2tan (problem 3.3.2)

Percentage Accurate: 62.7% → 99.5%

Time: 13.1s

Alternatives: 15

Speedup: 17.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.7% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\cos x}^{2}\\ t_1 := \frac{{\sin x}^{2}}{t\_0}\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666 - \left(\mathsf{fma}\left(t\_1, 0.16666666666666666, \mathsf{fma}\left(t\_1, -0.5, -0.5\right)\right) - \frac{\mathsf{fma}\left(\sin x, \sin x, \frac{{\sin x}^{4}}{t\_0}\right)}{t\_0}\right), \varepsilon, \frac{\frac{{\sin x}^{3}}{t\_0} + \sin x}{\cos x}\right), \varepsilon, t\_1\right), \varepsilon, \varepsilon\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (cos x) 2.0)) (t_1 (/ (pow (sin x) 2.0) t_0)))
   (fma
    (fma
     (fma
      (-
       -0.16666666666666666
       (-
        (fma t_1 0.16666666666666666 (fma t_1 -0.5 -0.5))
        (/ (fma (sin x) (sin x) (/ (pow (sin x) 4.0) t_0)) t_0)))
      eps
      (/ (+ (/ (pow (sin x) 3.0) t_0) (sin x)) (cos x)))
     eps
     t_1)
    eps
    eps)))

C

double code(double x, double eps) {
	double t_0 = pow(cos(x), 2.0);
	double t_1 = pow(sin(x), 2.0) / t_0;
	return fma(fma(fma((-0.16666666666666666 - (fma(t_1, 0.16666666666666666, fma(t_1, -0.5, -0.5)) - (fma(sin(x), sin(x), (pow(sin(x), 4.0) / t_0)) / t_0))), eps, (((pow(sin(x), 3.0) / t_0) + sin(x)) / cos(x))), eps, t_1), eps, eps);
}

Julia

function code(x, eps)
	t_0 = cos(x) ^ 2.0
	t_1 = Float64((sin(x) ^ 2.0) / t_0)
	return fma(fma(fma(Float64(-0.16666666666666666 - Float64(fma(t_1, 0.16666666666666666, fma(t_1, -0.5, -0.5)) - Float64(fma(sin(x), sin(x), Float64((sin(x) ^ 4.0) / t_0)) / t_0))), eps, Float64(Float64(Float64((sin(x) ^ 3.0) / t_0) + sin(x)) / cos(x))), eps, t_1), eps, eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[Cos[x], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[x], $MachinePrecision], 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]}, N[(N[(N[(N[(-0.16666666666666666 - N[(N[(t$95$1 * 0.16666666666666666 + N[(t$95$1 * -0.5 + -0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[x], $MachinePrecision] * N[Sin[x], $MachinePrecision] + N[(N[Power[N[Sin[x], $MachinePrecision], 4.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + N[(N[(N[(N[Power[N[Sin[x], $MachinePrecision], 3.0], $MachinePrecision] / t$95$0), $MachinePrecision] + N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps + t$95$1), $MachinePrecision] * eps + eps), $MachinePrecision]]]

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

Alternative 2: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.2%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 59.2

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

4. Applied rewrites 59.2%

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

5. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. lower-cos.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Final simplification 100.0%

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

Alternative 3: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.2%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 59.2

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

4. Applied rewrites 59.2%

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

5. Taylor expanded in eps around inf

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

   1. lower-sin.f64 100.0

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

7. Applied rewrites 100.0%

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

Alternative 4: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right), x \cdot x, 1\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (/ (sin eps) (cos (+ eps x)))
  (fma
   (fma (fma -0.001388888888888889 (* x x) 0.041666666666666664) (* x x) -0.5)
   (* x x)
   1.0)))

C

double code(double x, double eps) {
	return (sin(eps) / cos((eps + x))) / fma(fma(fma(-0.001388888888888889, (x * x), 0.041666666666666664), (x * x), -0.5), (x * x), 1.0);
}

Julia

function code(x, eps)
	return Float64(Float64(sin(eps) / cos(Float64(eps + x))) / fma(fma(fma(-0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), -0.5), Float64(x * x), 1.0))
end

Wolfram

code[x_, eps_] := N[(N[(N[Sin[eps], $MachinePrecision] / N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.2%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 59.2

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

4. Applied rewrites 59.2%

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

5. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. lower-cos.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \left(x + \varepsilon\right)}}{1 + \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}} \]
9. Step-by-step derivation

10. Applied rewrites 99.7%

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \left(x + \varepsilon\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right), \color{blue}{x \cdot x}, 1\right)} \]

11. Final simplification 99.7%

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right), x \cdot x, 1\right)} \]
12. Add Preprocessing

Alternative 5: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right), x \cdot x, 1\right) \cdot \cos \left(\varepsilon + x\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (sin eps)
  (*
   (fma
    (fma (fma -0.001388888888888889 (* x x) 0.041666666666666664) (* x x) -0.5)
    (* x x)
    1.0)
   (cos (+ eps x)))))

C

double code(double x, double eps) {
	return sin(eps) / (fma(fma(fma(-0.001388888888888889, (x * x), 0.041666666666666664), (x * x), -0.5), (x * x), 1.0) * cos((eps + x)));
}

Julia

function code(x, eps)
	return Float64(sin(eps) / Float64(fma(fma(fma(-0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), -0.5), Float64(x * x), 1.0) * cos(Float64(eps + x))))
end

Wolfram

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] / N[(N[(N[(N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.2%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 59.2

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

4. Applied rewrites 59.2%

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

5. Taylor expanded in eps around inf

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

   1. lower-sin.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot {x}^{2}} + 1\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, {x}^{2}, 1\right)}} \]

   4. sub-neg N/A

      \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {x}^{2}, 1\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {x}^{2}, 1\right)} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{2}}, {x}^{2}, 1\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}, {x}^{2}, \frac{-1}{2}\right)}, {x}^{2}, 1\right)} \]

   8. +-commutative N/A

      \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{-1}{2}\right), {x}^{2}, 1\right)} \]

   9. lower-fma.f64 N/A

      \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{-1}{2}\right), {x}^{2}, 1\right)} \]

   10. unpow2 N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{-1}{2}\right), {x}^{2}, 1\right)} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{-1}{2}\right), {x}^{2}, 1\right)} \]

   12. unpow2 N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{-1}{2}\right), {x}^{2}, 1\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{-1}{2}\right), {x}^{2}, 1\right)} \]

   14. unpow2 N/A

       \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{-1}{2}\right), \color{blue}{x \cdot x}, 1\right)} \]

   15. lower-*.f64 99.7

       \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right), \color{blue}{x \cdot x}, 1\right)} \]

10. Applied rewrites 99.7%

    \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right), x \cdot x, 1\right)}} \]

11. Final simplification 99.7%

    \[\leadsto \frac{\sin \varepsilon}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, -0.5\right), x \cdot x, 1\right) \cdot \cos \left(\varepsilon + x\right)} \]
12. Add Preprocessing

Alternative 6: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (/ (sin eps) (cos (+ eps x)))
  (fma (fma 0.041666666666666664 (* x x) -0.5) (* x x) 1.0)))

C

double code(double x, double eps) {
	return (sin(eps) / cos((eps + x))) / fma(fma(0.041666666666666664, (x * x), -0.5), (x * x), 1.0);
}

Julia

function code(x, eps)
	return Float64(Float64(sin(eps) / cos(Float64(eps + x))) / fma(fma(0.041666666666666664, Float64(x * x), -0.5), Float64(x * x), 1.0))
end

Wolfram

code[x_, eps_] := N[(N[(N[Sin[eps], $MachinePrecision] / N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.2%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 59.2

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

4. Applied rewrites 59.2%

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

5. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. lower-cos.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \left(x + \varepsilon\right)}}{1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}} \]
9. Step-by-step derivation

10. Applied rewrites 99.5%

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \left(x + \varepsilon\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), \color{blue}{x \cdot x}, 1\right)} \]

11. Final simplification 99.5%

    \[\leadsto \frac{\frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)} \]
12. Add Preprocessing

Alternative 7: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \varepsilon}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right) \cdot \cos \left(\varepsilon + x\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (sin eps)
  (*
   (fma (fma 0.041666666666666664 (* x x) -0.5) (* x x) 1.0)
   (cos (+ eps x)))))

C

double code(double x, double eps) {
	return sin(eps) / (fma(fma(0.041666666666666664, (x * x), -0.5), (x * x), 1.0) * cos((eps + x)));
}

Julia

function code(x, eps)
	return Float64(sin(eps) / Float64(fma(fma(0.041666666666666664, Float64(x * x), -0.5), Float64(x * x), 1.0) * cos(Float64(eps + x))))
end

Wolfram

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] / N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.2%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 59.2

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

4. Applied rewrites 59.2%

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

5. Taylor expanded in eps around inf

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

   1. lower-sin.f64 100.0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right)}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2}} + 1\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)}} \]

   4. sub-neg N/A

      \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, {x}^{2}, 1\right)} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} + \color{blue}{\frac{-1}{2}}, {x}^{2}, 1\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{-1}{2}\right)}, {x}^{2}, 1\right)} \]

   7. unpow2 N/A

      \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{-1}{2}\right), {x}^{2}, 1\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{-1}{2}\right), {x}^{2}, 1\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{-1}{2}\right), \color{blue}{x \cdot x}, 1\right)} \]

   10. lower-*.f64 99.4

       \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), \color{blue}{x \cdot x}, 1\right)} \]

10. Applied rewrites 99.4%

    \[\leadsto \frac{\sin \varepsilon}{\cos \left(\varepsilon + x\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right)}} \]

11. Final simplification 99.4%

    \[\leadsto \frac{\sin \varepsilon}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, -0.5\right), x \cdot x, 1\right) \cdot \cos \left(\varepsilon + x\right)} \]
12. Add Preprocessing

Alternative 8: 98.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{\cos \left(2 \cdot x\right) \cdot 0.5 + 0.5} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ eps (+ (* (cos (* 2.0 x)) 0.5) 0.5)))

C

double code(double x, double eps) {
	return eps / ((cos((2.0 * x)) * 0.5) + 0.5);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps / ((cos((2.0d0 * x)) * 0.5d0) + 0.5d0)
end function

Java

public static double code(double x, double eps) {
	return eps / ((Math.cos((2.0 * x)) * 0.5) + 0.5);
}

Python

def code(x, eps):
	return eps / ((math.cos((2.0 * x)) * 0.5) + 0.5)

Julia

function code(x, eps)
	return Float64(eps / Float64(Float64(cos(Float64(2.0 * x)) * 0.5) + 0.5))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps / ((cos((2.0 * x)) * 0.5) + 0.5);
end

Wolfram

code[x_, eps_] := N[(eps / N[(N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.2%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 59.2

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

4. Applied rewrites 59.2%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 99.2

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

7. Applied rewrites 99.2%

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

9. Applied rewrites 99.2%

   \[\leadsto \frac{\varepsilon}{0.5 + \color{blue}{\cos \left(2 \cdot x\right) \cdot 0.5}} \]

10. Final simplification 99.2%

    \[\leadsto \frac{\varepsilon}{\cos \left(2 \cdot x\right) \cdot 0.5 + 0.5} \]
11. Add Preprocessing

Alternative 9: 98.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.044444444444444446, x \cdot x, 0.3333333333333333\right), x \cdot x, -1\right), x \cdot x, 1\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  eps
  (fma
   (fma (fma -0.044444444444444446 (* x x) 0.3333333333333333) (* x x) -1.0)
   (* x x)
   1.0)))

C

double code(double x, double eps) {
	return eps / fma(fma(fma(-0.044444444444444446, (x * x), 0.3333333333333333), (x * x), -1.0), (x * x), 1.0);
}

Julia

function code(x, eps)
	return Float64(eps / fma(fma(fma(-0.044444444444444446, Float64(x * x), 0.3333333333333333), Float64(x * x), -1.0), Float64(x * x), 1.0))
end

Wolfram

code[x_, eps_] := N[(eps / N[(N[(N[(-0.044444444444444446 * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.2%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 59.2

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

4. Applied rewrites 59.2%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 99.2

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\varepsilon}{1 + \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{3} + \frac{-2}{45} \cdot {x}^{2}\right) - 1\right)}} \]
9. Step-by-step derivation

10. Applied rewrites 99.2%

    \[\leadsto \frac{\varepsilon}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.044444444444444446, x \cdot x, 0.3333333333333333\right), x \cdot x, -1\right), \color{blue}{x \cdot x}, 1\right)} \]
11. Add Preprocessing

Alternative 10: 98.3% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.2%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 59.2

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

4. Applied rewrites 59.2%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 99.2

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.0%

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

Alternative 11: 98.3% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.2%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 59.2

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

4. Applied rewrites 59.2%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 99.2

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.0%

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

Alternative 12: 98.1% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.2%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 59.2

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

4. Applied rewrites 59.2%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 99.2

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.8%

    \[\leadsto \frac{\varepsilon}{1 - \color{blue}{x \cdot x}} \]
11. Add Preprocessing

Alternative 13: 98.2% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.2%

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

   1. lift--.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. frac-sub N/A

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

   7. lower-/.f64 N/A

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

   8. sin-diff N/A

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

   9. lower-sin.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lower-cos.f64 59.2

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

4. Applied rewrites 59.2%

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

5. Taylor expanded in eps around 0

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-cos.f64 99.2

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in x around 0

   \[\leadsto \varepsilon + \color{blue}{\varepsilon \cdot {x}^{2}} \]
9. Step-by-step derivation

10. Applied rewrites 98.8%

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

Alternative 14: 97.8% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.0%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 98.0%

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

12. Final simplification 98.0%

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

Alternative 15: 5.4% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 59.2%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. distribute-neg-frac2 N/A

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

   7. div-inv N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. inv-pow N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-cos.f64 59.2

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 59.2

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

4. Applied rewrites 59.2%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{\sin x}{\cos x} + \frac{\sin x}{\cos x}} \]
6. Step-by-step derivation

   1. distribute-lft1-in N/A

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

   2. metadata-eval N/A

      \[\leadsto \color{blue}{0} \cdot \frac{\sin x}{\cos x} \]

   3. mul0-lft 5.3

      \[\leadsto \color{blue}{0} \]

7. Applied rewrites 5.3%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024277 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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