10. Applied rewrites 99.8%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), x, 0.3333333333333333\right) \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\tan x, \tan x, 1\right) \cdot \mathsf{fma}\left(\varepsilon, \tan x, 1\right)\right) \cdot \varepsilon \]
11. Add Preprocessing

Alternative 6: 99.5% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (fma
   (* eps eps)
   0.3333333333333333
   (* (fma eps (tan x) 1.0) (fma (tan x) (tan x) 1.0)))
  eps))

C

double code(double x, double eps) {
	return fma((eps * eps), 0.3333333333333333, (fma(eps, tan(x), 1.0) * fma(tan(x), tan(x), 1.0))) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(Float64(eps * eps), 0.3333333333333333, Float64(fma(eps, tan(x), 1.0) * fma(tan(x), tan(x), 1.0))) * eps)
end

Wolfram

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

Derivation

1. Initial program 60.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 100.0%

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.8%

    \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.3333333333333333, \mathsf{fma}\left(\varepsilon, \tan x, 1\right) \cdot \mathsf{fma}\left(\tan x, \tan x, 1\right)\right) \cdot \varepsilon \]
11. Add Preprocessing

Alternative 7: 98.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 1.3333333333333333, 1\right), x, \mathsf{fma}\left({\varepsilon}^{3}, 0.6666666666666666, \varepsilon\right)\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.3333333333333333, 1\right) \cdot \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (*
   eps
   (fma
    (fma (* eps eps) 1.3333333333333333 1.0)
    x
    (fma (pow eps 3.0) 0.6666666666666666 eps)))
  x
  (* (fma (* eps eps) 0.3333333333333333 1.0) eps)))

C

double code(double x, double eps) {
	return fma((eps * fma(fma((eps * eps), 1.3333333333333333, 1.0), x, fma(pow(eps, 3.0), 0.6666666666666666, eps))), x, (fma((eps * eps), 0.3333333333333333, 1.0) * eps));
}

Julia

function code(x, eps)
	return fma(Float64(eps * fma(fma(Float64(eps * eps), 1.3333333333333333, 1.0), x, fma((eps ^ 3.0), 0.6666666666666666, eps))), x, Float64(fma(Float64(eps * eps), 0.3333333333333333, 1.0) * eps))
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(N[(N[(eps * eps), $MachinePrecision] * 1.3333333333333333 + 1.0), $MachinePrecision] * x + N[(N[Power[eps, 3.0], $MachinePrecision] * 0.6666666666666666 + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + N[(N[(N[(eps * eps), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), x, 0.3333333333333333\right) \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot \left(\frac{\sin x \cdot \varepsilon}{\cos x} + 1\right)\right) \cdot \varepsilon \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 99.2%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 1.3333333333333333, 1\right), x, \mathsf{fma}\left({\varepsilon}^{3}, 0.6666666666666666, \varepsilon\right)\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.3333333333333333, 1\right)\right) \cdot \varepsilon \]

12. Taylor expanded in x around 0

    \[\leadsto \varepsilon \cdot \left(1 + \frac{1}{3} \cdot {\varepsilon}^{2}\right) + \color{blue}{x \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + \varepsilon \cdot \left(\varepsilon + \frac{2}{3} \cdot {\varepsilon}^{3}\right)\right)} \]
13. Step-by-step derivation

14. Applied rewrites 99.2%

    \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 1.3333333333333333, 1\right), x, \mathsf{fma}\left({\varepsilon}^{3}, 0.6666666666666666, \varepsilon\right)\right), \color{blue}{x}, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.3333333333333333, 1\right) \cdot \varepsilon\right) \]
15. Add Preprocessing

Alternative 8: 98.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \varepsilon, x, \mathsf{fma}\left(x \cdot x, 1.3333333333333333, 0.3333333333333333\right)\right), \varepsilon, x\right), \varepsilon, \mathsf{fma}\left(x, x, 1\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (fma
   (fma
    (fma
     (* 0.6666666666666666 eps)
     x
     (fma (* x x) 1.3333333333333333 0.3333333333333333))
    eps
    x)
   eps
   (fma x x 1.0))
  eps))

C

double code(double x, double eps) {
	return fma(fma(fma((0.6666666666666666 * eps), x, fma((x * x), 1.3333333333333333, 0.3333333333333333)), eps, x), eps, fma(x, x, 1.0)) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(fma(fma(Float64(0.6666666666666666 * eps), x, fma(Float64(x * x), 1.3333333333333333, 0.3333333333333333)), eps, x), eps, fma(x, x, 1.0)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(0.6666666666666666 * eps), $MachinePrecision] * x + N[(N[(x * x), $MachinePrecision] * 1.3333333333333333 + 0.3333333333333333), $MachinePrecision]), $MachinePrecision] * eps + x), $MachinePrecision] * eps + N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 60.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), x, 0.3333333333333333\right) \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot \left(\frac{\sin x \cdot \varepsilon}{\cos x} + 1\right)\right) \cdot \varepsilon \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 99.2%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 1.3333333333333333, 1\right), x, \mathsf{fma}\left({\varepsilon}^{3}, 0.6666666666666666, \varepsilon\right)\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.3333333333333333, 1\right)\right) \cdot \varepsilon \]

12. Taylor expanded in eps around 0

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

14. Applied rewrites 99.2%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \varepsilon, x, \mathsf{fma}\left(x \cdot x, 1.3333333333333333, 0.3333333333333333\right)\right), \varepsilon, x\right), \varepsilon, \mathsf{fma}\left(x, x, 1\right)\right) \cdot \varepsilon \]
15. Add Preprocessing

Alternative 9: 98.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 1.3333333333333333, 0.3333333333333333\right), \varepsilon, x\right), \varepsilon, \mathsf{fma}\left(x, x, 1\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (fma
   (fma (fma (* x x) 1.3333333333333333 0.3333333333333333) eps x)
   eps
   (fma x x 1.0))
  eps))

C

double code(double x, double eps) {
	return fma(fma(fma((x * x), 1.3333333333333333, 0.3333333333333333), eps, x), eps, fma(x, x, 1.0)) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(fma(fma(Float64(x * x), 1.3333333333333333, 0.3333333333333333), eps, x), eps, fma(x, x, 1.0)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 1.3333333333333333 + 0.3333333333333333), $MachinePrecision] * eps + x), $MachinePrecision] * eps + N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 60.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), x, 0.3333333333333333\right) \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot \left(\frac{\sin x \cdot \varepsilon}{\cos x} + 1\right)\right) \cdot \varepsilon \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 99.2%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 1.3333333333333333, 1\right), x, \mathsf{fma}\left({\varepsilon}^{3}, 0.6666666666666666, \varepsilon\right)\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.3333333333333333, 1\right)\right) \cdot \varepsilon \]

12. Taylor expanded in eps around 0

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

14. Applied rewrites 99.2%

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

Alternative 10: 98.2% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, \varepsilon, 1.3333333333333333 \cdot x\right), x, 0.3333333333333333\right) \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\frac{\sin x}{{\cos x}^{2}}, \sin x, 1\right) \cdot \left(\frac{\sin x \cdot \varepsilon}{\cos x} + 1\right)\right) \cdot \varepsilon \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 99.2%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 1.3333333333333333, 1\right), x, \mathsf{fma}\left({\varepsilon}^{3}, 0.6666666666666666, \varepsilon\right)\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.3333333333333333, 1\right)\right) \cdot \varepsilon \]

12. Taylor expanded in eps around 0

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

14. Applied rewrites 99.1%

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

Alternative 11: 98.1% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.2

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.9%

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

10. Applied rewrites 98.9%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 98.9%

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

Alternative 12: 97.7% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ 1 \cdot \varepsilon \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 1.0 * eps

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = 1.0 * eps;
end

Wolfram

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

Derivation

1. Initial program 60.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 99.2

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.9%

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

10. Applied rewrites 98.9%

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

11. Taylor expanded in x around 0

    \[\leadsto 1 \cdot \varepsilon \]
12. Step-by-step derivation

13. Applied rewrites 98.4%

    \[\leadsto 1 \cdot \varepsilon \]
14. Add Preprocessing

Developer Target 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024318 
(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)))