tan-example (used to crash)

Percentage Accurate: 79.5% → 99.7%

Time: 28.4s

Alternatives: 13

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: 79.5% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y \cdot \tan z\\ x + \left(\frac{\tan y + \tan z}{1 - {t\_0}^{2}} \cdot \left(1 + t\_0\right) - \tan a\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (* (tan y) (tan z))))
   (+
    x
    (-
     (* (/ (+ (tan y) (tan z)) (- 1.0 (pow t_0 2.0))) (+ 1.0 t_0))
     (tan a)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) * tan(z);
	return x + ((((tan(y) + tan(z)) / (1.0 - pow(t_0, 2.0))) * (1.0 + t_0)) - 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
    real(8) :: t_0
    t_0 = tan(y) * tan(z)
    code = x + ((((tan(y) + tan(z)) / (1.0d0 - (t_0 ** 2.0d0))) * (1.0d0 + t_0)) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan(y) * Math.tan(z);
	return x + ((((Math.tan(y) + Math.tan(z)) / (1.0 - Math.pow(t_0, 2.0))) * (1.0 + t_0)) - Math.tan(a));
}

Python

def code(x, y, z, a):
	t_0 = math.tan(y) * math.tan(z)
	return x + ((((math.tan(y) + math.tan(z)) / (1.0 - math.pow(t_0, 2.0))) * (1.0 + t_0)) - math.tan(a))

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) * tan(z))
	return Float64(x + Float64(Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - (t_0 ^ 2.0))) * Float64(1.0 + t_0)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	t_0 = tan(y) * tan(z);
	tmp = x + ((((tan(y) + tan(z)) / (1.0 - (t_0 ^ 2.0))) * (1.0 + t_0)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]}, N[(x + N[(N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 81.7%

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

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

   3. associate-/r/ N/A

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

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

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

4. Applied egg-rr 99.7%

   \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - {\left(\tan y \cdot \tan z\right)}^{2}} \cdot \left(1 + \tan y \cdot \tan z\right)} - \tan a\right) \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. tan-sum N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. tan-lowering-tan.f64 99.7%

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

4. Applied egg-rr 99.7%

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

   1. *-commutative N/A

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

   2. tan-quot N/A

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

   3. tan-quot N/A

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

   4. frac-times N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. cos-lowering-cos.f64 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

   4. *-commutative N/A

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

   5. associate-/r* N/A

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

   6. tan-quot N/A

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

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

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

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

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

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

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

   10. sin-lowering-sin.f64 99.7%

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

8. Applied egg-rr 99.7%

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

Alternative 3: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 4: 88.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;a \leq -0.0002:\\ \;\;\;\;x + \left(t\_0 - \frac{\sin a}{\cos a}\right)\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{-32}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t\_0 - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (<= a -0.0002)
     (+ x (- t_0 (/ (sin a) (cos a))))
     (if (<= a 1.65e-32)
       (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) a))
       (+ x (- t_0 (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if (a <= -0.0002) {
		tmp = x + (t_0 - (sin(a) / cos(a)));
	} else if (a <= 1.65e-32) {
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - a);
	} else {
		tmp = x + (t_0 - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan((y + z))
    if (a <= (-0.0002d0)) then
        tmp = x + (t_0 - (sin(a) / cos(a)))
    else if (a <= 1.65d-32) then
        tmp = x + (((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z)))) - a)
    else
        tmp = x + (t_0 - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan((y + z));
	double tmp;
	if (a <= -0.0002) {
		tmp = x + (t_0 - (Math.sin(a) / Math.cos(a)));
	} else if (a <= 1.65e-32) {
		tmp = x + (((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z)))) - a);
	} else {
		tmp = x + (t_0 - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z))
	tmp = 0
	if a <= -0.0002:
		tmp = x + (t_0 - (math.sin(a) / math.cos(a)))
	elif a <= 1.65e-32:
		tmp = x + (((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z)))) - a)
	else:
		tmp = x + (t_0 - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if (a <= -0.0002)
		tmp = Float64(x + Float64(t_0 - Float64(sin(a) / cos(a))));
	elseif (a <= 1.65e-32)
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - a));
	else
		tmp = Float64(x + Float64(t_0 - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan((y + z));
	tmp = 0.0;
	if (a <= -0.0002)
		tmp = x + (t_0 - (sin(a) / cos(a)));
	elseif (a <= 1.65e-32)
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - a);
	else
		tmp = x + (t_0 - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, -0.0002], N[(x + N[(t$95$0 - N[(N[Sin[a], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.65e-32], 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] - a), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.0000000000000001e-4

   1. Initial program 87.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 87.8%

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

   4. Applied egg-rr 87.8%

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

   if -2.0000000000000001e-4 < a < 1.65000000000000013e-32

   1. Initial program 81.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 81.1%

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

      1. tan-sum N/A

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

      2. tan-quot N/A

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

      3. un-div-inv N/A

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

      4. +-commutative N/A

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

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

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

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

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

      7. un-div-inv N/A

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

      8. tan-quot N/A

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

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

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

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

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

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

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

      12. tan-quot N/A

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

      13. un-div-inv N/A

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

      14. *-commutative N/A

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

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

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

      16. un-div-inv N/A

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

      17. tan-quot N/A

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

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

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

      19. tan-lowering-tan.f64 99.7%

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

   7. Applied egg-rr 99.7%

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

   if 1.65000000000000013e-32 < a

   1. Initial program 76.7%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0002:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \frac{\sin a}{\cos a}\right)\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{-32}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(\tan y - \tan a\right)\\ \mathbf{if}\;\tan a \leq -0.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-26}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + a \cdot \left(-1 - a \cdot \left(a \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (- (tan y) (tan a)))))
   (if (<= (tan a) -0.05)
     t_0
     (if (<= (tan a) 2e-26)
       (+ x (+ (tan (+ y z)) (* a (- -1.0 (* a (* a 0.3333333333333333))))))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x + (tan(y) - tan(a));
	double tmp;
	if (tan(a) <= -0.05) {
		tmp = t_0;
	} else if (tan(a) <= 2e-26) {
		tmp = x + (tan((y + z)) + (a * (-1.0 - (a * (a * 0.3333333333333333)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (tan(y) - tan(a))
    if (tan(a) <= (-0.05d0)) then
        tmp = t_0
    else if (tan(a) <= 2d-26) then
        tmp = x + (tan((y + z)) + (a * ((-1.0d0) - (a * (a * 0.3333333333333333d0)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = x + (Math.tan(y) - Math.tan(a));
	double tmp;
	if (Math.tan(a) <= -0.05) {
		tmp = t_0;
	} else if (Math.tan(a) <= 2e-26) {
		tmp = x + (Math.tan((y + z)) + (a * (-1.0 - (a * (a * 0.3333333333333333)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = x + (math.tan(y) - math.tan(a))
	tmp = 0
	if math.tan(a) <= -0.05:
		tmp = t_0
	elif math.tan(a) <= 2e-26:
		tmp = x + (math.tan((y + z)) + (a * (-1.0 - (a * (a * 0.3333333333333333)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(x + Float64(tan(y) - tan(a)))
	tmp = 0.0
	if (tan(a) <= -0.05)
		tmp = t_0;
	elseif (tan(a) <= 2e-26)
		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

function tmp_2 = code(x, y, z, a)
	t_0 = x + (tan(y) - tan(a));
	tmp = 0.0;
	if (tan(a) <= -0.05)
		tmp = t_0;
	elseif (tan(a) <= 2e-26)
		tmp = x + (tan((y + z)) + (a * (-1.0 - (a * (a * 0.3333333333333333)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.05], t$95$0, If[LessEqual[N[Tan[a], $MachinePrecision], 2e-26], 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 (tan.f64 a) < -0.050000000000000003 or 2.0000000000000001e-26 < (tan.f64 a)

   1. Initial program 81.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 65.3%

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

   if -0.050000000000000003 < (tan.f64 a) < 2.0000000000000001e-26

   1. Initial program 81.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 81.5%

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

   5. Simplified 81.5%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.05:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-26}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + a \cdot \left(-1 - a \cdot \left(a \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4.5 \cdot 10^{-20}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 4.5e-20) (+ x (- (tan y) (tan a))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 4.5e-20) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (tan(z) - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= 4.5d-20) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (tan(z) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 4.5e-20) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 4.5e-20:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 4.5e-20)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 4.5e-20)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 4.5e-20], 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 z < 4.5000000000000001e-20

   1. Initial program 88.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 77.5%

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

   if 4.5000000000000001e-20 < z

   1. Initial program 59.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 58.0%

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

Alternative 7: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

Alternative 8: 60.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -0.125:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.16:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + a \cdot \left(-1 - a \cdot \left(a \cdot \left(0.3333333333333333 + \left(a \cdot a\right) \cdot 0.13333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -0.125)
     t_0
     (if (<= a 0.16)
       (+
        x
        (+
         (tan (+ y z))
         (*
          a
          (-
           -1.0
           (*
            a
            (* a (+ 0.3333333333333333 (* (* a a) 0.13333333333333333))))))))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -0.125) {
		tmp = t_0;
	} else if (a <= 0.16) {
		tmp = x + (tan((y + z)) + (a * (-1.0 - (a * (a * (0.3333333333333333 + ((a * a) * 0.13333333333333333)))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - tan(a)
    if (a <= (-0.125d0)) then
        tmp = t_0
    else if (a <= 0.16d0) 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

public static double code(double x, double y, double z, double a) {
	double t_0 = x - Math.tan(a);
	double tmp;
	if (a <= -0.125) {
		tmp = t_0;
	} else if (a <= 0.16) {
		tmp = x + (Math.tan((y + z)) + (a * (-1.0 - (a * (a * (0.3333333333333333 + ((a * a) * 0.13333333333333333)))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = x - math.tan(a)
	tmp = 0
	if a <= -0.125:
		tmp = t_0
	elif a <= 0.16:
		tmp = x + (math.tan((y + z)) + (a * (-1.0 - (a * (a * (0.3333333333333333 + ((a * a) * 0.13333333333333333)))))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -0.125)
		tmp = t_0;
	elseif (a <= 0.16)
		tmp = Float64(x + Float64(tan(Float64(y + z)) + Float64(a * Float64(-1.0 - Float64(a * Float64(a * Float64(0.3333333333333333 + Float64(Float64(a * a) * 0.13333333333333333))))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = x - tan(a);
	tmp = 0.0;
	if (a <= -0.125)
		tmp = t_0;
	elseif (a <= 0.16)
		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

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.125], t$95$0, If[LessEqual[a, 0.16], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] + N[(a * N[(-1.0 - N[(a * N[(a * N[(0.3333333333333333 + N[(N[(a * a), $MachinePrecision] * 0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.125 or 0.160000000000000003 < a

   1. Initial program 81.5%

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

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

      2. tan-quot N/A

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

      3. div-inv N/A

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

      4. fma-define N/A

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

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

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

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

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

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

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

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

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

      9. tan-lowering-tan.f64 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-+r- N/A

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

      2. +-commutative N/A

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

      3. un-div-inv N/A

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

      4. tan-quot N/A

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

      5. tan-sum N/A

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

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

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. tan-lowering-tan.f64 81.3%

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

   8. Applied egg-rr 81.3%

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

   9. Taylor expanded in x around inf

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

   11. Simplified 49.7%

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

   if -0.125 < a < 0.160000000000000003

   1. Initial program 82.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 82.0%

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

   5. Simplified 82.0%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.125:\\ \;\;\;\;x - \tan a\\ \mathbf{elif}\;a \leq 0.16:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + a \cdot \left(-1 - a \cdot \left(a \cdot \left(0.3333333333333333 + \left(a \cdot a\right) \cdot 0.13333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \tan a\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -0.06:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.125:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + a \cdot \left(-1 - a \cdot \left(a \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -0.06)
     t_0
     (if (<= a 0.125)
       (+ x (+ (tan (+ y z)) (* a (- -1.0 (* a (* a 0.3333333333333333))))))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -0.06) {
		tmp = t_0;
	} else if (a <= 0.125) {
		tmp = x + (tan((y + z)) + (a * (-1.0 - (a * (a * 0.3333333333333333)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - tan(a)
    if (a <= (-0.06d0)) then
        tmp = t_0
    else if (a <= 0.125d0) then
        tmp = x + (tan((y + z)) + (a * ((-1.0d0) - (a * (a * 0.3333333333333333d0)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = x - Math.tan(a);
	double tmp;
	if (a <= -0.06) {
		tmp = t_0;
	} else if (a <= 0.125) {
		tmp = x + (Math.tan((y + z)) + (a * (-1.0 - (a * (a * 0.3333333333333333)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = x - math.tan(a)
	tmp = 0
	if a <= -0.06:
		tmp = t_0
	elif a <= 0.125:
		tmp = x + (math.tan((y + z)) + (a * (-1.0 - (a * (a * 0.3333333333333333)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -0.06)
		tmp = t_0;
	elseif (a <= 0.125)
		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

function tmp_2 = code(x, y, z, a)
	t_0 = x - tan(a);
	tmp = 0.0;
	if (a <= -0.06)
		tmp = t_0;
	elseif (a <= 0.125)
		tmp = x + (tan((y + z)) + (a * (-1.0 - (a * (a * 0.3333333333333333)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.06], t$95$0, If[LessEqual[a, 0.125], 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 < -0.059999999999999998 or 0.125 < a

   1. Initial program 81.5%

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

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

      2. tan-quot N/A

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

      3. div-inv N/A

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

      4. fma-define N/A

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

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

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

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

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

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

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

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

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

      9. tan-lowering-tan.f64 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-+r- N/A

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

      2. +-commutative N/A

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

      3. un-div-inv N/A

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

      4. tan-quot N/A

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

      5. tan-sum N/A

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

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

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. tan-lowering-tan.f64 81.3%

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

   8. Applied egg-rr 81.3%

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

   9. Taylor expanded in x around inf

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

   11. Simplified 49.7%

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

   if -0.059999999999999998 < a < 0.125

   1. Initial program 82.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 82.0%

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

   5. Simplified 82.0%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.06:\\ \;\;\;\;x - \tan a\\ \mathbf{elif}\;a \leq 0.125:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + a \cdot \left(-1 - a \cdot \left(a \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \tan a\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -0.0142:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.009:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -0.0142) t_0 (if (<= a 0.009) (+ x (- (tan (+ y z)) a)) t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -0.0142) {
		tmp = t_0;
	} else if (a <= 0.009) {
		tmp = x + (tan((y + z)) - a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - tan(a)
    if (a <= (-0.0142d0)) then
        tmp = t_0
    else if (a <= 0.009d0) then
        tmp = x + (tan((y + z)) - a)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = x - Math.tan(a);
	double tmp;
	if (a <= -0.0142) {
		tmp = t_0;
	} else if (a <= 0.009) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = x - math.tan(a)
	tmp = 0
	if a <= -0.0142:
		tmp = t_0
	elif a <= 0.009:
		tmp = x + (math.tan((y + z)) - a)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -0.0142)
		tmp = t_0;
	elseif (a <= 0.009)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = x - tan(a);
	tmp = 0.0;
	if (a <= -0.0142)
		tmp = t_0;
	elseif (a <= 0.009)
		tmp = x + (tan((y + z)) - a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.0142], t$95$0, If[LessEqual[a, 0.009], 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 < -0.014200000000000001 or 0.00899999999999999932 < a

   1. Initial program 81.5%

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

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

      2. tan-quot N/A

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

      3. div-inv N/A

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

      4. fma-define N/A

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

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

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

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

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

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

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

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

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

      9. tan-lowering-tan.f64 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-+r- N/A

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

      2. +-commutative N/A

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

      3. un-div-inv N/A

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

      4. tan-quot N/A

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

      5. tan-sum N/A

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

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

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. tan-lowering-tan.f64 81.3%

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

   8. Applied egg-rr 81.3%

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

   9. Taylor expanded in x around inf

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

   11. Simplified 49.7%

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

   if -0.014200000000000001 < a < 0.00899999999999999932

   1. Initial program 82.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 81.4%

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

Alternative 11: 50.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -0.0046:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.0125:\\ \;\;\;\;x + \left(\tan y - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -0.0046) t_0 (if (<= a 0.0125) (+ x (- (tan y) a)) t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -0.0046) {
		tmp = t_0;
	} else if (a <= 0.0125) {
		tmp = x + (tan(y) - a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - tan(a)
    if (a <= (-0.0046d0)) then
        tmp = t_0
    else if (a <= 0.0125d0) then
        tmp = x + (tan(y) - a)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = x - Math.tan(a);
	double tmp;
	if (a <= -0.0046) {
		tmp = t_0;
	} else if (a <= 0.0125) {
		tmp = x + (Math.tan(y) - a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = x - math.tan(a)
	tmp = 0
	if a <= -0.0046:
		tmp = t_0
	elif a <= 0.0125:
		tmp = x + (math.tan(y) - a)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -0.0046)
		tmp = t_0;
	elseif (a <= 0.0125)
		tmp = Float64(x + Float64(tan(y) - a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = x - tan(a);
	tmp = 0.0;
	if (a <= -0.0046)
		tmp = t_0;
	elseif (a <= 0.0125)
		tmp = x + (tan(y) - a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.0046], t$95$0, If[LessEqual[a, 0.0125], N[(x + N[(N[Tan[y], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0045999999999999999 or 0.012500000000000001 < a

   1. Initial program 81.5%

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

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

      2. tan-quot N/A

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

      3. div-inv N/A

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

      4. fma-define N/A

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

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

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

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

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

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

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

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

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

      9. tan-lowering-tan.f64 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-+r- N/A

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

      2. +-commutative N/A

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

      3. un-div-inv N/A

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

      4. tan-quot N/A

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

      5. tan-sum N/A

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

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

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. tan-lowering-tan.f64 81.3%

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

   8. Applied egg-rr 81.3%

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

   9. Taylor expanded in x around inf

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

   11. Simplified 49.7%

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

   if -0.0045999999999999999 < a < 0.012500000000000001

   1. Initial program 82.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 81.4%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 65.1%

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

Alternative 12: 41.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.7%

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

   1. tan-sum N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. tan-lowering-tan.f64 99.7%

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

4. Applied egg-rr 99.7%

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

   1. +-commutative N/A

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

   2. tan-quot N/A

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

   3. div-inv N/A

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

   4. fma-define N/A

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

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

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

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

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

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

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

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

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

   9. tan-lowering-tan.f64 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-+r- N/A

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

   2. +-commutative N/A

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

   3. un-div-inv N/A

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

   4. tan-quot N/A

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

   5. tan-sum N/A

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

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

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

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

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

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

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

   9. +-commutative N/A

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

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

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

   11. tan-lowering-tan.f64 81.6%

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

8. Applied egg-rr 81.6%

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

9. Taylor expanded in x around inf

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

11. Simplified 47.3%

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

Alternative 13: 31.2% accurate, 207.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y, z, a):
	return x

Julia

function code(x, y, z, a)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, a_] := x

Derivation

1. Initial program 81.7%

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

3. Taylor expanded in x around inf

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

5. Simplified 32.6%

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

Reproduce

herbie shell --seed 2024139 
(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))))