2tan (problem 3.3.2)

Percentage Accurate: 62.5% → 100.0%

Time: 12.4s

Alternatives: 7

Speedup: 17.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.5% 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: 100.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-rev 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 60.7

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

4. Applied rewrites 60.7%

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

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 99.9

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

7. Applied rewrites 99.9%

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

   1. lift-cos.f64 N/A

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

   2. sin-+PI/2-rev N/A

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

   3. lower-sin.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. lower-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-/.f64 99.9

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

9. Applied rewrites 99.9%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-sin.f64 N/A

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

    4. lift-+.f64 N/A

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

    5. sin-sum N/A

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

    6. distribute-rgt-in N/A

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

    7. lower-fma.f64 N/A

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

11. Applied rewrites 100.0%

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

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{PI}\left(\right)}{2}\\ \frac{\sin \varepsilon}{\sin \left(x + \left(t\_0 + \varepsilon\right)\right) \cdot \sin \left(t\_0 + x\right)} \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (PI) 2.0)))
   (/ (sin eps) (* (sin (+ x (+ t_0 eps))) (sin (+ t_0 x))))))

Derivation

1. Initial program 60.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-rev 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 60.7

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

4. Applied rewrites 60.7%

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

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 99.9

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

7. Applied rewrites 99.9%

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

   1. lift-cos.f64 N/A

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

   2. sin-+PI/2-rev N/A

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

   3. lower-sin.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. lower-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-/.f64 99.9

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

9. Applied rewrites 99.9%

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

    1. lift-cos.f64 N/A

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

    2. sin-+PI/2-rev N/A

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

    3. lower-sin.f64 N/A

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

    4. lift-PI.f64 N/A

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

    5. lift-/.f64 N/A

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

    6. +-commutative N/A

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

    7. lower-+.f64 99.9

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

11. Applied rewrites 99.9%

    \[\leadsto \frac{\sin \varepsilon}{\sin \left(x + \left(\frac{\mathsf{PI}\left(\right)}{2} + \varepsilon\right)\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2} + x\right)}} \]
12. Add Preprocessing

Alternative 3: 99.9% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (/ (sin eps) (* (sin (fma (PI) 0.5 (+ x eps))) (cos x))))

Derivation

1. Initial program 60.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-rev 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 60.7

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

4. Applied rewrites 60.7%

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

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 99.9

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

7. Applied rewrites 99.9%

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

   1. lift-cos.f64 N/A

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

   2. sin-+PI/2-rev N/A

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

   3. lower-sin.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. lower-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-/.f64 99.9

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

9. Applied rewrites 99.9%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. +-commutative N/A

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

    3. associate-+l+ N/A

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

    4. +-commutative N/A

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

    5. *-commutative N/A

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

    6. lower-fma.f64 N/A

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

    7. lower-PI.f64 N/A

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

    8. +-commutative N/A

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

    9. lower-+.f64 99.9

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

12. Applied rewrites 99.9%

    \[\leadsto \frac{\sin \varepsilon}{\sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, x + \varepsilon\right)\right)} \cdot \cos x} \]
13. 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 60.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-rev 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 60.7

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

4. Applied rewrites 60.7%

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

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 99.9

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

7. Applied rewrites 99.9%

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

Alternative 5: 98.9% 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 60.7%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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-rev 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 60.7

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

4. Applied rewrites 60.7%

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

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 99.5

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

7. Applied rewrites 99.5%

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

Alternative 6: 98.0% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 99.2%

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

Alternative 7: 98.0% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 99.2%

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

Developer Target 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024339 
(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)))