2tan (problem 3.3.2)

Percentage Accurate: 62.6% → 99.6%

Time: 15.5s

Alternatives: 12

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: 62.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ t_1 := {\sin x}^{2}\\ t_2 := \frac{t\_1}{t\_0}\\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \left(\frac{t\_1 + \frac{{\sin x}^{4}}{t\_0}}{t\_0} - \mathsf{fma}\left(0.16666666666666666, t\_2, \mathsf{fma}\left(-0.5, t\_2, -0.5\right)\right)\right) + -0.16666666666666666, \frac{\sin x + \frac{{\sin x}^{3}}{t\_0}}{\cos x}\right), t\_2\right), \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0)) (t_1 (pow (sin x) 2.0)) (t_2 (/ t_1 t_0)))
   (fma
    eps
    (fma
     eps
     (fma
      eps
      (+
       (-
        (/ (+ t_1 (/ (pow (sin x) 4.0) t_0)) t_0)
        (fma 0.16666666666666666 t_2 (fma -0.5 t_2 -0.5)))
       -0.16666666666666666)
      (/ (+ (sin x) (/ (pow (sin x) 3.0) t_0)) (cos x)))
     t_2)
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	double t_1 = pow(sin(x), 2.0);
	double t_2 = t_1 / t_0;
	return fma(eps, fma(eps, fma(eps, ((((t_1 + (pow(sin(x), 4.0) / t_0)) / t_0) - fma(0.16666666666666666, t_2, fma(-0.5, t_2, -0.5))) + -0.16666666666666666), ((sin(x) + (pow(sin(x), 3.0) / t_0)) / cos(x))), t_2), eps);
}

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	t_1 = sin(x) ^ 2.0
	t_2 = Float64(t_1 / t_0)
	return fma(eps, fma(eps, fma(eps, Float64(Float64(Float64(Float64(t_1 + Float64((sin(x) ^ 4.0) / t_0)) / t_0) - fma(0.16666666666666666, t_2, fma(-0.5, t_2, -0.5))) + -0.16666666666666666), Float64(Float64(sin(x) + Float64((sin(x) ^ 3.0) / t_0)) / cos(x))), t_2), eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / t$95$0), $MachinePrecision]}, N[(eps * N[(eps * N[(eps * N[(N[(N[(N[(t$95$1 + N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(0.16666666666666666 * t$95$2 + N[(-0.5 * t$95$2 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + eps), $MachinePrecision]]]]

Derivation

1. Initial program 62.0%

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

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \tan x + {\tan x}^{3}, {\tan x}^{2}\right), \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma eps (fma eps (+ (tan x) (pow (tan x) 3.0)) (pow (tan x) 2.0)) eps))

C

double code(double x, double eps) {
	return fma(eps, fma(eps, (tan(x) + pow(tan(x), 3.0)), pow(tan(x), 2.0)), eps);
}

Julia

function code(x, eps)
	return fma(eps, fma(eps, Float64(tan(x) + (tan(x) ^ 3.0)), (tan(x) ^ 2.0)), eps)
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * N[(N[Tan[x], $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]

Derivation

1. Initial program 62.0%

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

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 100.0%

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

10. Applied rewrites 100.0%

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

11. Final simplification 100.0%

    \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \tan x + {\tan x}^{3}, {\tan x}^{2}\right), \varepsilon\right) \]
12. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x \]

   4. lower-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right)} - \tan x \]

   5. lower-cos.f64 62.0

      \[\leadsto \frac{\sin \left(x + \varepsilon\right)}{\color{blue}{\cos \left(x + \varepsilon\right)}} - \tan x \]

4. Applied rewrites 62.0%

   \[\leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x \]

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
6. 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. mul-1-neg N/A

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

   3. remove-double-neg N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon + 1 \cdot \varepsilon} \]

   6. *-lft-identity N/A

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

   7. lower-fma.f64 N/A

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

   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 99.8

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 99.8%

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

11. Applied rewrites 99.8%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x \]

   4. lower-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right)} - \tan x \]

   5. lower-cos.f64 62.0

      \[\leadsto \frac{\sin \left(x + \varepsilon\right)}{\color{blue}{\cos \left(x + \varepsilon\right)}} - \tan x \]

4. Applied rewrites 62.0%

   \[\leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x \]

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
6. 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. mul-1-neg N/A

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

   3. remove-double-neg N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon + 1 \cdot \varepsilon} \]

   6. *-lft-identity N/A

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

   7. lower-fma.f64 N/A

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

   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 99.8

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 99.8%

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

Alternative 5: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\right)}^{2}, \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (pow
   (fma
    (fma
     (* x x)
     (fma (* x x) 0.05396825396825397 0.13333333333333333)
     0.3333333333333333)
    (* x (* x x))
    x)
   2.0)
  eps
  eps))

C

double code(double x, double eps) {
	return fma(pow(fma(fma((x * x), fma((x * x), 0.05396825396825397, 0.13333333333333333), 0.3333333333333333), (x * (x * x)), x), 2.0), eps, eps);
}

Julia

function code(x, eps)
	return fma((fma(fma(Float64(x * x), fma(Float64(x * x), 0.05396825396825397, 0.13333333333333333), 0.3333333333333333), Float64(x * Float64(x * x)), x) ^ 2.0), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[Power[N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.05396825396825397 + 0.13333333333333333), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], 2.0], $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 62.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x \]

   4. lower-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right)} - \tan x \]

   5. lower-cos.f64 62.0

      \[\leadsto \frac{\sin \left(x + \varepsilon\right)}{\color{blue}{\cos \left(x + \varepsilon\right)}} - \tan x \]

4. Applied rewrites 62.0%

   \[\leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x \]

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
6. 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. mul-1-neg N/A

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

   3. remove-double-neg N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon + 1 \cdot \varepsilon} \]

   6. *-lft-identity N/A

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

   7. lower-fma.f64 N/A

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

   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 99.8

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 99.8%

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

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left({\left(x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {x}^{2}\right)\right)\right)\right)}^{2}, \varepsilon, \varepsilon\right) \]
11. Step-by-step derivation

12. Applied rewrites 99.3%

    \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)\right)}^{2}, \varepsilon, \varepsilon\right) \]
13. Add Preprocessing

Alternative 6: 98.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.19682539682539682, 0.37777777777777777\right), 0.6666666666666666\right), 1\right), \varepsilon, \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (*
   (* x x)
   (fma
    (* x x)
    (fma
     x
     (* x (fma (* x x) 0.19682539682539682 0.37777777777777777))
     0.6666666666666666)
    1.0))
  eps
  eps))

C

double code(double x, double eps) {
	return fma(((x * x) * fma((x * x), fma(x, (x * fma((x * x), 0.19682539682539682, 0.37777777777777777)), 0.6666666666666666), 1.0)), eps, eps);
}

Julia

function code(x, eps)
	return fma(Float64(Float64(x * x) * fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 0.19682539682539682, 0.37777777777777777)), 0.6666666666666666), 1.0)), eps, eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.19682539682539682 + 0.37777777777777777), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * eps + eps), $MachinePrecision]

Derivation

1. Initial program 62.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x \]

   4. lower-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right)} - \tan x \]

   5. lower-cos.f64 62.0

      \[\leadsto \frac{\sin \left(x + \varepsilon\right)}{\color{blue}{\cos \left(x + \varepsilon\right)}} - \tan x \]

4. Applied rewrites 62.0%

   \[\leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x \]

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
6. 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. mul-1-neg N/A

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

   3. remove-double-neg N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon + 1 \cdot \varepsilon} \]

   6. *-lft-identity N/A

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

   7. lower-fma.f64 N/A

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

   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 99.8

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{17}{45} + \frac{62}{315} \cdot {x}^{2}\right)\right)\right), \varepsilon, \varepsilon\right) \]
9. Step-by-step derivation

10. Applied rewrites 99.2%

    \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.19682539682539682, 0.37777777777777777\right), 0.6666666666666666\right), 1\right), \varepsilon, \varepsilon\right) \]
11. Add Preprocessing

Alternative 7: 98.6% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 99.2%

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

Alternative 8: 98.5% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x \]

   4. lower-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right)} - \tan x \]

   5. lower-cos.f64 62.0

      \[\leadsto \frac{\sin \left(x + \varepsilon\right)}{\color{blue}{\cos \left(x + \varepsilon\right)}} - \tan x \]

4. Applied rewrites 62.0%

   \[\leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x \]

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
6. 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. mul-1-neg N/A

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

   3. remove-double-neg N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon + 1 \cdot \varepsilon} \]

   6. *-lft-identity N/A

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

   7. lower-fma.f64 N/A

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

   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 99.8

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.1%

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

Alternative 9: 98.5% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.0%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 99.0%

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

12. Taylor expanded in x around 0

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

14. Applied rewrites 99.0%

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

Alternative 10: 98.5% accurate, 13.8× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.0%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 99.0%

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

Alternative 11: 98.4% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.0%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x \]

   4. lower-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin \left(x + \varepsilon\right)}}{\cos \left(x + \varepsilon\right)} - \tan x \]

   5. lower-cos.f64 62.0

      \[\leadsto \frac{\sin \left(x + \varepsilon\right)}{\color{blue}{\cos \left(x + \varepsilon\right)}} - \tan x \]

4. Applied rewrites 62.0%

   \[\leadsto \color{blue}{\frac{\sin \left(x + \varepsilon\right)}{\cos \left(x + \varepsilon\right)}} - \tan x \]

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 - -1 \cdot \frac{{\sin x}^{2}}{{\cos x}^{2}}\right)} \]
6. 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. mul-1-neg N/A

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

   3. remove-double-neg N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\frac{{\sin x}^{2}}{{\cos x}^{2}} \cdot \varepsilon + 1 \cdot \varepsilon} \]

   6. *-lft-identity N/A

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

   7. lower-fma.f64 N/A

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

   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 99.8

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.9%

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

Alternative 12: 98.0% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.5%

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

Developer Target 1: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 62.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024237 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (/ (sin eps) (* (cos x) (cos (+ x eps)))))

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

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

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