tan-example (used to crash)

Percentage Accurate: 79.8% → 99.7%

Time: 24.8s

Alternatives: 22

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: 79.8% 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

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+
  x
  (+
   (/ (tan y) (- 1.0 (/ (* (tan z) (sin y)) (cos y))))
   (- (/ (tan z) (- 1.0 (* (tan y) (tan z)))) (tan a)))))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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) / (1.0d0 - ((tan(z) * sin(y)) / cos(y)))) + ((tan(z) / (1.0d0 - (tan(y) * tan(z)))) - tan(a)))
end function

Java

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

Python

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

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(x + Float64(Float64(tan(y) / Float64(1.0 - Float64(Float64(tan(z) * sin(y)) / cos(y)))) + Float64(Float64(tan(z) / Float64(1.0 - Float64(tan(y) * tan(z)))) - tan(a))))
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = x + ((tan(y) / (1.0 - ((tan(z) * sin(y)) / cos(y)))) + ((tan(z) / (1.0 - (tan(y) * tan(z)))) - tan(a)));
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(x + N[(N[(N[Tan[y], $MachinePrecision] / N[(1.0 - N[(N[(N[Tan[z], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] / N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Tan[z], $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. div-inv N/A

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

   2. fmm-def N/A

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

   3. fma-define N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate-+l+ N/A

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

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

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

6. Applied egg-rr 99.7%

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

   1. *-commutative N/A

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

   2. tan-quot N/A

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

   3. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

   8. cos-lowering-cos.f64 99.7%

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

8. Applied egg-rr 99.7%

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

Alternative 2: 99.7% accurate, 0.3× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+
  x
  (+
   (- (/ (tan z) (- 1.0 (* (tan y) (tan z)))) (tan a))
   (/ (tan y) (- 1.0 (/ (* (tan y) (sin z)) (cos z)))))))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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(z) / (1.0d0 - (tan(y) * tan(z)))) - tan(a)) + (tan(y) / (1.0d0 - ((tan(y) * sin(z)) / cos(z)))))
end function

Java

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

Python

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

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(x + Float64(Float64(Float64(tan(z) / Float64(1.0 - Float64(tan(y) * tan(z)))) - tan(a)) + Float64(tan(y) / Float64(1.0 - Float64(Float64(tan(y) * sin(z)) / cos(z))))))
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = x + (((tan(z) / (1.0 - (tan(y) * tan(z)))) - tan(a)) + (tan(y) / (1.0 - ((tan(y) * sin(z)) / cos(z)))));
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(x + N[(N[(N[(N[Tan[z], $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + N[(N[Tan[y], $MachinePrecision] / N[(1.0 - N[(N[(N[Tan[y], $MachinePrecision] * N[Sin[z], $MachinePrecision]), $MachinePrecision] / N[Cos[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. div-inv N/A

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

   2. fmm-def N/A

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

   3. fma-define N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate-+l+ N/A

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

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

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

6. Applied egg-rr 99.7%

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

   1. tan-quot N/A

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

   2. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

   7. cos-lowering-cos.f64 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 99.7% accurate, 0.3× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+
  x
  (+
   (/ (tan y) (- 1.0 (* (tan y) (tan z))))
   (- (/ (tan z) (+ 1.0 (/ (tan z) (/ -1.0 (tan y))))) (tan a)))))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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) / (1.0d0 - (tan(y) * tan(z)))) + ((tan(z) / (1.0d0 + (tan(z) / ((-1.0d0) / tan(y))))) - tan(a)))
end function

Java

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

Python

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

Julia

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

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = x + ((tan(y) / (1.0 - (tan(y) * tan(z)))) + ((tan(z) / (1.0 + (tan(z) / (-1.0 / tan(y))))) - tan(a)));
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(x + N[(N[(N[Tan[y], $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Tan[z], $MachinePrecision] / N[(1.0 + N[(N[Tan[z], $MachinePrecision] / N[(-1.0 / N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. div-inv N/A

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

   2. fmm-def N/A

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

   3. fma-define N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate-+l+ N/A

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

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

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

6. Applied egg-rr 99.7%

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

   1. *-commutative N/A

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

   2. tan-quot N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

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

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

   7. clear-num N/A

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

   8. tan-quot N/A

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

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

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

   10. tan-lowering-tan.f64 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 4: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := 1 - \tan y \cdot \tan z\\ x + \left(\left(\frac{\tan z}{t\_0} - \tan a\right) + \frac{\tan y}{t\_0}\right) \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- 1.0 (* (tan y) (tan z)))))
   (+ x (+ (- (/ (tan z) t_0) (tan a)) (/ (tan y) t_0)))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = 1.0 - (tan(y) * tan(z));
	return x + (((tan(z) / t_0) - tan(a)) + (tan(y) / t_0));
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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
    t_0 = 1.0d0 - (tan(y) * tan(z))
    code = x + (((tan(z) / t_0) - tan(a)) + (tan(y) / t_0))
end function

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	double t_0 = 1.0 - (Math.tan(y) * Math.tan(z));
	return x + (((Math.tan(z) / t_0) - Math.tan(a)) + (Math.tan(y) / t_0));
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	t_0 = 1.0 - (math.tan(y) * math.tan(z))
	return x + (((math.tan(z) / t_0) - math.tan(a)) + (math.tan(y) / t_0))

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(1.0 - Float64(tan(y) * tan(z)))
	return Float64(x + Float64(Float64(Float64(tan(z) / t_0) - tan(a)) + Float64(tan(y) / t_0)))
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	t_0 = 1.0 - (tan(y) * tan(z));
	tmp = x + (((tan(z) / t_0) - tan(a)) + (tan(y) / t_0));
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x + N[(N[(N[(N[Tan[z], $MachinePrecision] / t$95$0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + N[(N[Tan[y], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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. div-inv N/A

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

   2. fmm-def N/A

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

   3. fma-define N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate-+l+ N/A

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

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

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 5: 89.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -0.005:\\ \;\;\;\;x + \left(\frac{t\_0}{1 + \frac{\tan y}{\frac{-1 - -0.3333333333333333 \cdot \left(z \cdot z\right)}{z}}} - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 10^{-58}:\\ \;\;\;\;x + \left(\frac{t\_0}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{t\_0}{1 + \frac{\tan y}{\frac{-1 - \left(z \cdot z\right) \cdot \left(-0.3333333333333333 + \left(z \cdot z\right) \cdot \left(-0.022222222222222223 + \left(z \cdot z\right) \cdot -0.0021164021164021165\right)\right)}{z}}} - \tan a\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (<= (tan a) -0.005)
     (+
      x
      (-
       (/
        t_0
        (+ 1.0 (/ (tan y) (/ (- -1.0 (* -0.3333333333333333 (* z z))) z))))
       (tan a)))
     (if (<= (tan a) 1e-58)
       (+ x (- (/ t_0 (- 1.0 (* (tan y) (tan z)))) a))
       (+
        x
        (-
         (/
          t_0
          (+
           1.0
           (/
            (tan y)
            (/
             (-
              -1.0
              (*
               (* z z)
               (+
                -0.3333333333333333
                (*
                 (* z z)
                 (+
                  -0.022222222222222223
                  (* (* z z) -0.0021164021164021165))))))
             z))))
         (tan a)))))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if (tan(a) <= -0.005) {
		tmp = x + ((t_0 / (1.0 + (tan(y) / ((-1.0 - (-0.3333333333333333 * (z * z))) / z)))) - tan(a));
	} else if (tan(a) <= 1e-58) {
		tmp = x + ((t_0 / (1.0 - (tan(y) * tan(z)))) - a);
	} else {
		tmp = x + ((t_0 / (1.0 + (tan(y) / ((-1.0 - ((z * z) * (-0.3333333333333333 + ((z * z) * (-0.022222222222222223 + ((z * z) * -0.0021164021164021165)))))) / z)))) - tan(a));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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) <= (-0.005d0)) then
        tmp = x + ((t_0 / (1.0d0 + (tan(y) / (((-1.0d0) - ((-0.3333333333333333d0) * (z * z))) / z)))) - tan(a))
    else if (tan(a) <= 1d-58) then
        tmp = x + ((t_0 / (1.0d0 - (tan(y) * tan(z)))) - a)
    else
        tmp = x + ((t_0 / (1.0d0 + (tan(y) / (((-1.0d0) - ((z * z) * ((-0.3333333333333333d0) + ((z * z) * ((-0.022222222222222223d0) + ((z * z) * (-0.0021164021164021165d0))))))) / z)))) - tan(a))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < a;
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) <= -0.005) {
		tmp = x + ((t_0 / (1.0 + (Math.tan(y) / ((-1.0 - (-0.3333333333333333 * (z * z))) / z)))) - Math.tan(a));
	} else if (Math.tan(a) <= 1e-58) {
		tmp = x + ((t_0 / (1.0 - (Math.tan(y) * Math.tan(z)))) - a);
	} else {
		tmp = x + ((t_0 / (1.0 + (Math.tan(y) / ((-1.0 - ((z * z) * (-0.3333333333333333 + ((z * z) * (-0.022222222222222223 + ((z * z) * -0.0021164021164021165)))))) / z)))) - Math.tan(a));
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	t_0 = math.tan(y) + math.tan(z)
	tmp = 0
	if math.tan(a) <= -0.005:
		tmp = x + ((t_0 / (1.0 + (math.tan(y) / ((-1.0 - (-0.3333333333333333 * (z * z))) / z)))) - math.tan(a))
	elif math.tan(a) <= 1e-58:
		tmp = x + ((t_0 / (1.0 - (math.tan(y) * math.tan(z)))) - a)
	else:
		tmp = x + ((t_0 / (1.0 + (math.tan(y) / ((-1.0 - ((z * z) * (-0.3333333333333333 + ((z * z) * (-0.022222222222222223 + ((z * z) * -0.0021164021164021165)))))) / z)))) - math.tan(a))
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if (tan(a) <= -0.005)
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 + Float64(tan(y) / Float64(Float64(-1.0 - Float64(-0.3333333333333333 * Float64(z * z))) / z)))) - tan(a)));
	elseif (tan(a) <= 1e-58)
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))) - a));
	else
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 + Float64(tan(y) / Float64(Float64(-1.0 - Float64(Float64(z * z) * Float64(-0.3333333333333333 + Float64(Float64(z * z) * Float64(-0.022222222222222223 + Float64(Float64(z * z) * -0.0021164021164021165)))))) / z)))) - tan(a)));
	end
	return tmp
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp_2 = code(x, y, z, a)
	t_0 = tan(y) + tan(z);
	tmp = 0.0;
	if (tan(a) <= -0.005)
		tmp = x + ((t_0 / (1.0 + (tan(y) / ((-1.0 - (-0.3333333333333333 * (z * z))) / z)))) - tan(a));
	elseif (tan(a) <= 1e-58)
		tmp = x + ((t_0 / (1.0 - (tan(y) * tan(z)))) - a);
	else
		tmp = x + ((t_0 / (1.0 + (tan(y) / ((-1.0 - ((z * z) * (-0.3333333333333333 + ((z * z) * (-0.022222222222222223 + ((z * z) * -0.0021164021164021165)))))) / z)))) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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], -0.005], N[(x + N[(N[(t$95$0 / N[(1.0 + N[(N[Tan[y], $MachinePrecision] / N[(N[(-1.0 - N[(-0.3333333333333333 * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 1e-58], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(t$95$0 / N[(1.0 + N[(N[Tan[y], $MachinePrecision] / N[(N[(-1.0 - N[(N[(z * z), $MachinePrecision] * N[(-0.3333333333333333 + N[(N[(z * z), $MachinePrecision] * N[(-0.022222222222222223 + N[(N[(z * z), $MachinePrecision] * -0.0021164021164021165), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0050000000000000001

   1. Initial program 75.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.7%

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

   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. tan-quot N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. clear-num N/A

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

      7. tan-quot N/A

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

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

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

      9. tan-lowering-tan.f64 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in z around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 76.1%

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

   9. Simplified 76.1%

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

   if -0.0050000000000000001 < (tan.f64 a) < 1e-58

   1. Initial program 77.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. Taylor expanded in a around 0

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

   7. Simplified 99.7%

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

   if 1e-58 < (tan.f64 a)

   1. Initial program 79.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. tan-quot N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. clear-num N/A

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

      7. tan-quot N/A

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

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

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

      9. tan-lowering-tan.f64 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in z around 0

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

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

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

   9. Simplified 79.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.005:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 + \frac{\tan y}{\frac{-1 - -0.3333333333333333 \cdot \left(z \cdot z\right)}{z}}} - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 10^{-58}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 + \frac{\tan y}{\frac{-1 - \left(z \cdot z\right) \cdot \left(-0.3333333333333333 + \left(z \cdot z\right) \cdot \left(-0.022222222222222223 + \left(z \cdot z\right) \cdot -0.0021164021164021165\right)\right)}{z}}} - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -0.005:\\ \;\;\;\;x + \left(\frac{t\_0}{1 + \frac{\tan y}{\frac{-1 - -0.3333333333333333 \cdot \left(z \cdot z\right)}{z}}} - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 10^{-58}:\\ \;\;\;\;x + \left(\frac{t\_0}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{t\_0}{1 + \frac{\tan y}{\frac{-1 - \left(z \cdot z\right) \cdot \left(-0.3333333333333333 + z \cdot \left(z \cdot -0.022222222222222223\right)\right)}{z}}} - \tan a\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (<= (tan a) -0.005)
     (+
      x
      (-
       (/
        t_0
        (+ 1.0 (/ (tan y) (/ (- -1.0 (* -0.3333333333333333 (* z z))) z))))
       (tan a)))
     (if (<= (tan a) 1e-58)
       (+ x (- (/ t_0 (- 1.0 (* (tan y) (tan z)))) a))
       (+
        x
        (-
         (/
          t_0
          (+
           1.0
           (/
            (tan y)
            (/
             (-
              -1.0
              (*
               (* z z)
               (+ -0.3333333333333333 (* z (* z -0.022222222222222223)))))
             z))))
         (tan a)))))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if (tan(a) <= -0.005) {
		tmp = x + ((t_0 / (1.0 + (tan(y) / ((-1.0 - (-0.3333333333333333 * (z * z))) / z)))) - tan(a));
	} else if (tan(a) <= 1e-58) {
		tmp = x + ((t_0 / (1.0 - (tan(y) * tan(z)))) - a);
	} else {
		tmp = x + ((t_0 / (1.0 + (tan(y) / ((-1.0 - ((z * z) * (-0.3333333333333333 + (z * (z * -0.022222222222222223))))) / z)))) - tan(a));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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) <= (-0.005d0)) then
        tmp = x + ((t_0 / (1.0d0 + (tan(y) / (((-1.0d0) - ((-0.3333333333333333d0) * (z * z))) / z)))) - tan(a))
    else if (tan(a) <= 1d-58) then
        tmp = x + ((t_0 / (1.0d0 - (tan(y) * tan(z)))) - a)
    else
        tmp = x + ((t_0 / (1.0d0 + (tan(y) / (((-1.0d0) - ((z * z) * ((-0.3333333333333333d0) + (z * (z * (-0.022222222222222223d0)))))) / z)))) - tan(a))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < a;
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) <= -0.005) {
		tmp = x + ((t_0 / (1.0 + (Math.tan(y) / ((-1.0 - (-0.3333333333333333 * (z * z))) / z)))) - Math.tan(a));
	} else if (Math.tan(a) <= 1e-58) {
		tmp = x + ((t_0 / (1.0 - (Math.tan(y) * Math.tan(z)))) - a);
	} else {
		tmp = x + ((t_0 / (1.0 + (Math.tan(y) / ((-1.0 - ((z * z) * (-0.3333333333333333 + (z * (z * -0.022222222222222223))))) / z)))) - Math.tan(a));
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	t_0 = math.tan(y) + math.tan(z)
	tmp = 0
	if math.tan(a) <= -0.005:
		tmp = x + ((t_0 / (1.0 + (math.tan(y) / ((-1.0 - (-0.3333333333333333 * (z * z))) / z)))) - math.tan(a))
	elif math.tan(a) <= 1e-58:
		tmp = x + ((t_0 / (1.0 - (math.tan(y) * math.tan(z)))) - a)
	else:
		tmp = x + ((t_0 / (1.0 + (math.tan(y) / ((-1.0 - ((z * z) * (-0.3333333333333333 + (z * (z * -0.022222222222222223))))) / z)))) - math.tan(a))
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if (tan(a) <= -0.005)
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 + Float64(tan(y) / Float64(Float64(-1.0 - Float64(-0.3333333333333333 * Float64(z * z))) / z)))) - tan(a)));
	elseif (tan(a) <= 1e-58)
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))) - a));
	else
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 + Float64(tan(y) / Float64(Float64(-1.0 - Float64(Float64(z * z) * Float64(-0.3333333333333333 + Float64(z * Float64(z * -0.022222222222222223))))) / z)))) - tan(a)));
	end
	return tmp
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp_2 = code(x, y, z, a)
	t_0 = tan(y) + tan(z);
	tmp = 0.0;
	if (tan(a) <= -0.005)
		tmp = x + ((t_0 / (1.0 + (tan(y) / ((-1.0 - (-0.3333333333333333 * (z * z))) / z)))) - tan(a));
	elseif (tan(a) <= 1e-58)
		tmp = x + ((t_0 / (1.0 - (tan(y) * tan(z)))) - a);
	else
		tmp = x + ((t_0 / (1.0 + (tan(y) / ((-1.0 - ((z * z) * (-0.3333333333333333 + (z * (z * -0.022222222222222223))))) / z)))) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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], -0.005], N[(x + N[(N[(t$95$0 / N[(1.0 + N[(N[Tan[y], $MachinePrecision] / N[(N[(-1.0 - N[(-0.3333333333333333 * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 1e-58], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(t$95$0 / N[(1.0 + N[(N[Tan[y], $MachinePrecision] / N[(N[(-1.0 - N[(N[(z * z), $MachinePrecision] * N[(-0.3333333333333333 + N[(z * N[(z * -0.022222222222222223), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0050000000000000001

   1. Initial program 75.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.7%

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

   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. tan-quot N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. clear-num N/A

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

      7. tan-quot N/A

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

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

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

      9. tan-lowering-tan.f64 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in z around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 76.1%

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

   9. Simplified 76.1%

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

   if -0.0050000000000000001 < (tan.f64 a) < 1e-58

   1. Initial program 77.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. Taylor expanded in a around 0

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

   7. Simplified 99.7%

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

   if 1e-58 < (tan.f64 a)

   1. Initial program 79.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. tan-quot N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. clear-num N/A

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

      7. tan-quot N/A

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

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

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

      9. tan-lowering-tan.f64 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in z around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

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

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

      14. *-lowering-*.f64 79.7%

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

   9. Simplified 79.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.005:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 + \frac{\tan y}{\frac{-1 - -0.3333333333333333 \cdot \left(z \cdot z\right)}{z}}} - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 10^{-58}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 + \frac{\tan y}{\frac{-1 - \left(z \cdot z\right) \cdot \left(-0.3333333333333333 + z \cdot \left(z \cdot -0.022222222222222223\right)\right)}{z}}} - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -0.005:\\ \;\;\;\;x + \left(t\_0 - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 10^{-58}:\\ \;\;\;\;x + \left(\frac{t\_0}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, t\_0, x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (<= (tan a) -0.005)
     (+ x (- t_0 (tan a)))
     (if (<= (tan a) 1e-58)
       (+ x (- (/ t_0 (- 1.0 (* (tan y) (tan z)))) a))
       (fma 1.0 t_0 (- x (tan a)))))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if (tan(a) <= -0.005) {
		tmp = x + (t_0 - tan(a));
	} else if (tan(a) <= 1e-58) {
		tmp = x + ((t_0 / (1.0 - (tan(y) * tan(z)))) - a);
	} else {
		tmp = fma(1.0, t_0, (x - tan(a)));
	}
	return tmp;
}

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if (tan(a) <= -0.005)
		tmp = Float64(x + Float64(t_0 - tan(a)));
	elseif (tan(a) <= 1e-58)
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))) - a));
	else
		tmp = fma(1.0, t_0, Float64(x - tan(a)));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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], -0.005], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 1e-58], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(1.0 * t$95$0 + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0050000000000000001

   1. Initial program 75.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 99.7%

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

   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. Taylor expanded in y around 0

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

   7. Simplified 75.4%

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

   if -0.0050000000000000001 < (tan.f64 a) < 1e-58

   1. Initial program 77.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. Taylor expanded in a around 0

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

   7. Simplified 99.7%

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

   if 1e-58 < (tan.f64 a)

   1. Initial program 79.5%

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

      1. +-commutative N/A

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

      2. sub-neg N/A

         \[\leadsto \left(\tan \left(y + z\right) + \left(\mathsf{neg}\left(\tan a\right)\right)\right) + x \]

      3. associate-+l+ N/A

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

      4. tan-sum N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. fma-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      17. +-commutative N/A

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

      18. sub-neg N/A

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

      19. --lowering--.f64 N/A

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

      20. tan-lowering-tan.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 79.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, \tan y + \tan z, x - \tan a\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.005:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 10^{-58}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \tan y + \tan z, x - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.7% accurate, 0.4× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+ x (- (/ 1.0 (/ (- 1.0 (* (tan y) (tan z))) (+ (tan y) (tan z)))) (tan a))))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 + ((1.0d0 / ((1.0d0 - (tan(y) * tan(z))) / (tan(y) + tan(z)))) - tan(a))
end function

Java

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

Python

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

Julia

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

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = x + ((1.0 / ((1.0 - (tan(y) * tan(z))) / (tan(y) + tan(z)))) - tan(a));
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(x + N[(N[(1.0 / N[(N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. tan-quot N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

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

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

   6. clear-num N/A

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

   7. tan-quot N/A

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

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

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

   9. tan-lowering-tan.f64 99.7%

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

6. Applied egg-rr 99.7%

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

   1. clear-num N/A

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

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

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

   3. metadata-eval N/A

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

   4. associate-/r/ N/A

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

   5. associate-*l/ N/A

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

   6. div-sub N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   18. tan-lowering-tan.f64 99.7%

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

8. Applied egg-rr 99.7%

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

Alternative 9: 99.7% accurate, 0.4× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) (tan a))))

C

assert(x < y && y < z && z < a);
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

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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

assert x < y && y < z && z < a;
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

[x, y, z, a] = sort([x, y, z, a])
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

x, y, z, a = sort([x, y, z, a])
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

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - tan(a));
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 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. Add Preprocessing

Alternative 10: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \left(\frac{t\_0}{1 + \frac{\tan y}{\frac{-1 - -0.3333333333333333 \cdot \left(z \cdot z\right)}{z}}} - \tan a\right)\\ \mathbf{if}\;a \leq -0.0004:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-54}:\\ \;\;\;\;x + \left(\frac{t\_0}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z)))
        (t_1
         (+
          x
          (-
           (/
            t_0
            (+ 1.0 (/ (tan y) (/ (- -1.0 (* -0.3333333333333333 (* z z))) z))))
           (tan a)))))
   (if (<= a -0.0004)
     t_1
     (if (<= a 2.8e-54) (+ x (- (/ t_0 (- 1.0 (* (tan y) (tan z)))) a)) t_1))))

C

assert(x < y && y < z && z < a);
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(y) / ((-1.0 - (-0.3333333333333333 * (z * z))) / z)))) - tan(a));
	double tmp;
	if (a <= -0.0004) {
		tmp = t_1;
	} else if (a <= 2.8e-54) {
		tmp = x + ((t_0 / (1.0 - (tan(y) * tan(z)))) - a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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(y) / (((-1.0d0) - ((-0.3333333333333333d0) * (z * z))) / z)))) - tan(a))
    if (a <= (-0.0004d0)) then
        tmp = t_1
    else if (a <= 2.8d-54) then
        tmp = x + ((t_0 / (1.0d0 - (tan(y) * tan(z)))) - a)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < a;
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(y) / ((-1.0 - (-0.3333333333333333 * (z * z))) / z)))) - Math.tan(a));
	double tmp;
	if (a <= -0.0004) {
		tmp = t_1;
	} else if (a <= 2.8e-54) {
		tmp = x + ((t_0 / (1.0 - (Math.tan(y) * Math.tan(z)))) - a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	t_0 = math.tan(y) + math.tan(z)
	t_1 = x + ((t_0 / (1.0 + (math.tan(y) / ((-1.0 - (-0.3333333333333333 * (z * z))) / z)))) - math.tan(a))
	tmp = 0
	if a <= -0.0004:
		tmp = t_1
	elif a <= 2.8e-54:
		tmp = x + ((t_0 / (1.0 - (math.tan(y) * math.tan(z)))) - a)
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
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(y) / Float64(Float64(-1.0 - Float64(-0.3333333333333333 * Float64(z * z))) / z)))) - tan(a)))
	tmp = 0.0
	if (a <= -0.0004)
		tmp = t_1;
	elseif (a <= 2.8e-54)
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))) - a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp_2 = code(x, y, z, a)
	t_0 = tan(y) + tan(z);
	t_1 = x + ((t_0 / (1.0 + (tan(y) / ((-1.0 - (-0.3333333333333333 * (z * z))) / z)))) - tan(a));
	tmp = 0.0;
	if (a <= -0.0004)
		tmp = t_1;
	elseif (a <= 2.8e-54)
		tmp = x + ((t_0 / (1.0 - (tan(y) * tan(z)))) - a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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[y], $MachinePrecision] / N[(N[(-1.0 - N[(-0.3333333333333333 * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.0004], t$95$1, If[LessEqual[a, 2.8e-54], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.00000000000000019e-4 or 2.8000000000000002e-54 < a

   1. Initial program 77.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. tan-quot N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. clear-num N/A

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

      7. tan-quot N/A

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

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

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

      9. tan-lowering-tan.f64 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in z around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 78.1%

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

   9. Simplified 78.1%

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

   if -4.00000000000000019e-4 < a < 2.8000000000000002e-54

   1. Initial program 77.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. Taylor expanded in a around 0

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

   7. Simplified 99.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0004:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 + \frac{\tan y}{\frac{-1 - -0.3333333333333333 \cdot \left(z \cdot z\right)}{z}}} - \tan a\right)\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-54}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 + \frac{\tan y}{\frac{-1 - -0.3333333333333333 \cdot \left(z \cdot z\right)}{z}}} - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.8% accurate, 0.7× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+ x (- (tan (+ y z)) (/ (sin a) (cos a)))))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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)) - (sin(a) / cos(a)))
end function

Java

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

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	return x + (math.tan((y + z)) - (math.sin(a) / math.cos(a)))

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - Float64(sin(a) / cos(a))))
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - (sin(a) / cos(a)));
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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-quot N/A

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

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

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

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

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

   4. cos-lowering-cos.f64 77.3%

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

4. Applied egg-rr 77.3%

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

Alternative 12: 80.1% accurate, 0.7× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a) :precision binary64 (+ x (- (+ (tan y) (tan z)) (tan a))))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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

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

Python

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

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(x + Float64(Float64(tan(y) + tan(z)) - tan(a)))
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = x + ((tan(y) + tan(z)) - tan(a));
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 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. Taylor expanded in y around 0

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

7. Simplified 77.7%

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

8. Final simplification 77.7%

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

Alternative 13: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := x + \left(\tan y - \tan a\right)\\ \mathbf{if}\;a \leq -0.086:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{-41}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + a \cdot \left(-1 - \left(a \cdot a\right) \cdot \left(0.3333333333333333 + \left(a \cdot a\right) \cdot \left(0.13333333333333333 + \left(a \cdot a\right) \cdot 0.05396825396825397\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (- (tan y) (tan a)))))
   (if (<= a -0.086)
     t_0
     (if (<= a 1.15e-41)
       (+
        x
        (+
         (tan (+ y z))
         (*
          a
          (-
           -1.0
           (*
            (* a a)
            (+
             0.3333333333333333
             (*
              (* a a)
              (+ 0.13333333333333333 (* (* a a) 0.05396825396825397)))))))))
       t_0))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = x + (tan(y) - tan(a));
	double tmp;
	if (a <= -0.086) {
		tmp = t_0;
	} else if (a <= 1.15e-41) {
		tmp = x + (tan((y + z)) + (a * (-1.0 - ((a * a) * (0.3333333333333333 + ((a * a) * (0.13333333333333333 + ((a * a) * 0.05396825396825397))))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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(y) - tan(a))
    if (a <= (-0.086d0)) then
        tmp = t_0
    else if (a <= 1.15d-41) then
        tmp = x + (tan((y + z)) + (a * ((-1.0d0) - ((a * a) * (0.3333333333333333d0 + ((a * a) * (0.13333333333333333d0 + ((a * a) * 0.05396825396825397d0))))))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	double t_0 = x + (Math.tan(y) - Math.tan(a));
	double tmp;
	if (a <= -0.086) {
		tmp = t_0;
	} else if (a <= 1.15e-41) {
		tmp = x + (Math.tan((y + z)) + (a * (-1.0 - ((a * a) * (0.3333333333333333 + ((a * a) * (0.13333333333333333 + ((a * a) * 0.05396825396825397))))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	t_0 = x + (math.tan(y) - math.tan(a))
	tmp = 0
	if a <= -0.086:
		tmp = t_0
	elif a <= 1.15e-41:
		tmp = x + (math.tan((y + z)) + (a * (-1.0 - ((a * a) * (0.3333333333333333 + ((a * a) * (0.13333333333333333 + ((a * a) * 0.05396825396825397))))))))
	else:
		tmp = t_0
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(x + Float64(tan(y) - tan(a)))
	tmp = 0.0
	if (a <= -0.086)
		tmp = t_0;
	elseif (a <= 1.15e-41)
		tmp = Float64(x + Float64(tan(Float64(y + z)) + Float64(a * Float64(-1.0 - Float64(Float64(a * a) * Float64(0.3333333333333333 + Float64(Float64(a * a) * Float64(0.13333333333333333 + Float64(Float64(a * a) * 0.05396825396825397)))))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp_2 = code(x, y, z, a)
	t_0 = x + (tan(y) - tan(a));
	tmp = 0.0;
	if (a <= -0.086)
		tmp = t_0;
	elseif (a <= 1.15e-41)
		tmp = x + (tan((y + z)) + (a * (-1.0 - ((a * a) * (0.3333333333333333 + ((a * a) * (0.13333333333333333 + ((a * a) * 0.05396825396825397))))))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.086], t$95$0, If[LessEqual[a, 1.15e-41], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] + N[(a * N[(-1.0 - N[(N[(a * a), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(a * a), $MachinePrecision] * N[(0.13333333333333333 + N[(N[(a * a), $MachinePrecision] * 0.05396825396825397), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.085999999999999993 or 1.15000000000000005e-41 < a

   1. Initial program 76.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 57.2%

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

   if -0.085999999999999993 < a < 1.15000000000000005e-41

   1. Initial program 78.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \color{blue}{\left(a \cdot \left(1 + {a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)\right)}\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \color{blue}{\left(1 + {a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{1}{3}} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{1}{3}} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \color{blue}{\left({a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{2}{15}} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{2}{15}} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{2}{15}, \color{blue}{\left(\frac{17}{315} \cdot {a}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{2}{15}, \left({a}^{2} \cdot \color{blue}{\frac{17}{315}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{2}{15}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\frac{17}{315}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{2}{15}, \mathsf{*.f64}\left(\left(a \cdot a\right), \frac{17}{315}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 78.0%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{2}{15}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \frac{17}{315}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 78.0%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \left(a \cdot a\right) \cdot \left(0.3333333333333333 + \left(a \cdot a\right) \cdot \left(0.13333333333333333 + \left(a \cdot a\right) \cdot 0.05396825396825397\right)\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.086:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{-41}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + a \cdot \left(-1 - \left(a \cdot a\right) \cdot \left(0.3333333333333333 + \left(a \cdot a\right) \cdot \left(0.13333333333333333 + \left(a \cdot a\right) \cdot 0.05396825396825397\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;y + z \leq -1 \cdot 10^{-10}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -1e-10) (+ x (- (tan y) (tan a))) (+ x (- (tan z) (tan a)))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -1e-10) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (tan(z) - tan(a));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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) <= (-1d-10)) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (tan(z) - tan(a))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -1e-10) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -1e-10:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -1e-10)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -1e-10)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -1e-10], 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) < -1.00000000000000004e-10

   1. Initial program 68.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 46.8%

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

   if -1.00000000000000004e-10 < (+.f64 y z)

   1. Initial program 83.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 67.2%

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

Alternative 15: 79.8% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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

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

Python

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

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 77.3%

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

Alternative 16: 60.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -0.47:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + a \cdot \left(-1 - \left(a \cdot a\right) \cdot \left(0.3333333333333333 + \left(a \cdot a\right) \cdot \left(0.13333333333333333 + \left(a \cdot a\right) \cdot 0.05396825396825397\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -0.47)
     t_0
     (if (<= a 6.5e-6)
       (+
        x
        (+
         (tan (+ y z))
         (*
          a
          (-
           -1.0
           (*
            (* a a)
            (+
             0.3333333333333333
             (*
              (* a a)
              (+ 0.13333333333333333 (* (* a a) 0.05396825396825397)))))))))
       t_0))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -0.47) {
		tmp = t_0;
	} else if (a <= 6.5e-6) {
		tmp = x + (tan((y + z)) + (a * (-1.0 - ((a * a) * (0.3333333333333333 + ((a * a) * (0.13333333333333333 + ((a * a) * 0.05396825396825397))))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 <= (-0.47d0)) then
        tmp = t_0
    else if (a <= 6.5d-6) then
        tmp = x + (tan((y + z)) + (a * ((-1.0d0) - ((a * a) * (0.3333333333333333d0 + ((a * a) * (0.13333333333333333d0 + ((a * a) * 0.05396825396825397d0))))))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	double t_0 = x - Math.tan(a);
	double tmp;
	if (a <= -0.47) {
		tmp = t_0;
	} else if (a <= 6.5e-6) {
		tmp = x + (Math.tan((y + z)) + (a * (-1.0 - ((a * a) * (0.3333333333333333 + ((a * a) * (0.13333333333333333 + ((a * a) * 0.05396825396825397))))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	t_0 = x - math.tan(a)
	tmp = 0
	if a <= -0.47:
		tmp = t_0
	elif a <= 6.5e-6:
		tmp = x + (math.tan((y + z)) + (a * (-1.0 - ((a * a) * (0.3333333333333333 + ((a * a) * (0.13333333333333333 + ((a * a) * 0.05396825396825397))))))))
	else:
		tmp = t_0
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -0.47)
		tmp = t_0;
	elseif (a <= 6.5e-6)
		tmp = Float64(x + Float64(tan(Float64(y + z)) + Float64(a * Float64(-1.0 - Float64(Float64(a * a) * Float64(0.3333333333333333 + Float64(Float64(a * a) * Float64(0.13333333333333333 + Float64(Float64(a * a) * 0.05396825396825397)))))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp_2 = code(x, y, z, a)
	t_0 = x - tan(a);
	tmp = 0.0;
	if (a <= -0.47)
		tmp = t_0;
	elseif (a <= 6.5e-6)
		tmp = x + (tan((y + z)) + (a * (-1.0 - ((a * a) * (0.3333333333333333 + ((a * a) * (0.13333333333333333 + ((a * a) * 0.05396825396825397))))))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.47], t$95$0, If[LessEqual[a, 6.5e-6], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] + N[(a * N[(-1.0 - N[(N[(a * a), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(a * a), $MachinePrecision] * N[(0.13333333333333333 + N[(N[(a * a), $MachinePrecision] * 0.05396825396825397), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.46999999999999997 or 6.4999999999999996e-6 < a

   1. Initial program 76.1%

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

      1. associate-+r- N/A

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

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

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

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

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

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

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

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

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

      6. tan-lowering-tan.f64 76.0%

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

   4. Applied egg-rr 76.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 39.4%

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

   if -0.46999999999999997 < a < 6.4999999999999996e-6

   1. Initial program 78.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \color{blue}{\left(a \cdot \left(1 + {a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)\right)}\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \color{blue}{\left(1 + {a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{1}{3}} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{1}{3}} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \color{blue}{\left({a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{2}{15}} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{2}{15}} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{2}{15}, \color{blue}{\left(\frac{17}{315} \cdot {a}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{2}{15}, \left({a}^{2} \cdot \color{blue}{\frac{17}{315}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{2}{15}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\frac{17}{315}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{2}{15}, \mathsf{*.f64}\left(\left(a \cdot a\right), \frac{17}{315}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 78.4%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{2}{15}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \frac{17}{315}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 78.4%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \left(a \cdot a\right) \cdot \left(0.3333333333333333 + \left(a \cdot a\right) \cdot \left(0.13333333333333333 + \left(a \cdot a\right) \cdot 0.05396825396825397\right)\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.47:\\ \;\;\;\;x - \tan a\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + a \cdot \left(-1 - \left(a \cdot a\right) \cdot \left(0.3333333333333333 + \left(a \cdot a\right) \cdot \left(0.13333333333333333 + \left(a \cdot a\right) \cdot 0.05396825396825397\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \tan a\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 60.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -0.58:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + a \cdot \left(-1 - \left(a \cdot a\right) \cdot \left(0.3333333333333333 + \left(a \cdot a\right) \cdot 0.13333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -0.58)
     t_0
     (if (<= a 6.5e-6)
       (+
        x
        (+
         (tan (+ y z))
         (*
          a
          (-
           -1.0
           (*
            (* a a)
            (+ 0.3333333333333333 (* (* a a) 0.13333333333333333)))))))
       t_0))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -0.58) {
		tmp = t_0;
	} else if (a <= 6.5e-6) {
		tmp = x + (tan((y + z)) + (a * (-1.0 - ((a * a) * (0.3333333333333333 + ((a * a) * 0.13333333333333333))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 <= (-0.58d0)) then
        tmp = t_0
    else if (a <= 6.5d-6) then
        tmp = x + (tan((y + z)) + (a * ((-1.0d0) - ((a * a) * (0.3333333333333333d0 + ((a * a) * 0.13333333333333333d0))))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	double t_0 = x - Math.tan(a);
	double tmp;
	if (a <= -0.58) {
		tmp = t_0;
	} else if (a <= 6.5e-6) {
		tmp = x + (Math.tan((y + z)) + (a * (-1.0 - ((a * a) * (0.3333333333333333 + ((a * a) * 0.13333333333333333))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	t_0 = x - math.tan(a)
	tmp = 0
	if a <= -0.58:
		tmp = t_0
	elif a <= 6.5e-6:
		tmp = x + (math.tan((y + z)) + (a * (-1.0 - ((a * a) * (0.3333333333333333 + ((a * a) * 0.13333333333333333))))))
	else:
		tmp = t_0
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -0.58)
		tmp = t_0;
	elseif (a <= 6.5e-6)
		tmp = Float64(x + Float64(tan(Float64(y + z)) + Float64(a * Float64(-1.0 - Float64(Float64(a * a) * Float64(0.3333333333333333 + Float64(Float64(a * a) * 0.13333333333333333)))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp_2 = code(x, y, z, a)
	t_0 = x - tan(a);
	tmp = 0.0;
	if (a <= -0.58)
		tmp = t_0;
	elseif (a <= 6.5e-6)
		tmp = x + (tan((y + z)) + (a * (-1.0 - ((a * a) * (0.3333333333333333 + ((a * a) * 0.13333333333333333))))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.58], t$95$0, If[LessEqual[a, 6.5e-6], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] + N[(a * N[(-1.0 - N[(N[(a * a), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(a * a), $MachinePrecision] * 0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.57999999999999996 or 6.4999999999999996e-6 < a

   1. Initial program 76.1%

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

      1. associate-+r- N/A

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

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

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

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

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

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

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

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

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

      6. tan-lowering-tan.f64 76.0%

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

   4. Applied egg-rr 76.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 39.4%

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

   if -0.57999999999999996 < a < 6.4999999999999996e-6

   1. Initial program 78.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \color{blue}{\left(a \cdot \left(1 + {a}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right)\right)\right)}\right)\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \color{blue}{\left(1 + {a}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \color{blue}{\left({a}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right)}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(a \cdot a\right), \left(\color{blue}{\frac{1}{3}} + \frac{2}{15} \cdot {a}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{\frac{1}{3}} + \frac{2}{15} \cdot {a}^{2}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \color{blue}{\left(\frac{2}{15} \cdot {a}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({a}^{2} \cdot \color{blue}{\frac{2}{15}}\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({a}^{2}\right), \color{blue}{\frac{2}{15}}\right)\right)\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(a \cdot a\right), \frac{2}{15}\right)\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 78.3%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{tan.f64}\left(\mathsf{+.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, a\right), \frac{2}{15}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 78.3%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + \left(a \cdot a\right) \cdot \left(0.3333333333333333 + \left(a \cdot a\right) \cdot 0.13333333333333333\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.58:\\ \;\;\;\;x - \tan a\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + a \cdot \left(-1 - \left(a \cdot a\right) \cdot \left(0.3333333333333333 + \left(a \cdot a\right) \cdot 0.13333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \tan a\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 60.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -0.33:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + a \cdot \left(-1 - a \cdot \left(a \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -0.33)
     t_0
     (if (<= a 6.5e-6)
       (+ x (+ (tan (+ y z)) (* a (- -1.0 (* a (* a 0.3333333333333333))))))
       t_0))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -0.33) {
		tmp = t_0;
	} else if (a <= 6.5e-6) {
		tmp = x + (tan((y + z)) + (a * (-1.0 - (a * (a * 0.3333333333333333)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 <= (-0.33d0)) then
        tmp = t_0
    else if (a <= 6.5d-6) then
        tmp = x + (tan((y + z)) + (a * ((-1.0d0) - (a * (a * 0.3333333333333333d0)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	double t_0 = x - Math.tan(a);
	double tmp;
	if (a <= -0.33) {
		tmp = t_0;
	} else if (a <= 6.5e-6) {
		tmp = x + (Math.tan((y + z)) + (a * (-1.0 - (a * (a * 0.3333333333333333)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	t_0 = x - math.tan(a)
	tmp = 0
	if a <= -0.33:
		tmp = t_0
	elif a <= 6.5e-6:
		tmp = x + (math.tan((y + z)) + (a * (-1.0 - (a * (a * 0.3333333333333333)))))
	else:
		tmp = t_0
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -0.33)
		tmp = t_0;
	elseif (a <= 6.5e-6)
		tmp = Float64(x + Float64(tan(Float64(y + z)) + Float64(a * Float64(-1.0 - Float64(a * Float64(a * 0.3333333333333333))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp_2 = code(x, y, z, a)
	t_0 = x - tan(a);
	tmp = 0.0;
	if (a <= -0.33)
		tmp = t_0;
	elseif (a <= 6.5e-6)
		tmp = x + (tan((y + z)) + (a * (-1.0 - (a * (a * 0.3333333333333333)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.33], t$95$0, If[LessEqual[a, 6.5e-6], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] + N[(a * N[(-1.0 - N[(a * N[(a * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.330000000000000016 or 6.4999999999999996e-6 < a

   1. Initial program 76.1%

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

      1. associate-+r- N/A

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

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

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

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

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

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

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

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

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

      6. tan-lowering-tan.f64 76.0%

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

   4. Applied egg-rr 76.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 39.4%

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

   if -0.330000000000000016 < a < 6.4999999999999996e-6

   1. Initial program 78.5%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 78.0%

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

   5. Simplified 78.0%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a \cdot \left(1 + a \cdot \left(a \cdot 0.3333333333333333\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.33:\\ \;\;\;\;x - \tan a\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + a \cdot \left(-1 - a \cdot \left(a \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \tan a\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 60.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -0.095:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;\left(x + \tan \left(y + z\right)\right) - a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -0.095) t_0 (if (<= a 6.5e-6) (- (+ x (tan (+ y z))) a) t_0))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -0.095) {
		tmp = t_0;
	} else if (a <= 6.5e-6) {
		tmp = (x + tan((y + z))) - a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 <= (-0.095d0)) then
        tmp = t_0
    else if (a <= 6.5d-6) then
        tmp = (x + tan((y + z))) - a
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	double t_0 = x - Math.tan(a);
	double tmp;
	if (a <= -0.095) {
		tmp = t_0;
	} else if (a <= 6.5e-6) {
		tmp = (x + Math.tan((y + z))) - a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	t_0 = x - math.tan(a)
	tmp = 0
	if a <= -0.095:
		tmp = t_0
	elif a <= 6.5e-6:
		tmp = (x + math.tan((y + z))) - a
	else:
		tmp = t_0
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -0.095)
		tmp = t_0;
	elseif (a <= 6.5e-6)
		tmp = Float64(Float64(x + tan(Float64(y + z))) - a);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp_2 = code(x, y, z, a)
	t_0 = x - tan(a);
	tmp = 0.0;
	if (a <= -0.095)
		tmp = t_0;
	elseif (a <= 6.5e-6)
		tmp = (x + tan((y + z))) - a;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.095], t$95$0, If[LessEqual[a, 6.5e-6], N[(N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.095000000000000001 or 6.4999999999999996e-6 < a

   1. Initial program 76.1%

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

      1. associate-+r- N/A

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

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

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

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

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

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

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

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

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

      6. tan-lowering-tan.f64 76.0%

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

   4. Applied egg-rr 76.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 39.4%

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

   if -0.095000000000000001 < a < 6.4999999999999996e-6

   1. Initial program 78.5%

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

      1. associate-+r- N/A

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

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

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

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

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

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

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

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

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

      6. tan-lowering-tan.f64 78.5%

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

   4. Applied egg-rr 78.5%

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

   5. Taylor expanded in a around 0

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

   7. Simplified 77.7%

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

Alternative 20: 60.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -0.068:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -0.068) t_0 (if (<= a 6.5e-6) (+ x (- (tan (+ y z)) a)) t_0))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -0.068) {
		tmp = t_0;
	} else if (a <= 6.5e-6) {
		tmp = x + (tan((y + z)) - a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 <= (-0.068d0)) then
        tmp = t_0
    else if (a <= 6.5d-6) then
        tmp = x + (tan((y + z)) - a)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	double t_0 = x - Math.tan(a);
	double tmp;
	if (a <= -0.068) {
		tmp = t_0;
	} else if (a <= 6.5e-6) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	t_0 = x - math.tan(a)
	tmp = 0
	if a <= -0.068:
		tmp = t_0
	elif a <= 6.5e-6:
		tmp = x + (math.tan((y + z)) - a)
	else:
		tmp = t_0
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -0.068)
		tmp = t_0;
	elseif (a <= 6.5e-6)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp_2 = code(x, y, z, a)
	t_0 = x - tan(a);
	tmp = 0.0;
	if (a <= -0.068)
		tmp = t_0;
	elseif (a <= 6.5e-6)
		tmp = x + (tan((y + z)) - a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.068], t$95$0, If[LessEqual[a, 6.5e-6], 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 < -0.068000000000000005 or 6.4999999999999996e-6 < a

   1. Initial program 76.1%

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

      1. associate-+r- N/A

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

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

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

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

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

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

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

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

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

      6. tan-lowering-tan.f64 76.0%

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

   4. Applied egg-rr 76.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 39.4%

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

   if -0.068000000000000005 < a < 6.4999999999999996e-6

   1. Initial program 78.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 77.7%

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

Alternative 21: 41.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ x - \tan a \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a) :precision binary64 (- x (tan a)))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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

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

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	return x - math.tan(a)

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(x - tan(a))
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = x - tan(a);
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.3%

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

   1. associate-+r- N/A

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

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

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

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

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

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

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

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

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

   6. tan-lowering-tan.f64 77.3%

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

4. Applied egg-rr 77.3%

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

5. Taylor expanded in x around inf

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

7. Simplified 39.5%

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

Alternative 22: 32.0% accurate, 207.0× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ x \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a) :precision binary64 x)

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	return x;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	return x;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	return x

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return x
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = x;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := x

Derivation

1. Initial program 77.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 30.5%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024159 
(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))))