tan-example (used to crash)

Percentage Accurate: 79.1% → 99.7%

Time: 23.3s

Alternatives: 15

Speedup: 1.0×

Specification

\[\left(\left(\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.751224 \cdot 10^{+308}\right)\right) \land \left(-1.776707 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right)\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.751224 \cdot 10^{+308}\right)\]

Math

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

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (tan((y + z)) - tan(a));
}

Fortran

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

Java

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

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 79.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (tan((y + z)) - tan(a));
}

Fortran

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

Java

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

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan z}{\frac{\cos y}{\sin y}}} - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+
  x
  (- (/ (+ (tan y) (tan z)) (- 1.0 (/ (tan z) (/ (cos y) (sin y))))) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (((tan(y) + tan(z)) / (1.0 - (tan(z) / (cos(y) / sin(y))))) - tan(a));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (((tan(y) + tan(z)) / (1.0d0 - (tan(z) / (cos(y) / sin(y))))) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + (((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(z) / (Math.cos(y) / Math.sin(y))))) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(z) / (math.cos(y) / math.sin(y))))) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(z) / Float64(cos(y) / sin(y))))) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(z) / (cos(y) / sin(y))))) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[z], $MachinePrecision] / N[(N[Cos[y], $MachinePrecision] / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.3%

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

   1. tan-sum N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right), \mathsf{tan.f64}\left(\color{blue}{a}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\tan y + \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(\color{blue}{a}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\tan y, \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan y \cdot \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\tan y, \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   9. tan-lowering-tan.f64 99.7%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

4. Applied egg-rr 99.7%

   \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \tan y\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   2. tan-quot N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \frac{\sin y}{\cos y}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \frac{1}{\frac{\cos y}{\sin y}}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   4. un-div-inv N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\frac{\tan z}{\frac{\cos y}{\sin y}}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\tan z, \left(\frac{\cos y}{\sin y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{\cos y}{\sin y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   7. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{1}{\frac{\sin y}{\cos y}}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   8. tan-quot N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{1}{\tan y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(1, \tan y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   10. tan-lowering-tan.f64 99.7%

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(y\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\frac{\tan z}{\frac{1}{\tan y}}}} - \tan a\right) \]

7. Taylor expanded in y around inf

   \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \color{blue}{\left(\frac{\cos y}{\sin y}\right)}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\cos y, \sin y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(y\right), \sin y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   3. sin-lowering-sin.f64 99.7%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(y\right), \mathsf{sin.f64}\left(y\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

9. Simplified 99.7%

   \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan z}{\color{blue}{\frac{\cos y}{\sin y}}}} - \tan a\right) \]
10. Add Preprocessing

Alternative 2: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -1 \cdot 10^{-16}:\\ \;\;\;\;x + \left(\frac{t\_0}{1 + \frac{\tan z}{\frac{-1 - -0.3333333333333333 \cdot \left(y \cdot y\right)}{y}}} - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-50}:\\ \;\;\;\;x + \frac{t\_0}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{t\_0}{1 + \frac{\tan z}{\frac{-1 - \left(y \cdot y\right) \cdot \left(-0.3333333333333333 + \left(y \cdot y\right) \cdot \left(-0.022222222222222223 + y \cdot \left(y \cdot -0.0021164021164021165\right)\right)\right)}{y}}} - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (<= (tan a) -1e-16)
     (+
      x
      (-
       (/
        t_0
        (+ 1.0 (/ (tan z) (/ (- -1.0 (* -0.3333333333333333 (* y y))) y))))
       (tan a)))
     (if (<= (tan a) 5e-50)
       (+ x (/ t_0 (- 1.0 (* (tan y) (tan z)))))
       (+
        x
        (-
         (/
          t_0
          (+
           1.0
           (/
            (tan z)
            (/
             (-
              -1.0
              (*
               (* y y)
               (+
                -0.3333333333333333
                (*
                 (* y y)
                 (+
                  -0.022222222222222223
                  (* y (* y -0.0021164021164021165)))))))
             y))))
         (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if (tan(a) <= -1e-16) {
		tmp = x + ((t_0 / (1.0 + (tan(z) / ((-1.0 - (-0.3333333333333333 * (y * y))) / y)))) - tan(a));
	} else if (tan(a) <= 5e-50) {
		tmp = x + (t_0 / (1.0 - (tan(y) * tan(z))));
	} else {
		tmp = x + ((t_0 / (1.0 + (tan(z) / ((-1.0 - ((y * y) * (-0.3333333333333333 + ((y * y) * (-0.022222222222222223 + (y * (y * -0.0021164021164021165))))))) / y)))) - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(y) + tan(z)
    if (tan(a) <= (-1d-16)) then
        tmp = x + ((t_0 / (1.0d0 + (tan(z) / (((-1.0d0) - ((-0.3333333333333333d0) * (y * y))) / y)))) - tan(a))
    else if (tan(a) <= 5d-50) then
        tmp = x + (t_0 / (1.0d0 - (tan(y) * tan(z))))
    else
        tmp = x + ((t_0 / (1.0d0 + (tan(z) / (((-1.0d0) - ((y * y) * ((-0.3333333333333333d0) + ((y * y) * ((-0.022222222222222223d0) + (y * (y * (-0.0021164021164021165d0)))))))) / y)))) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan(y) + Math.tan(z);
	double tmp;
	if (Math.tan(a) <= -1e-16) {
		tmp = x + ((t_0 / (1.0 + (Math.tan(z) / ((-1.0 - (-0.3333333333333333 * (y * y))) / y)))) - Math.tan(a));
	} else if (Math.tan(a) <= 5e-50) {
		tmp = x + (t_0 / (1.0 - (Math.tan(y) * Math.tan(z))));
	} else {
		tmp = x + ((t_0 / (1.0 + (Math.tan(z) / ((-1.0 - ((y * y) * (-0.3333333333333333 + ((y * y) * (-0.022222222222222223 + (y * (y * -0.0021164021164021165))))))) / y)))) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan(y) + math.tan(z)
	tmp = 0
	if math.tan(a) <= -1e-16:
		tmp = x + ((t_0 / (1.0 + (math.tan(z) / ((-1.0 - (-0.3333333333333333 * (y * y))) / y)))) - math.tan(a))
	elif math.tan(a) <= 5e-50:
		tmp = x + (t_0 / (1.0 - (math.tan(y) * math.tan(z))))
	else:
		tmp = x + ((t_0 / (1.0 + (math.tan(z) / ((-1.0 - ((y * y) * (-0.3333333333333333 + ((y * y) * (-0.022222222222222223 + (y * (y * -0.0021164021164021165))))))) / y)))) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if (tan(a) <= -1e-16)
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 + Float64(tan(z) / Float64(Float64(-1.0 - Float64(-0.3333333333333333 * Float64(y * y))) / y)))) - tan(a)));
	elseif (tan(a) <= 5e-50)
		tmp = Float64(x + Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))));
	else
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 + Float64(tan(z) / Float64(Float64(-1.0 - Float64(Float64(y * y) * Float64(-0.3333333333333333 + Float64(Float64(y * y) * Float64(-0.022222222222222223 + Float64(y * Float64(y * -0.0021164021164021165))))))) / y)))) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan(y) + tan(z);
	tmp = 0.0;
	if (tan(a) <= -1e-16)
		tmp = x + ((t_0 / (1.0 + (tan(z) / ((-1.0 - (-0.3333333333333333 * (y * y))) / y)))) - tan(a));
	elseif (tan(a) <= 5e-50)
		tmp = x + (t_0 / (1.0 - (tan(y) * tan(z))));
	else
		tmp = x + ((t_0 / (1.0 + (tan(z) / ((-1.0 - ((y * y) * (-0.3333333333333333 + ((y * y) * (-0.022222222222222223 + (y * (y * -0.0021164021164021165))))))) / y)))) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -1e-16], N[(x + N[(N[(t$95$0 / N[(1.0 + N[(N[Tan[z], $MachinePrecision] / N[(N[(-1.0 - N[(-0.3333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 5e-50], N[(x + N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(t$95$0 / N[(1.0 + N[(N[Tan[z], $MachinePrecision] / N[(N[(-1.0 - N[(N[(y * y), $MachinePrecision] * N[(-0.3333333333333333 + N[(N[(y * y), $MachinePrecision] * N[(-0.022222222222222223 + N[(y * N[(y * -0.0021164021164021165), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -9.9999999999999998e-17

   1. Initial program 77.3%

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

      1. tan-sum N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right), \mathsf{tan.f64}\left(\color{blue}{a}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\tan y + \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(\color{blue}{a}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\tan y, \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan y \cdot \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\tan y, \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      9. tan-lowering-tan.f64 99.7%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \tan y\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      2. tan-quot N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \frac{\sin y}{\cos y}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \frac{1}{\frac{\cos y}{\sin y}}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\frac{\tan z}{\frac{\cos y}{\sin y}}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\tan z, \left(\frac{\cos y}{\sin y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{\cos y}{\sin y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{1}{\frac{\sin y}{\cos y}}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      8. tan-quot N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{1}{\tan y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(1, \tan y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      10. tan-lowering-tan.f64 99.7%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(y\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\frac{\tan z}{\frac{1}{\tan y}}}} - \tan a\right) \]

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \color{blue}{\left(\frac{1 + \frac{-1}{3} \cdot {y}^{2}}{y}\right)}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\left(1 + \frac{-1}{3} \cdot {y}^{2}\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{3} \cdot {y}^{2}\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{3}, \left({y}^{2}\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{3}, \left(y \cdot y\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      5. *-lowering-*.f64 77.8%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   9. Simplified 77.8%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan z}{\color{blue}{\frac{1 + -0.3333333333333333 \cdot \left(y \cdot y\right)}{y}}}} - \tan a\right) \]

   if -9.9999999999999998e-17 < (tan.f64 a) < 4.99999999999999968e-50

   1. Initial program 73.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\sin \left(y + z\right), \cos \left(y + z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\left(y + z\right)\right), \cos \left(y + z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \cos \left(y + z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{cos.f64}\left(\left(y + z\right)\right)\right), x\right) \]

      7. +-lowering-+.f64 73.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(y, z\right)\right)\right), x\right) \]

   5. Simplified 73.7%

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

      1. tan-quot N/A

         \[\leadsto \mathsf{+.f64}\left(\tan \left(y + z\right), x\right) \]

      2. tan-sum N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\tan y + \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\tan y, \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \left(1 - \tan y \cdot \tan z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan y \cdot \tan z\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\tan y, \tan z\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right)\right)\right), x\right) \]

      10. tan-lowering-tan.f64 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right)\right)\right), x\right) \]

   7. Applied egg-rr 99.7%

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

   if 4.99999999999999968e-50 < (tan.f64 a)

   1. Initial program 80.0%

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

      1. tan-sum N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right), \mathsf{tan.f64}\left(\color{blue}{a}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\tan y + \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(\color{blue}{a}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\tan y, \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan y \cdot \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\tan y, \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      9. tan-lowering-tan.f64 99.5%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   4. Applied egg-rr 99.5%

      \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \tan y\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      2. tan-quot N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \frac{\sin y}{\cos y}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \frac{1}{\frac{\cos y}{\sin y}}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\frac{\tan z}{\frac{\cos y}{\sin y}}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\tan z, \left(\frac{\cos y}{\sin y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{\cos y}{\sin y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{1}{\frac{\sin y}{\cos y}}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      8. tan-quot N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{1}{\tan y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(1, \tan y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      10. tan-lowering-tan.f64 99.6%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(y\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\frac{\tan z}{\frac{1}{\tan y}}}} - \tan a\right) \]

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \color{blue}{\left(\frac{1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{-2}{945} \cdot {y}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{y}\right)}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{-2}{945} \cdot {y}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   9. Simplified 82.4%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan z}{\color{blue}{\frac{1 + \left(y \cdot y\right) \cdot \left(-0.3333333333333333 + \left(y \cdot y\right) \cdot \left(-0.022222222222222223 + y \cdot \left(y \cdot -0.0021164021164021165\right)\right)\right)}{y}}}} - \tan a\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -1 \cdot 10^{-16}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 + \frac{\tan z}{\frac{-1 - -0.3333333333333333 \cdot \left(y \cdot y\right)}{y}}} - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-50}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 + \frac{\tan z}{\frac{-1 - \left(y \cdot y\right) \cdot \left(-0.3333333333333333 + \left(y \cdot y\right) \cdot \left(-0.022222222222222223 + y \cdot \left(y \cdot -0.0021164021164021165\right)\right)\right)}{y}}} - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -1 \cdot 10^{-16}:\\ \;\;\;\;x + \left(\frac{t\_0}{1 + \frac{\tan z}{\frac{-1 - -0.3333333333333333 \cdot \left(y \cdot y\right)}{y}}} - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-50}:\\ \;\;\;\;x + \frac{t\_0}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{t\_0}{1 + \frac{\tan z}{\frac{-1 - \left(y \cdot y\right) \cdot \left(-0.3333333333333333 + y \cdot \left(y \cdot -0.022222222222222223\right)\right)}{y}}} - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (<= (tan a) -1e-16)
     (+
      x
      (-
       (/
        t_0
        (+ 1.0 (/ (tan z) (/ (- -1.0 (* -0.3333333333333333 (* y y))) y))))
       (tan a)))
     (if (<= (tan a) 5e-50)
       (+ x (/ t_0 (- 1.0 (* (tan y) (tan z)))))
       (+
        x
        (-
         (/
          t_0
          (+
           1.0
           (/
            (tan z)
            (/
             (-
              -1.0
              (*
               (* y y)
               (+ -0.3333333333333333 (* y (* y -0.022222222222222223)))))
             y))))
         (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if (tan(a) <= -1e-16) {
		tmp = x + ((t_0 / (1.0 + (tan(z) / ((-1.0 - (-0.3333333333333333 * (y * y))) / y)))) - tan(a));
	} else if (tan(a) <= 5e-50) {
		tmp = x + (t_0 / (1.0 - (tan(y) * tan(z))));
	} else {
		tmp = x + ((t_0 / (1.0 + (tan(z) / ((-1.0 - ((y * y) * (-0.3333333333333333 + (y * (y * -0.022222222222222223))))) / y)))) - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(y) + tan(z)
    if (tan(a) <= (-1d-16)) then
        tmp = x + ((t_0 / (1.0d0 + (tan(z) / (((-1.0d0) - ((-0.3333333333333333d0) * (y * y))) / y)))) - tan(a))
    else if (tan(a) <= 5d-50) then
        tmp = x + (t_0 / (1.0d0 - (tan(y) * tan(z))))
    else
        tmp = x + ((t_0 / (1.0d0 + (tan(z) / (((-1.0d0) - ((y * y) * ((-0.3333333333333333d0) + (y * (y * (-0.022222222222222223d0)))))) / y)))) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan(y) + Math.tan(z);
	double tmp;
	if (Math.tan(a) <= -1e-16) {
		tmp = x + ((t_0 / (1.0 + (Math.tan(z) / ((-1.0 - (-0.3333333333333333 * (y * y))) / y)))) - Math.tan(a));
	} else if (Math.tan(a) <= 5e-50) {
		tmp = x + (t_0 / (1.0 - (Math.tan(y) * Math.tan(z))));
	} else {
		tmp = x + ((t_0 / (1.0 + (Math.tan(z) / ((-1.0 - ((y * y) * (-0.3333333333333333 + (y * (y * -0.022222222222222223))))) / y)))) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan(y) + math.tan(z)
	tmp = 0
	if math.tan(a) <= -1e-16:
		tmp = x + ((t_0 / (1.0 + (math.tan(z) / ((-1.0 - (-0.3333333333333333 * (y * y))) / y)))) - math.tan(a))
	elif math.tan(a) <= 5e-50:
		tmp = x + (t_0 / (1.0 - (math.tan(y) * math.tan(z))))
	else:
		tmp = x + ((t_0 / (1.0 + (math.tan(z) / ((-1.0 - ((y * y) * (-0.3333333333333333 + (y * (y * -0.022222222222222223))))) / y)))) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if (tan(a) <= -1e-16)
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 + Float64(tan(z) / Float64(Float64(-1.0 - Float64(-0.3333333333333333 * Float64(y * y))) / y)))) - tan(a)));
	elseif (tan(a) <= 5e-50)
		tmp = Float64(x + Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))));
	else
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 + Float64(tan(z) / Float64(Float64(-1.0 - Float64(Float64(y * y) * Float64(-0.3333333333333333 + Float64(y * Float64(y * -0.022222222222222223))))) / y)))) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan(y) + tan(z);
	tmp = 0.0;
	if (tan(a) <= -1e-16)
		tmp = x + ((t_0 / (1.0 + (tan(z) / ((-1.0 - (-0.3333333333333333 * (y * y))) / y)))) - tan(a));
	elseif (tan(a) <= 5e-50)
		tmp = x + (t_0 / (1.0 - (tan(y) * tan(z))));
	else
		tmp = x + ((t_0 / (1.0 + (tan(z) / ((-1.0 - ((y * y) * (-0.3333333333333333 + (y * (y * -0.022222222222222223))))) / y)))) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -1e-16], N[(x + N[(N[(t$95$0 / N[(1.0 + N[(N[Tan[z], $MachinePrecision] / N[(N[(-1.0 - N[(-0.3333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 5e-50], N[(x + N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(t$95$0 / N[(1.0 + N[(N[Tan[z], $MachinePrecision] / N[(N[(-1.0 - N[(N[(y * y), $MachinePrecision] * N[(-0.3333333333333333 + N[(y * N[(y * -0.022222222222222223), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -9.9999999999999998e-17

   1. Initial program 77.3%

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

      1. tan-sum N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right), \mathsf{tan.f64}\left(\color{blue}{a}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\tan y + \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(\color{blue}{a}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\tan y, \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan y \cdot \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\tan y, \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      9. tan-lowering-tan.f64 99.7%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \tan y\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      2. tan-quot N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \frac{\sin y}{\cos y}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \frac{1}{\frac{\cos y}{\sin y}}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\frac{\tan z}{\frac{\cos y}{\sin y}}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\tan z, \left(\frac{\cos y}{\sin y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{\cos y}{\sin y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{1}{\frac{\sin y}{\cos y}}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      8. tan-quot N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{1}{\tan y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(1, \tan y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      10. tan-lowering-tan.f64 99.7%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(y\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\frac{\tan z}{\frac{1}{\tan y}}}} - \tan a\right) \]

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \color{blue}{\left(\frac{1 + \frac{-1}{3} \cdot {y}^{2}}{y}\right)}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\left(1 + \frac{-1}{3} \cdot {y}^{2}\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{3} \cdot {y}^{2}\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{3}, \left({y}^{2}\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{3}, \left(y \cdot y\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      5. *-lowering-*.f64 77.8%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   9. Simplified 77.8%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan z}{\color{blue}{\frac{1 + -0.3333333333333333 \cdot \left(y \cdot y\right)}{y}}}} - \tan a\right) \]

   if -9.9999999999999998e-17 < (tan.f64 a) < 4.99999999999999968e-50

   1. Initial program 73.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\sin \left(y + z\right), \cos \left(y + z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\left(y + z\right)\right), \cos \left(y + z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \cos \left(y + z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{cos.f64}\left(\left(y + z\right)\right)\right), x\right) \]

      7. +-lowering-+.f64 73.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(y, z\right)\right)\right), x\right) \]

   5. Simplified 73.7%

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

      1. tan-quot N/A

         \[\leadsto \mathsf{+.f64}\left(\tan \left(y + z\right), x\right) \]

      2. tan-sum N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\tan y + \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\tan y, \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \left(1 - \tan y \cdot \tan z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan y \cdot \tan z\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\tan y, \tan z\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right)\right)\right), x\right) \]

      10. tan-lowering-tan.f64 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right)\right)\right), x\right) \]

   7. Applied egg-rr 99.7%

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

   if 4.99999999999999968e-50 < (tan.f64 a)

   1. Initial program 80.0%

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

      1. tan-sum N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right), \mathsf{tan.f64}\left(\color{blue}{a}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\tan y + \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(\color{blue}{a}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\tan y, \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan y \cdot \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\tan y, \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      9. tan-lowering-tan.f64 99.5%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   4. Applied egg-rr 99.5%

      \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \tan y\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      2. tan-quot N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \frac{\sin y}{\cos y}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \frac{1}{\frac{\cos y}{\sin y}}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\frac{\tan z}{\frac{\cos y}{\sin y}}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\tan z, \left(\frac{\cos y}{\sin y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{\cos y}{\sin y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{1}{\frac{\sin y}{\cos y}}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      8. tan-quot N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{1}{\tan y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(1, \tan y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      10. tan-lowering-tan.f64 99.6%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(y\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\frac{\tan z}{\frac{1}{\tan y}}}} - \tan a\right) \]

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \color{blue}{\left(\frac{1 + {y}^{2} \cdot \left(\frac{-1}{45} \cdot {y}^{2} - \frac{1}{3}\right)}{y}\right)}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\left(1 + {y}^{2} \cdot \left(\frac{-1}{45} \cdot {y}^{2} - \frac{1}{3}\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({y}^{2} \cdot \left(\frac{-1}{45} \cdot {y}^{2} - \frac{1}{3}\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \left(\frac{-1}{45} \cdot {y}^{2} - \frac{1}{3}\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\frac{-1}{45} \cdot {y}^{2} - \frac{1}{3}\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{-1}{45} \cdot {y}^{2} - \frac{1}{3}\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{-1}{45} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{-1}{45} \cdot {y}^{2} + \frac{-1}{3}\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{-1}{3} + \frac{-1}{45} \cdot {y}^{2}\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left(\frac{-1}{45} \cdot {y}^{2}\right)\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left({y}^{2} \cdot \frac{-1}{45}\right)\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left(\left(y \cdot y\right) \cdot \frac{-1}{45}\right)\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{3}, \left(y \cdot \left(y \cdot \frac{-1}{45}\right)\right)\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(y, \left(y \cdot \frac{-1}{45}\right)\right)\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      14. *-lowering-*.f64 82.3%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{45}\right)\right)\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   9. Simplified 82.3%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan z}{\color{blue}{\frac{1 + \left(y \cdot y\right) \cdot \left(-0.3333333333333333 + y \cdot \left(y \cdot -0.022222222222222223\right)\right)}{y}}}} - \tan a\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -1 \cdot 10^{-16}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 + \frac{\tan z}{\frac{-1 - -0.3333333333333333 \cdot \left(y \cdot y\right)}{y}}} - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-50}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 + \frac{\tan z}{\frac{-1 - \left(y \cdot y\right) \cdot \left(-0.3333333333333333 + y \cdot \left(y \cdot -0.022222222222222223\right)\right)}{y}}} - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \left(\frac{t\_0}{1 + \frac{\tan z}{\frac{-1 - -0.3333333333333333 \cdot \left(y \cdot y\right)}{y}}} - \tan a\right)\\ \mathbf{if}\;\tan a \leq -1 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-15}:\\ \;\;\;\;x + \frac{t\_0}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z)))
        (t_1
         (+
          x
          (-
           (/
            t_0
            (+ 1.0 (/ (tan z) (/ (- -1.0 (* -0.3333333333333333 (* y y))) y))))
           (tan a)))))
   (if (<= (tan a) -1e-16)
     t_1
     (if (<= (tan a) 5e-15) (+ x (/ t_0 (- 1.0 (* (tan y) (tan z))))) t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = x + ((t_0 / (1.0 + (tan(z) / ((-1.0 - (-0.3333333333333333 * (y * y))) / y)))) - tan(a));
	double tmp;
	if (tan(a) <= -1e-16) {
		tmp = t_1;
	} else if (tan(a) <= 5e-15) {
		tmp = x + (t_0 / (1.0 - (tan(y) * tan(z))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = tan(y) + tan(z)
    t_1 = x + ((t_0 / (1.0d0 + (tan(z) / (((-1.0d0) - ((-0.3333333333333333d0) * (y * y))) / y)))) - tan(a))
    if (tan(a) <= (-1d-16)) then
        tmp = t_1
    else if (tan(a) <= 5d-15) then
        tmp = x + (t_0 / (1.0d0 - (tan(y) * tan(z))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan(y) + Math.tan(z);
	double t_1 = x + ((t_0 / (1.0 + (Math.tan(z) / ((-1.0 - (-0.3333333333333333 * (y * y))) / y)))) - Math.tan(a));
	double tmp;
	if (Math.tan(a) <= -1e-16) {
		tmp = t_1;
	} else if (Math.tan(a) <= 5e-15) {
		tmp = x + (t_0 / (1.0 - (Math.tan(y) * Math.tan(z))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan(y) + math.tan(z)
	t_1 = x + ((t_0 / (1.0 + (math.tan(z) / ((-1.0 - (-0.3333333333333333 * (y * y))) / y)))) - math.tan(a))
	tmp = 0
	if math.tan(a) <= -1e-16:
		tmp = t_1
	elif math.tan(a) <= 5e-15:
		tmp = x + (t_0 / (1.0 - (math.tan(y) * math.tan(z))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(x + Float64(Float64(t_0 / Float64(1.0 + Float64(tan(z) / Float64(Float64(-1.0 - Float64(-0.3333333333333333 * Float64(y * y))) / y)))) - tan(a)))
	tmp = 0.0
	if (tan(a) <= -1e-16)
		tmp = t_1;
	elseif (tan(a) <= 5e-15)
		tmp = Float64(x + Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan(y) + tan(z);
	t_1 = x + ((t_0 / (1.0 + (tan(z) / ((-1.0 - (-0.3333333333333333 * (y * y))) / y)))) - tan(a));
	tmp = 0.0;
	if (tan(a) <= -1e-16)
		tmp = t_1;
	elseif (tan(a) <= 5e-15)
		tmp = x + (t_0 / (1.0 - (tan(y) * tan(z))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(N[(t$95$0 / N[(1.0 + N[(N[Tan[z], $MachinePrecision] / N[(N[(-1.0 - N[(-0.3333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -1e-16], t$95$1, If[LessEqual[N[Tan[a], $MachinePrecision], 5e-15], N[(x + N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -9.9999999999999998e-17 or 4.99999999999999999e-15 < (tan.f64 a)

   1. Initial program 78.6%

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

      1. tan-sum N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right), \mathsf{tan.f64}\left(\color{blue}{a}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\tan y + \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(\color{blue}{a}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\tan y, \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan y \cdot \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\tan y, \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      9. tan-lowering-tan.f64 99.6%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   4. Applied egg-rr 99.6%

      \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \tan y\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      2. tan-quot N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \frac{\sin y}{\cos y}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \frac{1}{\frac{\cos y}{\sin y}}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\frac{\tan z}{\frac{\cos y}{\sin y}}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\tan z, \left(\frac{\cos y}{\sin y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{\cos y}{\sin y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{1}{\frac{\sin y}{\cos y}}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      8. tan-quot N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{1}{\tan y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(1, \tan y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      10. tan-lowering-tan.f64 99.6%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(y\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\frac{\tan z}{\frac{1}{\tan y}}}} - \tan a\right) \]

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \color{blue}{\left(\frac{1 + \frac{-1}{3} \cdot {y}^{2}}{y}\right)}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\left(1 + \frac{-1}{3} \cdot {y}^{2}\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{3} \cdot {y}^{2}\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{3}, \left({y}^{2}\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{3}, \left(y \cdot y\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      5. *-lowering-*.f64 79.4%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(y, y\right)\right)\right), y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   9. Simplified 79.4%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan z}{\color{blue}{\frac{1 + -0.3333333333333333 \cdot \left(y \cdot y\right)}{y}}}} - \tan a\right) \]

   if -9.9999999999999998e-17 < (tan.f64 a) < 4.99999999999999999e-15

   1. Initial program 74.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\sin \left(y + z\right), \cos \left(y + z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\left(y + z\right)\right), \cos \left(y + z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \cos \left(y + z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{cos.f64}\left(\left(y + z\right)\right)\right), x\right) \]

      7. +-lowering-+.f64 74.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(y, z\right)\right)\right), x\right) \]

   5. Simplified 74.0%

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

      1. tan-quot N/A

         \[\leadsto \mathsf{+.f64}\left(\tan \left(y + z\right), x\right) \]

      2. tan-sum N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\tan y + \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\tan y, \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \left(1 - \tan y \cdot \tan z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan y \cdot \tan z\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\tan y, \tan z\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right)\right)\right), x\right) \]

      10. tan-lowering-tan.f64 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right)\right)\right), x\right) \]

   7. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} + x \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -1 \cdot 10^{-16}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 + \frac{\tan z}{\frac{-1 - -0.3333333333333333 \cdot \left(y \cdot y\right)}{y}}} - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-15}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 + \frac{\tan z}{\frac{-1 - -0.3333333333333333 \cdot \left(y \cdot y\right)}{y}}} - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \left(t\_0 - \tan a\right)\\ \mathbf{if}\;\tan a \leq -1 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-15}:\\ \;\;\;\;x + \frac{t\_0}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ x (- t_0 (tan a)))))
   (if (<= (tan a) -1e-16)
     t_1
     (if (<= (tan a) 5e-15) (+ x (/ t_0 (- 1.0 (* (tan y) (tan z))))) t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = x + (t_0 - tan(a));
	double tmp;
	if (tan(a) <= -1e-16) {
		tmp = t_1;
	} else if (tan(a) <= 5e-15) {
		tmp = x + (t_0 / (1.0 - (tan(y) * tan(z))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = tan(y) + tan(z)
    t_1 = x + (t_0 - tan(a))
    if (tan(a) <= (-1d-16)) then
        tmp = t_1
    else if (tan(a) <= 5d-15) then
        tmp = x + (t_0 / (1.0d0 - (tan(y) * tan(z))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan(y) + Math.tan(z);
	double t_1 = x + (t_0 - Math.tan(a));
	double tmp;
	if (Math.tan(a) <= -1e-16) {
		tmp = t_1;
	} else if (Math.tan(a) <= 5e-15) {
		tmp = x + (t_0 / (1.0 - (Math.tan(y) * Math.tan(z))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan(y) + math.tan(z)
	t_1 = x + (t_0 - math.tan(a))
	tmp = 0
	if math.tan(a) <= -1e-16:
		tmp = t_1
	elif math.tan(a) <= 5e-15:
		tmp = x + (t_0 / (1.0 - (math.tan(y) * math.tan(z))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(x + Float64(t_0 - tan(a)))
	tmp = 0.0
	if (tan(a) <= -1e-16)
		tmp = t_1;
	elseif (tan(a) <= 5e-15)
		tmp = Float64(x + Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan(y) + tan(z);
	t_1 = x + (t_0 - tan(a));
	tmp = 0.0;
	if (tan(a) <= -1e-16)
		tmp = t_1;
	elseif (tan(a) <= 5e-15)
		tmp = x + (t_0 / (1.0 - (tan(y) * tan(z))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -1e-16], t$95$1, If[LessEqual[N[Tan[a], $MachinePrecision], 5e-15], N[(x + N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -9.9999999999999998e-17 or 4.99999999999999999e-15 < (tan.f64 a)

   1. Initial program 78.6%

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

      1. tan-sum N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right), \mathsf{tan.f64}\left(\color{blue}{a}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\tan y + \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(\color{blue}{a}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\tan y, \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan y \cdot \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\tan y, \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      9. tan-lowering-tan.f64 99.6%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   4. Applied egg-rr 99.6%

      \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \tan y\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      2. tan-quot N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \frac{\sin y}{\cos y}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \frac{1}{\frac{\cos y}{\sin y}}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\frac{\tan z}{\frac{\cos y}{\sin y}}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\tan z, \left(\frac{\cos y}{\sin y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{\cos y}{\sin y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{1}{\frac{\sin y}{\cos y}}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      8. tan-quot N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{1}{\tan y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(1, \tan y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

      10. tan-lowering-tan.f64 99.6%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(y\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\frac{\tan z}{\frac{1}{\tan y}}}} - \tan a\right) \]

   7. Taylor expanded in z around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \color{blue}{1}\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 79.3%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{1}} - \tan a\right) \]

   if -9.9999999999999998e-17 < (tan.f64 a) < 4.99999999999999999e-15

   1. Initial program 74.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\sin \left(y + z\right), \cos \left(y + z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\left(y + z\right)\right), \cos \left(y + z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \cos \left(y + z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{cos.f64}\left(\left(y + z\right)\right)\right), x\right) \]

      7. +-lowering-+.f64 74.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(y, z\right)\right)\right), x\right) \]

   5. Simplified 74.0%

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

      1. tan-quot N/A

         \[\leadsto \mathsf{+.f64}\left(\tan \left(y + z\right), x\right) \]

      2. tan-sum N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\tan y + \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\tan y, \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \left(1 - \tan y \cdot \tan z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan y \cdot \tan z\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\tan y, \tan z\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right)\right)\right), x\right) \]

      10. tan-lowering-tan.f64 99.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right)\right)\right), x\right) \]

   7. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} + x \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -1 \cdot 10^{-16}:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-15}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) (tan a))))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y, z, a)
	return Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - tan(a)))
end

MATLAB

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

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.3%

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

   1. tan-sum N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right), \mathsf{tan.f64}\left(\color{blue}{a}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\tan y + \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(\color{blue}{a}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\tan y, \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan y \cdot \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\tan y, \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   9. tan-lowering-tan.f64 99.7%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

4. Applied egg-rr 99.7%

   \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
5. Add Preprocessing

Alternative 7: 59.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;\tan a \leq -5 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\tan a \leq 0.02:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= (tan a) -5e-6) t_0 (if (<= (tan a) 0.02) (+ x (tan (+ y z))) t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (tan(a) <= -5e-6) {
		tmp = t_0;
	} else if (tan(a) <= 0.02) {
		tmp = x + tan((y + z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - tan(a)
    if (tan(a) <= (-5d-6)) then
        tmp = t_0
    else if (tan(a) <= 0.02d0) then
        tmp = x + tan((y + z))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = x - Math.tan(a);
	double tmp;
	if (Math.tan(a) <= -5e-6) {
		tmp = t_0;
	} else if (Math.tan(a) <= 0.02) {
		tmp = x + Math.tan((y + z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = x - math.tan(a)
	tmp = 0
	if math.tan(a) <= -5e-6:
		tmp = t_0
	elif math.tan(a) <= 0.02:
		tmp = x + math.tan((y + z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (tan(a) <= -5e-6)
		tmp = t_0;
	elseif (tan(a) <= 0.02)
		tmp = Float64(x + tan(Float64(y + z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = x - tan(a);
	tmp = 0.0;
	if (tan(a) <= -5e-6)
		tmp = t_0;
	elseif (tan(a) <= 0.02)
		tmp = x + tan((y + z));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -5e-6], t$95$0, If[LessEqual[N[Tan[a], $MachinePrecision], 0.02], N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -5.00000000000000041e-6 or 0.0200000000000000004 < (tan.f64 a)

   1. Initial program 77.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\sin y}{\cos y}\right), \color{blue}{\left(x - \frac{\sin a}{\cos a}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\sin y, \cos y\right), \left(\color{blue}{x} - \frac{\sin a}{\cos a}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \cos y\right), \left(x - \frac{\sin a}{\cos a}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \mathsf{cos.f64}\left(y\right)\right), \left(x - \frac{\sin a}{\cos a}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\sin a}{\cos a}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\sin a, \color{blue}{\cos a}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(a\right), \cos \color{blue}{a}\right)\right)\right) \]

      10. cos-lowering-cos.f64 60.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(a\right), \mathsf{cos.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 60.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\sin a}{\cos a}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\sin a, \color{blue}{\cos a}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(a\right), \cos \color{blue}{a}\right)\right) \]

      4. cos-lowering-cos.f64 45.5%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(a\right), \mathsf{cos.f64}\left(a\right)\right)\right) \]

   8. Simplified 45.5%

      \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\sin a}{\cos a}\right)}\right) \]

      2. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \tan a\right) \]

      3. tan-lowering-tan.f64 45.5%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{tan.f64}\left(a\right)\right) \]

   10. Applied egg-rr 45.5%

       \[\leadsto \color{blue}{x - \tan a} \]

   if -5.00000000000000041e-6 < (tan.f64 a) < 0.0200000000000000004

   1. Initial program 75.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\sin \left(y + z\right), \cos \left(y + z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\left(y + z\right)\right), \cos \left(y + z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \cos \left(y + z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{cos.f64}\left(\left(y + z\right)\right)\right), x\right) \]

      7. +-lowering-+.f64 73.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(y, z\right)\right)\right), x\right) \]

   5. Simplified 73.7%

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

      1. tan-quot N/A

         \[\leadsto \tan \left(y + z\right) + x \]

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

         \[\leadsto \mathsf{+.f64}\left(\tan \left(y + z\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{tan.f64}\left(\left(y + z\right)\right), x\right) \]

      4. +-lowering-+.f64 73.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), x\right) \]

   7. Applied egg-rr 73.7%

      \[\leadsto \color{blue}{\tan \left(y + z\right) + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -5 \cdot 10^{-6}:\\ \;\;\;\;x - \tan a\\ \mathbf{elif}\;\tan a \leq 0.02:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x - \tan a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.4% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (+ (tan y) (tan z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + ((tan(y) + tan(z)) - tan(a));
}

Fortran

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

Java

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

Python

def code(x, y, z, a):
	return x + ((math.tan(y) + math.tan(z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(Float64(tan(y) + tan(z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + ((tan(y) + tan(z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.3%

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

   1. tan-sum N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\right), \mathsf{tan.f64}\left(\color{blue}{a}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\tan y + \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(\color{blue}{a}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\tan y, \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \left(1 - \tan y \cdot \tan z\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan y \cdot \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\tan y, \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \tan z\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   9. tan-lowering-tan.f64 99.7%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

4. Applied egg-rr 99.7%

   \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \tan y\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   2. tan-quot N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \frac{\sin y}{\cos y}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\tan z \cdot \frac{1}{\frac{\cos y}{\sin y}}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   4. un-div-inv N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \left(\frac{\tan z}{\frac{\cos y}{\sin y}}\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\tan z, \left(\frac{\cos y}{\sin y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{\cos y}{\sin y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   7. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{1}{\frac{\sin y}{\cos y}}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   8. tan-quot N/A

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \left(\frac{1}{\tan y}\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(1, \tan y\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   10. tan-lowering-tan.f64 99.7%

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(z\right), \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(y\right)\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\frac{\tan z}{\frac{1}{\tan y}}}} - \tan a\right) \]

7. Taylor expanded in z around 0

   \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \color{blue}{1}\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]
8. Step-by-step derivation

9. Simplified 77.0%

   \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{1}} - \tan a\right) \]

10. Final simplification 77.0%

    \[\leadsto x + \left(\left(\tan y + \tan z\right) - \tan a\right) \]
11. Add Preprocessing

Alternative 9: 59.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2 \cdot 10^{-243}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -2e-243) (+ x (- (tan y) (tan a))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2e-243) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (tan(z) - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y + z) <= (-2d-243)) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (tan(z) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2e-243) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -2e-243:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -2e-243)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -2e-243)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -2e-243], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -1.99999999999999999e-243

   1. Initial program 73.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\color{blue}{y}\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 60.8%

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

   if -1.99999999999999999e-243 < (+.f64 y z)

   1. Initial program 79.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\color{blue}{z}\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 58.3%

      \[\leadsto x + \left(\tan \color{blue}{z} - \tan a\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 59.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq 0.05:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{1}{\tan \left(y + z\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) 0.05)
   (+ x (- (tan y) (tan a)))
   (+ x (/ 1.0 (/ 1.0 (tan (+ y z)))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 0.05) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (1.0 / (1.0 / tan((y + z))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y + z) <= 0.05d0) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (1.0d0 / (1.0d0 / tan((y + z))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 0.05) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (1.0 / (1.0 / Math.tan((y + z))));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= 0.05:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (1.0 / (1.0 / math.tan((y + z))))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= 0.05)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(1.0 / Float64(1.0 / tan(Float64(y + z)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= 0.05)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (1.0 / (1.0 / tan((y + z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], 0.05], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / N[(1.0 / N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < 0.050000000000000003

   1. Initial program 77.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\color{blue}{y}\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 66.5%

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

   if 0.050000000000000003 < (+.f64 y z)

   1. Initial program 73.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\sin \left(y + z\right), \cos \left(y + z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\left(y + z\right)\right), \cos \left(y + z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \cos \left(y + z\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{cos.f64}\left(\left(y + z\right)\right)\right), x\right) \]

      7. +-lowering-+.f64 46.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(y, z\right)\right)\right), x\right) \]

   5. Simplified 46.3%

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

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}}\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}\right)\right), x\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}}\right)\right), x\right) \]

      4. tan-quot N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{\tan \left(y + z\right)}\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \tan \left(y + z\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(\left(y + z\right)\right)\right)\right), x\right) \]

      7. +-lowering-+.f64 46.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right)\right)\right), x\right) \]

   7. Applied egg-rr 46.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\tan \left(y + z\right)}}} + x \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq 0.05:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{1}{\tan \left(y + z\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (tan((y + z)) - tan(a));
}

Fortran

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

Java

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

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 12: 60.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -1.35 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.06:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -1.35e-5) t_0 (if (<= a 0.06) (+ x (- (tan (+ y z)) a)) t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -1.35e-5) {
		tmp = t_0;
	} else if (a <= 0.06) {
		tmp = x + (tan((y + z)) - a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - tan(a)
    if (a <= (-1.35d-5)) then
        tmp = t_0
    else if (a <= 0.06d0) then
        tmp = x + (tan((y + z)) - a)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = x - Math.tan(a);
	double tmp;
	if (a <= -1.35e-5) {
		tmp = t_0;
	} else if (a <= 0.06) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = x - math.tan(a)
	tmp = 0
	if a <= -1.35e-5:
		tmp = t_0
	elif a <= 0.06:
		tmp = x + (math.tan((y + z)) - a)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -1.35e-5)
		tmp = t_0;
	elseif (a <= 0.06)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = x - tan(a);
	tmp = 0.0;
	if (a <= -1.35e-5)
		tmp = t_0;
	elseif (a <= 0.06)
		tmp = x + (tan((y + z)) - a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.35e-5], t$95$0, If[LessEqual[a, 0.06], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3499999999999999e-5 or 0.059999999999999998 < a

   1. Initial program 77.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\sin y}{\cos y}\right), \color{blue}{\left(x - \frac{\sin a}{\cos a}\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\sin y, \cos y\right), \left(\color{blue}{x} - \frac{\sin a}{\cos a}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \cos y\right), \left(x - \frac{\sin a}{\cos a}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \mathsf{cos.f64}\left(y\right)\right), \left(x - \frac{\sin a}{\cos a}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\sin a}{\cos a}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\sin a, \color{blue}{\cos a}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(a\right), \cos \color{blue}{a}\right)\right)\right) \]

      10. cos-lowering-cos.f64 60.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(a\right), \mathsf{cos.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 60.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\sin a}{\cos a}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\sin a, \color{blue}{\cos a}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(a\right), \cos \color{blue}{a}\right)\right) \]

      4. cos-lowering-cos.f64 44.8%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(a\right), \mathsf{cos.f64}\left(a\right)\right)\right) \]

   8. Simplified 44.8%

      \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\sin a}{\cos a}\right)}\right) \]

      2. tan-quot N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \tan a\right) \]

      3. tan-lowering-tan.f64 44.8%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{tan.f64}\left(a\right)\right) \]

   10. Applied egg-rr 44.8%

       \[\leadsto \color{blue}{x - \tan a} \]

   if -1.3499999999999999e-5 < a < 0.059999999999999998

   1. Initial program 75.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \color{blue}{a}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 75.5%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 40.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x - \tan a \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (- x (tan a)))

C

double code(double x, double y, double z, double a) {
	return x - tan(a);
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x - tan(a)
end function

Java

public static double code(double x, double y, double z, double a) {
	return x - Math.tan(a);
}

Python

def code(x, y, z, a):
	return x - math.tan(a)

Julia

function code(x, y, z, a)
	return Float64(x - tan(a))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x - tan(a);
end

Wolfram

code[x_, y_, z_, a_] := N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

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

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{\sin y}{\cos y}\right), \color{blue}{\left(x - \frac{\sin a}{\cos a}\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\sin y, \cos y\right), \left(\color{blue}{x} - \frac{\sin a}{\cos a}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \cos y\right), \left(x - \frac{\sin a}{\cos a}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \mathsf{cos.f64}\left(y\right)\right), \left(x - \frac{\sin a}{\cos a}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\sin a}{\cos a}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\sin a, \color{blue}{\cos a}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(a\right), \cos \color{blue}{a}\right)\right)\right) \]

   10. cos-lowering-cos.f64 59.7%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), \mathsf{cos.f64}\left(y\right)\right), \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(a\right), \mathsf{cos.f64}\left(a\right)\right)\right)\right) \]

5. Simplified 59.7%

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

6. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\sin a}{\cos a}\right)}\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\sin a, \color{blue}{\cos a}\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(a\right), \cos \color{blue}{a}\right)\right) \]

   4. cos-lowering-cos.f64 40.9%

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(a\right), \mathsf{cos.f64}\left(a\right)\right)\right) \]

8. Simplified 40.9%

   \[\leadsto \color{blue}{x - \frac{\sin a}{\cos a}} \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\sin a}{\cos a}\right)}\right) \]

   2. tan-quot N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \tan a\right) \]

   3. tan-lowering-tan.f64 40.9%

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{tan.f64}\left(a\right)\right) \]

10. Applied egg-rr 40.9%

    \[\leadsto \color{blue}{x - \tan a} \]
11. Add Preprocessing

Alternative 14: 31.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x + \sin z \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (+ x (sin z)))

C

double code(double x, double y, double z, double a) {
	return x + sin(z);
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + sin(z)
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + Math.sin(z);
}

Python

def code(x, y, z, a):
	return x + math.sin(z)

Julia

function code(x, y, z, a)
	return Float64(x + sin(z))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + sin(z);
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[Sin[z], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

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

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right), \color{blue}{x}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\sin \left(y + z\right), \cos \left(y + z\right)\right), x\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\left(y + z\right)\right), \cos \left(y + z\right)\right), x\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \cos \left(y + z\right)\right), x\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{cos.f64}\left(\left(y + z\right)\right)\right), x\right) \]

   7. +-lowering-+.f64 49.9%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(y, z\right)\right)\right), x\right) \]

5. Simplified 49.9%

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

6. Taylor expanded in y around inf

   \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{cos.f64}\left(\color{blue}{y}\right)\right), x\right) \]
7. Step-by-step derivation

8. Simplified 41.2%

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

9. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + \sin z} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \sin z + \color{blue}{x} \]

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

       \[\leadsto \mathsf{+.f64}\left(\sin z, \color{blue}{x}\right) \]

    3. sin-lowering-sin.f64 30.2%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{sin.f64}\left(z\right), x\right) \]

11. Simplified 30.2%

    \[\leadsto \color{blue}{\sin z + x} \]

12. Final simplification 30.2%

    \[\leadsto x + \sin z \]
13. Add Preprocessing

Alternative 15: 31.2% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x y z a) :precision binary64 x)

C

double code(double x, double y, double z, double a) {
	return x;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x
end function

Java

public static double code(double x, double y, double z, double a) {
	return x;
}

Python

def code(x, y, z, a):
	return x

Julia

function code(x, y, z, a)
	return x
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x;
end

Wolfram

code[x_, y_, z_, a_] := x

Derivation

1. Initial program 76.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 30.0%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(FPCore (x y z a)
  :name "tan-example (used to crash)"
  :precision binary64
  :pre (and (and (and (or (== x 0.0) (and (<= 0.5884142 x) (<= x 505.5909))) (or (and (<= -1.796658e+308 y) (<= y -9.425585e-310)) (and (<= 1.284938e-309 y) (<= y 1.751224e+308)))) (or (and (<= -1.776707e+308 z) (<= z -8.599796e-310)) (and (<= 3.293145e-311 z) (<= z 1.725154e+308)))) (or (and (<= -1.796658e+308 a) (<= a -9.425585e-310)) (and (<= 1.284938e-309 a) (<= a 1.751224e+308))))
  (+ x (- (tan (+ y z)) (tan a))))