2tan (problem 3.3.2)

Percentage Accurate: 62.7% → 100.0%

Time: 12.0s

Alternatives: 8

Speedup: 34.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.7% 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: 100.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 59.8

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

4. Applied rewrites 59.8%

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

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. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-+.f64 99.8

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 59.8

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

4. Applied rewrites 59.8%

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

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. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-+.f64 99.8

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 100.0%

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

10. Taylor expanded in x around inf

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

12. Applied rewrites 100.0%

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

Alternative 3: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 59.8

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

4. Applied rewrites 59.8%

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

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. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-+.f64 99.8

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

7. Applied rewrites 99.8%

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

Alternative 4: 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 59.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 59.8

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

4. Applied rewrites 59.8%

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

5. Taylor expanded in x around 0

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

   1. lower-sin.f64 99.8

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

7. Applied rewrites 99.8%

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

Alternative 5: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 59.8

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

4. Applied rewrites 59.8%

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

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. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-+.f64 99.8

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in eps around 0

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

10. Applied rewrites 99.0%

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

Alternative 6: 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 59.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 59.8

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

4. Applied rewrites 59.8%

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

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 98.5

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

7. Applied rewrites 98.5%

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

Alternative 7: 97.9% 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 59.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 97.3

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

5. Applied rewrites 97.3%

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

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 97.4%

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

Alternative 8: 97.9% accurate, 34.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.8%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 97.3

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

5. Applied rewrites 97.3%

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

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 97.4%

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

9. Taylor expanded in eps around inf

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

11. Applied rewrites 6.2%

    \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.3333333333333333\right) \cdot \varepsilon \]

12. Taylor expanded in eps around 0

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

14. Applied rewrites 97.4%

    \[\leadsto 1 \cdot \varepsilon \]
15. 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 2024327 
(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)))