2tan (problem 3.3.2)

Percentage Accurate: 62.4% → 99.9%

Time: 17.5s

Alternatives: 10

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: 62.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.6× speedup

Math

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

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(sin(eps) / Float64(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[Sin[eps], $MachinePrecision] / N[(N[Cos[x], $MachinePrecision] * N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.3%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-sub N/A

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

   4. associate-/r* N/A

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

   5. /-lowering-/.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. sin-diff N/A

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

   8. sin-lowering-sin.f64 N/A

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

   9. --lowering--.f64 N/A

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

   10. +-lowering-+.f64 N/A

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

   11. cos-lowering-cos.f64 N/A

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

   12. +-lowering-+.f64 N/A

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

   13. cos-lowering-cos.f64 61.3

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

4. Applied egg-rr 61.3%

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

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. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

   5. /-lowering-/.f64 N/A

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

   6. sin-lowering-sin.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. cos-lowering-cos.f64 N/A

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

   9. sub-neg N/A

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

   10. mul-1-neg N/A

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

   11. remove-double-neg N/A

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

   12. +-commutative N/A

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

   13. cos-lowering-cos.f64 N/A

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

   14. +-lowering-+.f64 100.0

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

7. Simplified 100.0%

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

Alternative 2: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), 1\right)}{\cos \left(\varepsilon + x\right)}}{\cos x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (/
   (*
    eps
    (fma
     (* eps eps)
     (fma (* eps eps) 0.008333333333333333 -0.16666666666666666)
     1.0))
   (cos (+ eps x)))
  (cos x)))

C

double code(double x, double eps) {
	return ((eps * fma((eps * eps), fma((eps * eps), 0.008333333333333333, -0.16666666666666666), 1.0)) / cos((eps + x))) / cos(x);
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(eps * fma(Float64(eps * eps), fma(Float64(eps * eps), 0.008333333333333333, -0.16666666666666666), 1.0)) / cos(Float64(eps + x))) / cos(x))
end

Wolfram

code[x_, eps_] := N[(N[(N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.3%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-sub N/A

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

   4. associate-/r* N/A

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

   5. /-lowering-/.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. sin-diff N/A

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

   8. sin-lowering-sin.f64 N/A

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

   9. --lowering--.f64 N/A

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

   10. +-lowering-+.f64 N/A

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

   11. cos-lowering-cos.f64 N/A

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

   12. +-lowering-+.f64 N/A

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

   13. cos-lowering-cos.f64 61.3

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

4. Applied egg-rr 61.3%

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

5. Taylor expanded in eps around 0

   \[\leadsto \frac{\frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)}}{\cos \left(x + \varepsilon\right)}}{\cos x} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\varepsilon \cdot \left(1 + {\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right)\right)}}{\cos \left(x + \varepsilon\right)}}{\cos x} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{\varepsilon \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}\right) + 1\right)}}{\cos \left(x + \varepsilon\right)}}{\cos x} \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\frac{\varepsilon \cdot \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}, 1\right)}}{\cos \left(x + \varepsilon\right)}}{\cos x} \]

   4. unpow2 N/A

      \[\leadsto \frac{\frac{\varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}, 1\right)}{\cos \left(x + \varepsilon\right)}}{\cos x} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{\frac{\varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{120} \cdot {\varepsilon}^{2} - \frac{1}{6}, 1\right)}{\cos \left(x + \varepsilon\right)}}{\cos x} \]

   6. sub-neg N/A

      \[\leadsto \frac{\frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\frac{1}{120} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right)}{\cos \left(x + \varepsilon\right)}}{\cos x} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{{\varepsilon}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), 1\right)}{\cos \left(x + \varepsilon\right)}}{\cos x} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, {\varepsilon}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, 1\right)}{\cos \left(x + \varepsilon\right)}}{\cos x} \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \color{blue}{\mathsf{fma}\left({\varepsilon}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, 1\right)}{\cos \left(x + \varepsilon\right)}}{\cos x} \]

   10. unpow2 N/A

       \[\leadsto \frac{\frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{120}, \frac{-1}{6}\right), 1\right)}{\cos \left(x + \varepsilon\right)}}{\cos x} \]

   11. *-lowering-*.f64 100.0

       \[\leadsto \frac{\frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, 0.008333333333333333, -0.16666666666666666\right), 1\right)}{\cos \left(x + \varepsilon\right)}}{\cos x} \]

7. Simplified 100.0%

   \[\leadsto \frac{\frac{\color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), 1\right)}}{\cos \left(x + \varepsilon\right)}}{\cos x} \]

8. Final simplification 100.0%

   \[\leadsto \frac{\frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.008333333333333333, -0.16666666666666666\right), 1\right)}{\cos \left(\varepsilon + x\right)}}{\cos x} \]
9. Add Preprocessing

Alternative 3: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.16666666666666666, 1\right)}{\cos \left(\varepsilon + x\right)}}{\cos x} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (/ (* eps (fma (* eps eps) -0.16666666666666666 1.0)) (cos (+ eps x)))
  (cos x)))

C

double code(double x, double eps) {
	return ((eps * fma((eps * eps), -0.16666666666666666, 1.0)) / cos((eps + x))) / cos(x);
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(eps * fma(Float64(eps * eps), -0.16666666666666666, 1.0)) / cos(Float64(eps + x))) / cos(x))
end

Wolfram

code[x_, eps_] := N[(N[(N[(eps * N[(N[(eps * eps), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.3%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-sub N/A

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

   4. associate-/r* N/A

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

   5. /-lowering-/.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. sin-diff N/A

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

   8. sin-lowering-sin.f64 N/A

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

   9. --lowering--.f64 N/A

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

   10. +-lowering-+.f64 N/A

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

   11. cos-lowering-cos.f64 N/A

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

   12. +-lowering-+.f64 N/A

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

   13. cos-lowering-cos.f64 61.3

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

4. Applied egg-rr 61.3%

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

5. Taylor expanded in eps around 0

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

   1. *-lowering-*.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. unpow2 N/A

      \[\leadsto \frac{\frac{\varepsilon \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \varepsilon}, \frac{-1}{6}, 1\right)}{\cos \left(x + \varepsilon\right)}}{\cos x} \]

   6. *-lowering-*.f64 99.9

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

   \[\leadsto \frac{\frac{\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -0.16666666666666666, 1\right)}{\cos \left(\varepsilon + x\right)}}{\cos x} \]
9. Add Preprocessing

Alternative 4: 99.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.3%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-sub N/A

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

   4. associate-/r* N/A

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

   5. /-lowering-/.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. sin-diff N/A

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

   8. sin-lowering-sin.f64 N/A

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

   9. --lowering--.f64 N/A

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

   10. +-lowering-+.f64 N/A

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

   11. cos-lowering-cos.f64 N/A

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

   12. +-lowering-+.f64 N/A

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

   13. cos-lowering-cos.f64 61.3

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

4. Applied egg-rr 61.3%

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

5. Taylor expanded in eps around 0

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

   \[\leadsto \frac{\frac{\varepsilon}{\cos \left(\varepsilon + x\right)}}{\cos x} \]
9. 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 61.3%

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

   1. tan-quot N/A

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

   2. tan-quot N/A

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

   3. frac-sub N/A

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

   4. associate-/r* N/A

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

   5. /-lowering-/.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. sin-diff N/A

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

   8. sin-lowering-sin.f64 N/A

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

   9. --lowering--.f64 N/A

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

   10. +-lowering-+.f64 N/A

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

   11. cos-lowering-cos.f64 N/A

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

   12. +-lowering-+.f64 N/A

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

   13. cos-lowering-cos.f64 61.3

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

4. Applied egg-rr 61.3%

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

5. Taylor expanded in eps around 0

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

   1. /-lowering-/.f64 N/A

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

   2. pow-lowering-pow.f64 N/A

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

   3. cos-lowering-cos.f64 99.3

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

7. Simplified 99.3%

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

Alternative 6: 98.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot 1.3333333333333333, 0.3333333333333333\right), x\right), x \cdot x\right), \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  eps
  (fma
   eps
   (fma eps (fma x (* x 1.3333333333333333) 0.3333333333333333) x)
   (* x x))
  eps))

C

double code(double x, double eps) {
	return fma(eps, fma(eps, fma(eps, fma(x, (x * 1.3333333333333333), 0.3333333333333333), x), (x * x)), eps);
}

Julia

function code(x, eps)
	return fma(eps, fma(eps, fma(eps, fma(x, Float64(x * 1.3333333333333333), 0.3333333333333333), x), Float64(x * x)), eps)
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * N[(eps * N[(x * N[(x * 1.3333333333333333), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + x), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]

Derivation

1. Initial program 61.3%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333, \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}, \left(\frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \left(0.16666666666666666 + \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right)\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \left(0.16666666666666666 + \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right)\right), \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right)} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + \frac{1}{3} \cdot {\varepsilon}^{2}}, \varepsilon\right) \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right), \frac{1}{3} \cdot {\varepsilon}^{2}\right)}, \varepsilon\right) \]

8. Simplified 98.9%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 1.3333333333333333, 1\right), \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right)\right), 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon\right) \]

9. Taylor expanded in eps around 0

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

    1. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(\varepsilon, x + \varepsilon \cdot \left(\frac{1}{3} + \frac{4}{3} \cdot {x}^{2}\right), {x}^{2}\right)}, \varepsilon\right) \]

    2. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\frac{1}{3} + \frac{4}{3} \cdot {x}^{2}\right) + x}, {x}^{2}\right), \varepsilon\right) \]

    3. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{1}{3} + \frac{4}{3} \cdot {x}^{2}, x\right)}, {x}^{2}\right), \varepsilon\right) \]

    4. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{4}{3} \cdot {x}^{2} + \frac{1}{3}}, x\right), {x}^{2}\right), \varepsilon\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{{x}^{2} \cdot \frac{4}{3}} + \frac{1}{3}, x\right), {x}^{2}\right), \varepsilon\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{\left(x \cdot x\right)} \cdot \frac{4}{3} + \frac{1}{3}, x\right), {x}^{2}\right), \varepsilon\right) \]

    7. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(x \cdot \frac{4}{3}\right)} + \frac{1}{3}, x\right), {x}^{2}\right), \varepsilon\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, x \cdot \color{blue}{\left(\frac{4}{3} \cdot x\right)} + \frac{1}{3}, x\right), {x}^{2}\right), \varepsilon\right) \]

    9. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \frac{4}{3} \cdot x, \frac{1}{3}\right)}, x\right), {x}^{2}\right), \varepsilon\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{4}{3}}, \frac{1}{3}\right), x\right), {x}^{2}\right), \varepsilon\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{4}{3}}, \frac{1}{3}\right), x\right), {x}^{2}\right), \varepsilon\right) \]

    12. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot \frac{4}{3}, \frac{1}{3}\right), x\right), \color{blue}{x \cdot x}\right), \varepsilon\right) \]

    13. *-lowering-*.f64 98.9

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

11. Simplified 98.9%

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

Alternative 7: 98.3% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.3%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333, \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}, \left(\frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \left(0.16666666666666666 + \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right)\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \left(0.16666666666666666 + \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right)\right), \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right)} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + \frac{1}{3} \cdot {\varepsilon}^{2}}, \varepsilon\right) \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right), \frac{1}{3} \cdot {\varepsilon}^{2}\right)}, \varepsilon\right) \]

8. Simplified 98.9%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 1.3333333333333333, 1\right), \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right)\right), 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon\right) \]

9. Taylor expanded in eps around 0

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

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, \color{blue}{x + \varepsilon}, \frac{1}{3} \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \varepsilon\right) \]

    2. +-lowering-+.f64 98.9

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

11. Simplified 98.9%

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

12. Final simplification 98.9%

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

Alternative 8: 98.2% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.3%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333, \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}, \left(\frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \left(0.16666666666666666 + \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right)\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \left(0.16666666666666666 + \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right)\right), \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right)} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + \frac{1}{3} \cdot {\varepsilon}^{2}}, \varepsilon\right) \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right), \frac{1}{3} \cdot {\varepsilon}^{2}\right)}, \varepsilon\right) \]

8. Simplified 98.9%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 1.3333333333333333, 1\right), \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right)\right), 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon\right) \]

9. Taylor expanded in eps around 0

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

    1. unpow2 N/A

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

    2. distribute-rgt-out N/A

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

    3. *-lowering-*.f64 N/A

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

    4. +-commutative N/A

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

    5. +-lowering-+.f64 98.9

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

11. Simplified 98.9%

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

12. Final simplification 98.9%

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

Alternative 9: 98.2% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.3%

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

3. Taylor expanded in eps around 0

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

5. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(0.3333333333333333, \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}, \left(\frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \left(0.16666666666666666 + \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right)\right) \cdot \frac{\sin x}{\cos x}\right), \frac{{\sin x}^{2} + \frac{{\sin x}^{4}}{{\cos x}^{2}}}{{\cos x}^{2}} - \left(0.16666666666666666 + \mathsf{fma}\left(0.16666666666666666, \frac{{\sin x}^{2}}{{\cos x}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\sin x}^{2}}{{\cos x}^{2}}, -0.5\right)\right)\right)\right), \frac{\sin x + \frac{{\sin x}^{3}}{{\cos x}^{2}}}{\cos x}\right), \frac{{\sin x}^{2}}{{\cos x}^{2}}\right), \varepsilon\right)} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(\varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right)\right) + \frac{1}{3} \cdot {\varepsilon}^{2}}, \varepsilon\right) \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \varepsilon \cdot \left(1 + \frac{2}{3} \cdot {\varepsilon}^{2}\right) + x \cdot \left(1 + \frac{4}{3} \cdot {\varepsilon}^{2}\right), \frac{1}{3} \cdot {\varepsilon}^{2}\right)}, \varepsilon\right) \]

8. Simplified 98.9%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 1.3333333333333333, 1\right), \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.6666666666666666, 1\right)\right), 0.3333333333333333 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, \varepsilon\right) \]

9. Taylor expanded in eps around 0

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

    1. unpow2 N/A

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

    2. *-lowering-*.f64 98.9

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

11. Simplified 98.9%

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

Alternative 10: 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 61.3%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 N/A

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

   2. sin-lowering-sin.f64 N/A

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

   3. cos-lowering-cos.f64 98.4

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

5. Simplified 98.4%

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

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon} \]
7. Step-by-step derivation

8. Simplified 98.4%

   \[\leadsto \color{blue}{\varepsilon} \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 62.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 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 2024198 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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

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

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