2tan (problem 3.3.2)

Percentage Accurate: 62.1% → 99.5%

Time: 28.7s

Alternatives: 11

Speedup: 17.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.1% 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: 99.5% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (sin x) 2.0)) (t_1 (pow (cos x) 2.0)))
   (fma
    (fma
     (*
      (fma
       (fma
        (fma (/ (sin x) t_1) (sin x) 1.0)
        -0.5
        (/
         (fma (fma t_0 (/ -1.0 t_1) -1.0) t_0 (* t_0 0.16666666666666666))
         t_1))
       -1.0
       -0.16666666666666666)
      eps)
     eps
     (/ (fma (fma (/ t_0 t_1) eps eps) (sin x) (/ t_0 (cos x))) (cos x)))
    eps
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(sin(x), 2.0);
	double t_1 = pow(cos(x), 2.0);
	return fma(fma((fma(fma(fma((sin(x) / t_1), sin(x), 1.0), -0.5, (fma(fma(t_0, (-1.0 / t_1), -1.0), t_0, (t_0 * 0.16666666666666666)) / t_1)), -1.0, -0.16666666666666666) * eps), eps, (fma(fma((t_0 / t_1), eps, eps), sin(x), (t_0 / cos(x))) / cos(x))), eps, eps);
}

Julia

function code(x, eps)
	t_0 = sin(x) ^ 2.0
	t_1 = cos(x) ^ 2.0
	return fma(fma(Float64(fma(fma(fma(Float64(sin(x) / t_1), sin(x), 1.0), -0.5, Float64(fma(fma(t_0, Float64(-1.0 / t_1), -1.0), t_0, Float64(t_0 * 0.16666666666666666)) / t_1)), -1.0, -0.16666666666666666) * eps), eps, Float64(fma(fma(Float64(t_0 / t_1), eps, eps), sin(x), Float64(t_0 / cos(x))) / cos(x))), eps, eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(N[(N[(N[(N[(N[(N[Sin[x], $MachinePrecision] / t$95$1), $MachinePrecision] * N[Sin[x], $MachinePrecision] + 1.0), $MachinePrecision] * -0.5 + N[(N[(N[(t$95$0 * N[(-1.0 / t$95$1), $MachinePrecision] + -1.0), $MachinePrecision] * t$95$0 + N[(t$95$0 * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * -1.0 + -0.16666666666666666), $MachinePrecision] * eps), $MachinePrecision] * eps + N[(N[(N[(N[(t$95$0 / t$95$1), $MachinePrecision] * eps + eps), $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(t$95$0 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]]]

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.2%

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

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x + \mathsf{fma}\left(\varepsilon, t\_0, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(-\mathsf{fma}\left(t\_0, \mathsf{fma}\left(-1, t\_0, -1\right) + -0.3333333333333333, -0.5\right)\right) + -0.16666666666666666\right)\right), \varepsilon, \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)))
   (fma
    (fma
     (tan x)
     (+ (tan x) (fma eps t_0 eps))
     (*
      (* eps eps)
      (+
       (- (fma t_0 (+ (fma -1.0 t_0 -1.0) -0.3333333333333333) -0.5))
       -0.16666666666666666)))
    eps
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(tan(x), 2.0);
	return fma(fma(tan(x), (tan(x) + fma(eps, t_0, eps)), ((eps * eps) * (-fma(t_0, (fma(-1.0, t_0, -1.0) + -0.3333333333333333), -0.5) + -0.16666666666666666))), eps, eps);
}

Julia

function code(x, eps)
	t_0 = tan(x) ^ 2.0
	return fma(fma(tan(x), Float64(tan(x) + fma(eps, t_0, eps)), Float64(Float64(eps * eps) * Float64(Float64(-fma(t_0, Float64(fma(-1.0, t_0, -1.0) + -0.3333333333333333), -0.5)) + -0.16666666666666666))), eps, eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(N[Tan[x], $MachinePrecision] * N[(N[Tan[x], $MachinePrecision] + N[(eps * t$95$0 + eps), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * N[((-N[(t$95$0 * N[(N[(-1.0 * t$95$0 + -1.0), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision]) + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]]

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sin x \cdot \tan x, \frac{-1}{\cos x}, -1\right), {\tan x}^{2}, \mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.5\right)\right)\right), -1, -0.16666666666666666\right), \mathsf{fma}\left(\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right), \tan x, {\tan x}^{2}\right)\right), \varepsilon, \varepsilon\right)} \]
8. Step-by-step derivation

9. Applied rewrites 99.2%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\tan x, \tan x + \mathsf{fma}\left(\varepsilon, {\tan x}^{2}, \varepsilon\right), \left(\varepsilon \cdot \varepsilon\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{\cos x}, \sin x \cdot \tan x, -1\right), {\tan x}^{2}, \mathsf{fma}\left({\tan x}^{2}, -0.3333333333333333, -0.5\right)\right), -1, -0.16666666666666666\right)\right), \varepsilon, \varepsilon\right) \]
10. Step-by-step derivation

11. Applied rewrites 99.2%

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

Alternative 3: 99.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (fma
   (tan x)
   (+ (tan x) (fma eps (pow (tan x) 2.0) eps))
   (* (* eps eps) 0.3333333333333333))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(fma(tan(x), (tan(x) + fma(eps, pow(tan(x), 2.0), eps)), ((eps * eps) * 0.3333333333333333)), eps, eps);
}

Julia

function code(x, eps)
	return fma(fma(tan(x), Float64(tan(x) + fma(eps, (tan(x) ^ 2.0), eps)), Float64(Float64(eps * eps) * 0.3333333333333333)), eps, eps)
end

Wolfram

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

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\sin x \cdot \tan x, \frac{-1}{\cos x}, -1\right), {\tan x}^{2}, \mathsf{fma}\left(0.16666666666666666, {\tan x}^{2}, \mathsf{fma}\left({\tan x}^{2}, -0.5, -0.5\right)\right)\right), -1, -0.16666666666666666\right), \mathsf{fma}\left(\mathsf{fma}\left({\tan x}^{2}, \varepsilon, \varepsilon\right), \tan x, {\tan x}^{2}\right)\right), \varepsilon, \varepsilon\right)} \]
8. Step-by-step derivation

9. Applied rewrites 99.2%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 99.0%

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

Alternative 4: 98.9% 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 60.4%

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

3. Taylor expanded in eps around 0

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

   1. fp-cancel-sub-sign-inv N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity 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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in eps around 0

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

   1. fp-cancel-sub-sign-inv N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity 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: 98.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\left(\mathsf{fma}\left(0.044444444444444446 \cdot \left(x \cdot x\right) - 0.3333333333333333, x \cdot x, 1\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\left(0.3333333333333333 \cdot x\right) \cdot x - 1, x \cdot x, 1\right)}, \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (/
   (*
    (*
     (fma (- (* 0.044444444444444446 (* x x)) 0.3333333333333333) (* x x) 1.0)
     x)
    x)
   (fma (- (* (* 0.3333333333333333 x) x) 1.0) (* x x) 1.0))
  eps
  eps))

C

double code(double x, double eps) {
	return fma((((fma(((0.044444444444444446 * (x * x)) - 0.3333333333333333), (x * x), 1.0) * x) * x) / fma((((0.3333333333333333 * x) * x) - 1.0), (x * x), 1.0)), eps, eps);
}

Julia

function code(x, eps)
	return fma(Float64(Float64(Float64(fma(Float64(Float64(0.044444444444444446 * Float64(x * x)) - 0.3333333333333333), Float64(x * x), 1.0) * x) * x) / fma(Float64(Float64(Float64(0.3333333333333333 * x) * x) - 1.0), Float64(x * x), 1.0)), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(N[(0.044444444444444446 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] / N[(N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in eps around 0

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

   1. fp-cancel-sub-sign-inv N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity 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

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

8. Applied rewrites 98.1%

   \[\leadsto \mathsf{fma}\left(\frac{\left(\mathsf{fma}\left(0.044444444444444446 \cdot \left(x \cdot x\right) - 0.3333333333333333, x \cdot x, 1\right) \cdot x\right) \cdot x}{{\cos x}^{2}}, \varepsilon, \varepsilon\right) \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.2%

    \[\leadsto \mathsf{fma}\left(\frac{\left(\mathsf{fma}\left(0.044444444444444446 \cdot \left(x \cdot x\right) - 0.3333333333333333, x \cdot x, 1\right) \cdot x\right) \cdot x}{\mathsf{fma}\left(\left(0.3333333333333333 \cdot x\right) \cdot x - 1, x \cdot x, 1\right)}, \varepsilon, \varepsilon\right) \]
12. Add Preprocessing

Alternative 7: 98.4% 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 60.4%

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

3. Taylor expanded in eps around 0

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

   1. fp-cancel-sub-sign-inv N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity 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 98.1%

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

Alternative 8: 98.3% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in eps around 0

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

   1. fp-cancel-sub-sign-inv N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity 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 98.1%

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

Alternative 9: 98.3% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.1%

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

11. Taylor expanded in eps around 0

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

13. Applied rewrites 98.1%

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

Alternative 10: 98.2% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in eps around 0

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

   1. fp-cancel-sub-sign-inv N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity 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 98.1%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.1%

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

Alternative 11: 6.4% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in eps around 0

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

   1. fp-cancel-sub-sign-inv N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity 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 98.1%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.1%

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

12. Taylor expanded in x around inf

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

14. Applied rewrites 6.6%

    \[\leadsto \left(\varepsilon \cdot x\right) \cdot x \]
15. 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 2024331 
(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)))