2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 99.6%

Time: 13.2s

Alternatives: 9

Speedup: 207.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% 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 60.3%

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

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

Alternative 2: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.3%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. sin-diff-rev N/A

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

   11. lower-sin.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 N/A

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

   20. lower-cos.f64 60.4

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

4. Applied rewrites 60.4%

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

5. Taylor expanded in x around 0

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

   1. lower-sin.f64 99.9

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

7. Applied rewrites 99.9%

   \[\leadsto \frac{\frac{\color{blue}{\sin \varepsilon}}{\cos \left(\varepsilon + x\right)}}{\cos x} \]
8. Add Preprocessing

Alternative 3: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.3%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. sin-diff-rev N/A

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

   11. lower-sin.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 N/A

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

   20. lower-cos.f64 60.4

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

4. Applied rewrites 60.4%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-cos.f64 99.9

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

7. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \cos x}} \]
8. Add Preprocessing

Alternative 4: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\varepsilon}{\cos x}}{\cos x} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ (/ eps (cos x)) (cos x)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.3%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. sin-diff-rev N/A

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

   11. lower-sin.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 N/A

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

   20. lower-cos.f64 60.4

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

4. Applied rewrites 60.4%

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

5. Taylor expanded in eps around 0

   \[\leadsto \frac{\color{blue}{\frac{\varepsilon}{\cos x}}}{\cos x} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\varepsilon}{\cos x}}}{\cos x} \]

   2. lower-cos.f64 99.2

      \[\leadsto \frac{\frac{\varepsilon}{\color{blue}{\cos x}}}{\cos x} \]

7. Applied rewrites 99.2%

   \[\leadsto \frac{\color{blue}{\frac{\varepsilon}{\cos x}}}{\cos x} \]
8. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{{\cos x}^{2}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps / math.pow(math.cos(x), 2.0)

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps / (cos(x) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 60.3%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. sin-diff-rev N/A

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

   11. lower-sin.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 N/A

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

   20. lower-cos.f64 60.4

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

4. Applied rewrites 60.4%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{\varepsilon}{{\cos x}^{2}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 99.2

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

7. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{\varepsilon}{{\cos x}^{2}}} \]
8. Add Preprocessing

Alternative 6: 98.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\varepsilon}{\cos x}}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ (/ eps (cos x)) (fma (* x x) -0.5 1.0)))

C

double code(double x, double eps) {
	return (eps / cos(x)) / fma((x * x), -0.5, 1.0);
}

Julia

function code(x, eps)
	return Float64(Float64(eps / cos(x)) / fma(Float64(x * x), -0.5, 1.0))
end

Wolfram

code[x_, eps_] := N[(N[(eps / N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.3%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. sin-diff-rev N/A

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

   11. lower-sin.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 N/A

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

   20. lower-cos.f64 60.4

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

4. Applied rewrites 60.4%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 60.0

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

7. Applied rewrites 60.0%

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

8. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-cos.f64 98.5

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

10. Applied rewrites 98.5%

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

Alternative 7: 97.8% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

   2. lower-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin \varepsilon}}{\cos \varepsilon} \]

   3. lower-cos.f64 97.9

      \[\leadsto \frac{\sin \varepsilon}{\color{blue}{\cos \varepsilon}} \]

5. Applied rewrites 97.9%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

6. Taylor expanded in eps around 0

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

8. Applied rewrites 97.9%

   \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.3333333333333333, 1\right) \cdot \color{blue}{\varepsilon} \]
9. Add Preprocessing

Alternative 8: 97.8% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 60.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

   2. lower-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin \varepsilon}}{\cos \varepsilon} \]

   3. lower-cos.f64 97.9

      \[\leadsto \frac{\sin \varepsilon}{\color{blue}{\cos \varepsilon}} \]

5. Applied rewrites 97.9%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
6. Step-by-step derivation

7. Applied rewrites 3.6%

   \[\leadsto \tan \left(\left(\varepsilon + \mathsf{PI}\left(\right)\right) + \mathsf{PI}\left(\right)\right) \]

8. Taylor expanded in eps around 0

   \[\leadsto \frac{\sin \left(2 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\cos \left(2 \cdot \mathsf{PI}\left(\right)\right)}} \]
9. Step-by-step derivation

10. Applied rewrites 5.6%

    \[\leadsto 0 \]

11. Taylor expanded in eps around 0

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

13. Applied rewrites 97.9%

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

Alternative 9: 5.5% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 60.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]

   2. lower-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin \varepsilon}}{\cos \varepsilon} \]

   3. lower-cos.f64 97.9

      \[\leadsto \frac{\sin \varepsilon}{\color{blue}{\cos \varepsilon}} \]

5. Applied rewrites 97.9%

   \[\leadsto \color{blue}{\frac{\sin \varepsilon}{\cos \varepsilon}} \]
6. Step-by-step derivation

7. Applied rewrites 3.6%

   \[\leadsto \tan \left(\left(\varepsilon + \mathsf{PI}\left(\right)\right) + \mathsf{PI}\left(\right)\right) \]

8. Taylor expanded in eps around 0

   \[\leadsto \frac{\sin \left(2 \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\cos \left(2 \cdot \mathsf{PI}\left(\right)\right)}} \]
9. Step-by-step derivation

10. Applied rewrites 5.6%

    \[\leadsto 0 \]
11. Add Preprocessing

Developer Target 1: 99.0% 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 2024326 
(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)))