2tan (problem 3.3.2)

Percentage Accurate: 62.7% → 99.5%

Time: 15.5s

Alternatives: 11

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: 62.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.1%

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

   1. tan-sum N/A

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x \]

   2. tan-quot N/A

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{\frac{\sin x}{\cos x}} \]

   3. clear-num N/A

      \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{\frac{1}{\frac{\cos x}{\sin x}}} \]

   4. frac-sub N/A

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

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

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

4. Applied egg-rr 61.9%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   8. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   16. *-commutative N/A

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

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

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.1%

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

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

   5. sin-diff N/A

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

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

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

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

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

   13. +-lowering-+.f64 63.1

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

4. Applied egg-rr 63.1%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

   5. *-commutative N/A

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 99.2

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

7. Simplified 99.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.16666666666666666, \varepsilon\right)}}{\cos x \cdot \cos \left(x + \varepsilon\right)} \]
8. Step-by-step derivation

   1. cos-sum N/A

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

   2. cancel-sign-sub-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. sin-lowering-sin.f64 99.2

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

9. Applied egg-rr 99.2%

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

10. Final simplification 99.2%

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

Alternative 3: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.1%

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

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

   5. sin-diff N/A

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

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

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

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

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

   13. +-lowering-+.f64 63.1

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

4. Applied egg-rr 63.1%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

   5. *-commutative N/A

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 99.2

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

7. Simplified 99.2%

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

8. Taylor expanded in eps around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

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

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

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

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

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

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

   6. sin-lowering-sin.f64 99.0

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

10. Simplified 99.0%

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

11. Taylor expanded in eps around 0

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

13. Simplified 99.2%

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

14. Final simplification 99.2%

    \[\leadsto \frac{\varepsilon}{\cos x \cdot \left(\cos x - \sin x \cdot \varepsilon\right)} \]
15. Add Preprocessing

Alternative 4: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.1%

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

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

   5. sin-diff N/A

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

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

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

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

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

   13. +-lowering-+.f64 63.1

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

4. Applied egg-rr 63.1%

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

5. Taylor expanded in eps around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

   5. *-commutative N/A

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 99.2

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

7. Simplified 99.2%

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

Alternative 5: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.1%

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

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

   5. sin-diff N/A

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

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

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

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

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

   13. +-lowering-+.f64 63.1

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

4. Applied egg-rr 63.1%

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

5. Taylor expanded in eps around 0

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

7. Simplified 99.1%

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

Alternative 6: 99.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps / (0.5 + (0.5 * cos((x + x))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps / (0.5d0 + (0.5d0 * cos((x + x))))
end function

Java

public static double code(double x, double eps) {
	return eps / (0.5 + (0.5 * Math.cos((x + x))));
}

Python

def code(x, eps):
	return eps / (0.5 + (0.5 * math.cos((x + x))))

Julia

function code(x, eps)
	return Float64(eps / Float64(0.5 + Float64(0.5 * cos(Float64(x + x)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps / (0.5 + (0.5 * cos((x + x))));
end

Wolfram

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

Derivation

1. Initial program 63.1%

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

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

   5. sin-diff N/A

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

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

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

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

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

   13. +-lowering-+.f64 63.1

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

4. Applied egg-rr 63.1%

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

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 98.5

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

7. Simplified 98.5%

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

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

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

   2. unpow2 N/A

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

   3. sqr-cos-a N/A

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

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

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

   5. cos-2 N/A

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

   6. cos-sum N/A

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

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

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

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

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

   9. +-lowering-+.f64 98.5

      \[\leadsto \frac{\varepsilon}{0.5 + 0.5 \cdot \cos \color{blue}{\left(x + x\right)}} \]

9. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{\varepsilon}{0.5 + 0.5 \cdot \cos \left(x + x\right)}} \]
10. Add Preprocessing

Alternative 7: 98.3% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.1%

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

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

   5. sin-diff N/A

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

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

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

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

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

   13. +-lowering-+.f64 63.1

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

4. Applied egg-rr 63.1%

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

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 98.5

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

7. Simplified 98.5%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

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

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 97.7

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

10. Simplified 97.7%

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

Alternative 8: 98.3% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot 0.6666666666666666, \varepsilon\right), \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (* x x) (fma (* x x) (* eps 0.6666666666666666) eps) eps))

C

double code(double x, double eps) {
	return fma((x * x), fma((x * x), (eps * 0.6666666666666666), eps), eps);
}

Julia

function code(x, eps)
	return fma(Float64(x * x), fma(Float64(x * x), Float64(eps * 0.6666666666666666), eps), eps)
end

Wolfram

code[x_, eps_] := N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(eps * 0.6666666666666666), $MachinePrecision] + eps), $MachinePrecision] + eps), $MachinePrecision]

Derivation

1. Initial program 63.1%

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

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

   5. sin-diff N/A

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

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

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

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

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

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

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

   9. *-commutative N/A

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

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

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

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

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

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

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

   13. +-lowering-+.f64 63.1

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

4. Applied egg-rr 63.1%

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

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 98.5

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

7. Simplified 98.5%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

10. Simplified 97.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \varepsilon \cdot 0.6666666666666666, \varepsilon\right), \varepsilon\right)} \]
11. Add Preprocessing

Alternative 9: 98.3% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.1%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. Simplified 99.2%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. +-commutative N/A

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

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

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

   5. +-commutative N/A

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

   6. distribute-lft-out N/A

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

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

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

   8. unpow2 N/A

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

   9. distribute-lft-out N/A

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

   10. +-commutative N/A

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

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

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

   12. +-commutative N/A

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

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

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 97.6

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

8. Simplified 97.6%

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

9. Taylor expanded in eps around 0

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

    1. +-commutative N/A

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

    2. distribute-lft-in N/A

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

    3. *-rgt-identity N/A

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

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

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

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

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

    6. +-commutative N/A

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

    7. distribute-lft-in N/A

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

    8. *-rgt-identity N/A

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

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

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

    10. *-commutative N/A

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

    11. unpow2 N/A

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

    12. associate-*l* N/A

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

    13. *-commutative N/A

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

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

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

    15. *-commutative N/A

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

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

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

    17. unpow2 N/A

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

    18. *-lowering-*.f64 97.6

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

11. Simplified 97.6%

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

12. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

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

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

    3. +-commutative N/A

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

    4. unpow2 N/A

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

    5. distribute-lft-out N/A

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

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

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

    7. +-lowering-+.f64 97.6

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

14. Simplified 97.6%

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

15. Final simplification 97.6%

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

Alternative 10: 98.2% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.1%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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. Simplified 99.2%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. +-commutative N/A

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

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

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

   5. +-commutative N/A

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

   6. distribute-lft-out N/A

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

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

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

   8. unpow2 N/A

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

   9. distribute-lft-out N/A

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

   10. +-commutative N/A

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

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

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

   12. +-commutative N/A

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

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

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 97.6

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

8. Simplified 97.6%

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

9. Taylor expanded in eps around 0

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

    1. +-commutative N/A

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

    2. distribute-lft-in N/A

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

    3. *-rgt-identity N/A

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

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

       \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon, {x}^{2}, \varepsilon\right)} \]

    5. unpow2 N/A

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

    6. *-lowering-*.f64 97.4

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

11. Simplified 97.4%

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

    1. *-commutative N/A

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

    2. distribute-lft1-in N/A

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

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

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

    4. accelerator-lowering-fma.f64 97.4

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

13. Applied egg-rr 97.4%

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

14. Final simplification 97.4%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, x, 1\right) \]
15. Add Preprocessing

Alternative 11: 97.7% 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 63.1%

   \[\tan \left(x + \varepsilon\right) - \tan x \]
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 96.7

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

5. Simplified 96.7%

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

6. Taylor expanded in eps around 0

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

8. Simplified 96.7%

   \[\leadsto \color{blue}{\varepsilon} \]
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: 62.8% 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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024199 
(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)))