2tan (problem 3.3.2)

Percentage Accurate: 62.4% → 99.4%

Time: 12.2s

Alternatives: 6

Speedup: 205.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.4% 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: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps + (eps * (tan(x) ** 2.0d0))) + ((eps ** 2.0d0) * (tan(x) + (tan(x) ** 3.0d0)))
end function

Java

public static double code(double x, double eps) {
	return (eps + (eps * Math.pow(Math.tan(x), 2.0))) + (Math.pow(eps, 2.0) * (Math.tan(x) + Math.pow(Math.tan(x), 3.0)));
}

Python

def code(x, eps):
	return (eps + (eps * math.pow(math.tan(x), 2.0))) + (math.pow(eps, 2.0) * (math.tan(x) + math.pow(math.tan(x), 3.0)))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = (eps + (eps * (tan(x) ^ 2.0))) + ((eps ^ 2.0) * (tan(x) + (tan(x) ^ 3.0)));
end

Wolfram

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

Derivation

1. Initial program 60.5%

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

3. Taylor expanded in eps around 0 99.1%

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

   1. cancel-sign-sub-inv 99.1%

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

   2. metadata-eval 99.1%

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

   3. *-un-lft-identity 99.1%

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

   4. distribute-rgt-in 99.1%

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

   5. *-un-lft-identity 99.1%

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

   6. unpow2 99.1%

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

   7. unpow2 99.1%

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

   8. frac-times 99.1%

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

   9. tan-quot 99.1%

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

   10. tan-quot 99.1%

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

   11. pow2 99.1%

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

5. Applied egg-rr 99.1%

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

   1. associate-/l* 99.1%

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

   2. *-commutative 99.1%

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

   3. *-un-lft-identity 99.1%

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

   4. times-frac 99.1%

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

7. Applied egg-rr 99.1%

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

   1. /-rgt-identity 99.1%

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

   2. *-commutative 99.1%

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

   3. distribute-lft-in 99.1%

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

   4. *-rgt-identity 99.1%

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

   5. unpow2 99.1%

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

   6. cube-mult 99.1%

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

9. Simplified 99.1%

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

10. Final simplification 99.1%

    \[\leadsto \left(\varepsilon + \varepsilon \cdot {\tan x}^{2}\right) + {\varepsilon}^{2} \cdot \left(\tan x + {\tan x}^{3}\right) \]
11. Add Preprocessing

Alternative 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \varepsilon + e^{\log \left(\varepsilon \cdot {\tan x}^{2}\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+ eps (exp (log (* eps (pow (tan x) 2.0))))))

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + exp(log((eps * (tan(x) ** 2.0d0))))
end function

Java

public static double code(double x, double eps) {
	return eps + Math.exp(Math.log((eps * Math.pow(Math.tan(x), 2.0))));
}

Python

def code(x, eps):
	return eps + math.exp(math.log((eps * math.pow(math.tan(x), 2.0))))

Julia

function code(x, eps)
	return Float64(eps + exp(log(Float64(eps * (tan(x) ^ 2.0)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps + exp(log((eps * (tan(x) ^ 2.0))));
end

Wolfram

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

Derivation

1. Initial program 60.5%

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

3. Taylor expanded in eps around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. metadata-eval 98.9%

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

   3. *-lft-identity 98.9%

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

5. Simplified 98.9%

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

   1. unpow2 98.9%

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

   2. unpow2 98.9%

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

   3. frac-times 98.9%

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

   4. tan-quot 98.9%

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

   5. tan-quot 98.9%

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

   6. *-un-lft-identity 98.9%

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

   7. pow2 98.9%

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

7. Applied egg-rr 98.9%

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

   1. *-lft-identity 98.9%

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

9. Simplified 98.9%

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

    1. distribute-rgt-in 98.9%

       \[\leadsto \color{blue}{1 \cdot \varepsilon + {\tan x}^{2} \cdot \varepsilon} \]

    2. *-un-lft-identity 98.9%

       \[\leadsto \color{blue}{\varepsilon} + {\tan x}^{2} \cdot \varepsilon \]

    3. +-commutative 98.9%

       \[\leadsto \color{blue}{{\tan x}^{2} \cdot \varepsilon + \varepsilon} \]

11. Applied egg-rr 98.9%

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

    1. add-exp-log 98.9%

       \[\leadsto \color{blue}{e^{\log \left({\tan x}^{2} \cdot \varepsilon\right)}} + \varepsilon \]

13. Applied egg-rr 98.9%

    \[\leadsto \color{blue}{e^{\log \left({\tan x}^{2} \cdot \varepsilon\right)}} + \varepsilon \]

14. Final simplification 98.9%

    \[\leadsto \varepsilon + e^{\log \left(\varepsilon \cdot {\tan x}^{2}\right)} \]
15. Add Preprocessing

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left({\tan x}^{2} + 1\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (+ (pow (tan x) 2.0) 1.0)))

C

double code(double x, double eps) {
	return eps * (pow(tan(x), 2.0) + 1.0);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((tan(x) ** 2.0d0) + 1.0d0)
end function

Java

public static double code(double x, double eps) {
	return eps * (Math.pow(Math.tan(x), 2.0) + 1.0);
}

Python

def code(x, eps):
	return eps * (math.pow(math.tan(x), 2.0) + 1.0)

Julia

function code(x, eps)
	return Float64(eps * Float64((tan(x) ^ 2.0) + 1.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((tan(x) ^ 2.0) + 1.0);
end

Wolfram

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

Derivation

1. Initial program 60.5%

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

3. Taylor expanded in eps around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. metadata-eval 98.9%

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

   3. *-lft-identity 98.9%

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

5. Simplified 98.9%

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

   1. unpow2 98.9%

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

   2. unpow2 98.9%

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

   3. frac-times 98.9%

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

   4. tan-quot 98.9%

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

   5. tan-quot 98.9%

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

   6. *-un-lft-identity 98.9%

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

   7. pow2 98.9%

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

7. Applied egg-rr 98.9%

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

   1. *-lft-identity 98.9%

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

9. Simplified 98.9%

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

10. Final simplification 98.9%

    \[\leadsto \varepsilon \cdot \left({\tan x}^{2} + 1\right) \]
11. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \varepsilon \cdot {\tan x}^{2} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + (eps * (tan(x) ** 2.0d0))
end function

Java

public static double code(double x, double eps) {
	return eps + (eps * Math.pow(Math.tan(x), 2.0));
}

Python

def code(x, eps):
	return eps + (eps * math.pow(math.tan(x), 2.0))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps + (eps * (tan(x) ^ 2.0));
end

Wolfram

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

Derivation

1. Initial program 60.5%

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

3. Taylor expanded in eps around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. metadata-eval 98.9%

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

   3. *-lft-identity 98.9%

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

5. Simplified 98.9%

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

   1. unpow2 98.9%

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

   2. unpow2 98.9%

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

   3. frac-times 98.9%

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

   4. tan-quot 98.9%

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

   5. tan-quot 98.9%

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

   6. *-un-lft-identity 98.9%

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

   7. pow2 98.9%

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

7. Applied egg-rr 98.9%

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

   1. *-lft-identity 98.9%

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

9. Simplified 98.9%

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

    1. distribute-rgt-in 98.9%

       \[\leadsto \color{blue}{1 \cdot \varepsilon + {\tan x}^{2} \cdot \varepsilon} \]

    2. *-un-lft-identity 98.9%

       \[\leadsto \color{blue}{\varepsilon} + {\tan x}^{2} \cdot \varepsilon \]

    3. +-commutative 98.9%

       \[\leadsto \color{blue}{{\tan x}^{2} \cdot \varepsilon + \varepsilon} \]

11. Applied egg-rr 98.9%

    \[\leadsto \color{blue}{{\tan x}^{2} \cdot \varepsilon + \varepsilon} \]

12. Final simplification 98.9%

    \[\leadsto \varepsilon + \varepsilon \cdot {\tan x}^{2} \]
13. Add Preprocessing

Alternative 5: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon + \varepsilon \cdot {x}^{2} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps + (eps * (x ** 2.0d0))
end function

Java

public static double code(double x, double eps) {
	return eps + (eps * Math.pow(x, 2.0));
}

Python

def code(x, eps):
	return eps + (eps * math.pow(x, 2.0))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps + (eps * (x ^ 2.0));
end

Wolfram

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

Derivation

1. Initial program 60.5%

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

3. Taylor expanded in eps around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. metadata-eval 98.9%

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

   3. *-lft-identity 98.9%

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

5. Simplified 98.9%

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

   1. unpow2 98.9%

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

   2. unpow2 98.9%

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

   3. frac-times 98.9%

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

   4. tan-quot 98.9%

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

   5. tan-quot 98.9%

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

   6. *-un-lft-identity 98.9%

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

   7. pow2 98.9%

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

7. Applied egg-rr 98.9%

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

   1. *-lft-identity 98.9%

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

9. Simplified 98.9%

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

10. Taylor expanded in x around 0 98.6%

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

11. Final simplification 98.6%

    \[\leadsto \varepsilon + \varepsilon \cdot {x}^{2} \]
12. Add Preprocessing

Alternative 6: 97.9% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 60.5%

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

3. Taylor expanded in eps around 0 98.9%

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

   1. cancel-sign-sub-inv 98.9%

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

   2. metadata-eval 98.9%

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

   3. *-lft-identity 98.9%

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

5. Simplified 98.9%

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

   1. unpow2 98.9%

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

   2. unpow2 98.9%

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

   3. frac-times 98.9%

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

   4. tan-quot 98.9%

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

   5. tan-quot 98.9%

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

   6. *-un-lft-identity 98.9%

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

   7. pow2 98.9%

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

7. Applied egg-rr 98.9%

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

   1. *-lft-identity 98.9%

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

9. Simplified 98.9%

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

10. Taylor expanded in x around 0 98.2%

    \[\leadsto \color{blue}{\varepsilon} \]

11. Final simplification 98.2%

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

Developer target: 99.9% accurate, 0.7× 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]

Reproduce

herbie shell --seed 2024039 
(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)))

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

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