tan-example (used to crash)

Percentage Accurate: 79.6% → 99.6%

Time: 25.7s

Alternatives: 20

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: 79.6% 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: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ x + \left(\frac{\tan y + \tan z}{1 - \frac{{\sin y}^{2}}{{\cos z}^{2}} \cdot \frac{{\sin z}^{2}}{{\cos y}^{2}}} \cdot \left(1 + \tan y \cdot \tan z\right) - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+
  x
  (-
   (*
    (/
     (+ (tan y) (tan z))
     (-
      1.0
      (*
       (/ (pow (sin y) 2.0) (pow (cos z) 2.0))
       (/ (pow (sin z) 2.0) (pow (cos y) 2.0)))))
    (+ 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 - ((pow(sin(y), 2.0) / pow(cos(z), 2.0)) * (pow(sin(z), 2.0) / pow(cos(y), 2.0))))) * (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 - (((sin(y) ** 2.0d0) / (cos(z) ** 2.0d0)) * ((sin(z) ** 2.0d0) / (cos(y) ** 2.0d0))))) * (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.pow(Math.sin(y), 2.0) / Math.pow(Math.cos(z), 2.0)) * (Math.pow(Math.sin(z), 2.0) / Math.pow(Math.cos(y), 2.0))))) * (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.pow(math.sin(y), 2.0) / math.pow(math.cos(z), 2.0)) * (math.pow(math.sin(z), 2.0) / math.pow(math.cos(y), 2.0))))) * (1.0 + (math.tan(y) * math.tan(z)))) - math.tan(a))

Julia

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

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[(N[Power[N[Sin[y], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[z], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[z], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Cos[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 80.4%

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

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

   1. *-commutative N/A

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

   2. times-frac N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(y\right), 2\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(z\right), 2\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(z\right), 2\right), \mathsf{pow.f64}\left(\cos y, 2\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

   13. cos-lowering-cos.f64 99.7%

       \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(y\right), 2\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(z\right), 2\right)\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{sin.f64}\left(z\right), 2\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(y\right), 2\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{tan.f64}\left(y\right), \mathsf{tan.f64}\left(z\right)\right)\right)\right), \mathsf{tan.f64}\left(a\right)\right)\right) \]

7. Simplified 99.7%

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

Alternative 2: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) * tan(z);
	return x + (((1.0 + t_0) * ((tan(y) + tan(z)) / (1.0 - pow(t_0, 2.0)))) - 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 + (((1.0d0 + t_0) * ((tan(y) + tan(z)) / (1.0d0 - (t_0 ** 2.0d0)))) - 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 + (((1.0 + t_0) * ((Math.tan(y) + Math.tan(z)) / (1.0 - Math.pow(t_0, 2.0)))) - Math.tan(a));
}

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, a)
	t_0 = tan(y) * tan(z);
	tmp = x + (((1.0 + t_0) * ((tan(y) + tan(z)) / (1.0 - (t_0 ^ 2.0)))) - 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[(1.0 + t$95$0), $MachinePrecision] * N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.4%

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

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

Alternative 3: 99.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ x (- (* (+ (tan y) (tan z)) (/ 1.0 (- 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 / (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 / (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 / (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 / (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(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 / (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[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.4%

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

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

   3. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. tan-lowering-tan.f64 99.7%

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 4: 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 80.4%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 5: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;a \leq -0.00018:\\ \;\;\;\;x + \left(t\_0 - \tan a\right)\\ \mathbf{elif}\;a \leq 0.00115:\\ \;\;\;\;x + \left(t\_0 \cdot \frac{1}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1}{\cos \left(y + z\right)} \cdot \sin \left(y + z\right) - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (<= a -0.00018)
     (+ x (- t_0 (tan a)))
     (if (<= a 0.00115)
       (+ x (- (* t_0 (/ 1.0 (- 1.0 (* (tan y) (tan z))))) a))
       (+ x (- (* (/ 1.0 (cos (+ y z))) (sin (+ y z))) (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if (a <= -0.00018) {
		tmp = x + (t_0 - tan(a));
	} else if (a <= 0.00115) {
		tmp = x + ((t_0 * (1.0 / (1.0 - (tan(y) * tan(z))))) - a);
	} else {
		tmp = x + (((1.0 / cos((y + z))) * sin((y + 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) :: t_0
    real(8) :: tmp
    t_0 = tan(y) + tan(z)
    if (a <= (-0.00018d0)) then
        tmp = x + (t_0 - tan(a))
    else if (a <= 0.00115d0) then
        tmp = x + ((t_0 * (1.0d0 / (1.0d0 - (tan(y) * tan(z))))) - a)
    else
        tmp = x + (((1.0d0 / cos((y + z))) * sin((y + z))) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan(y) + Math.tan(z);
	double tmp;
	if (a <= -0.00018) {
		tmp = x + (t_0 - Math.tan(a));
	} else if (a <= 0.00115) {
		tmp = x + ((t_0 * (1.0 / (1.0 - (Math.tan(y) * Math.tan(z))))) - a);
	} else {
		tmp = x + (((1.0 / Math.cos((y + z))) * Math.sin((y + z))) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan(y) + math.tan(z)
	tmp = 0
	if a <= -0.00018:
		tmp = x + (t_0 - math.tan(a))
	elif a <= 0.00115:
		tmp = x + ((t_0 * (1.0 / (1.0 - (math.tan(y) * math.tan(z))))) - a)
	else:
		tmp = x + (((1.0 / math.cos((y + z))) * math.sin((y + z))) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if (a <= -0.00018)
		tmp = Float64(x + Float64(t_0 - tan(a)));
	elseif (a <= 0.00115)
		tmp = Float64(x + Float64(Float64(t_0 * Float64(1.0 / Float64(1.0 - Float64(tan(y) * tan(z))))) - a));
	else
		tmp = Float64(x + Float64(Float64(Float64(1.0 / cos(Float64(y + z))) * sin(Float64(y + z))) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan(y) + tan(z);
	tmp = 0.0;
	if (a <= -0.00018)
		tmp = x + (t_0 - tan(a));
	elseif (a <= 0.00115)
		tmp = x + ((t_0 * (1.0 / (1.0 - (tan(y) * tan(z))))) - a);
	else
		tmp = x + (((1.0 / cos((y + z))) * sin((y + z))) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.00018], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.00115], N[(x + N[(N[(t$95$0 * N[(1.0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(1.0 / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.80000000000000011e-4

   1. Initial program 78.7%

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

      1. div-inv N/A

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around 0

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

   9. Simplified 79.5%

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

   if -1.80000000000000011e-4 < a < 0.00115

   1. Initial program 79.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 79.1%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a}\right) \]
   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. div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. tan-lowering-tan.f64 99.7%

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

   7. Applied egg-rr 99.7%

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

   if 0.00115 < a

   1. Initial program 85.2%

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

      1. tan-quot N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

      9. +-lowering-+.f64 85.2%

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

   4. Applied egg-rr 85.2%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00018:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 0.00115:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) \cdot \frac{1}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1}{\cos \left(y + z\right)} \cdot \sin \left(y + z\right) - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;a \leq -0.00038:\\ \;\;\;\;x + \left(t\_0 - \tan a\right)\\ \mathbf{elif}\;a \leq 0.00065:\\ \;\;\;\;x + \left(\frac{t\_0}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1}{\cos \left(y + z\right)} \cdot \sin \left(y + z\right) - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (<= a -0.00038)
     (+ x (- t_0 (tan a)))
     (if (<= a 0.00065)
       (+ x (- (/ t_0 (- 1.0 (* (tan y) (tan z)))) a))
       (+ x (- (* (/ 1.0 (cos (+ y z))) (sin (+ y z))) (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if (a <= -0.00038) {
		tmp = x + (t_0 - tan(a));
	} else if (a <= 0.00065) {
		tmp = x + ((t_0 / (1.0 - (tan(y) * tan(z)))) - a);
	} else {
		tmp = x + (((1.0 / cos((y + z))) * sin((y + 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) :: t_0
    real(8) :: tmp
    t_0 = tan(y) + tan(z)
    if (a <= (-0.00038d0)) then
        tmp = x + (t_0 - tan(a))
    else if (a <= 0.00065d0) then
        tmp = x + ((t_0 / (1.0d0 - (tan(y) * tan(z)))) - a)
    else
        tmp = x + (((1.0d0 / cos((y + z))) * sin((y + z))) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan(y) + Math.tan(z);
	double tmp;
	if (a <= -0.00038) {
		tmp = x + (t_0 - Math.tan(a));
	} else if (a <= 0.00065) {
		tmp = x + ((t_0 / (1.0 - (Math.tan(y) * Math.tan(z)))) - a);
	} else {
		tmp = x + (((1.0 / Math.cos((y + z))) * Math.sin((y + z))) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan(y) + math.tan(z)
	tmp = 0
	if a <= -0.00038:
		tmp = x + (t_0 - math.tan(a))
	elif a <= 0.00065:
		tmp = x + ((t_0 / (1.0 - (math.tan(y) * math.tan(z)))) - a)
	else:
		tmp = x + (((1.0 / math.cos((y + z))) * math.sin((y + z))) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if (a <= -0.00038)
		tmp = Float64(x + Float64(t_0 - tan(a)));
	elseif (a <= 0.00065)
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))) - a));
	else
		tmp = Float64(x + Float64(Float64(Float64(1.0 / cos(Float64(y + z))) * sin(Float64(y + z))) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan(y) + tan(z);
	tmp = 0.0;
	if (a <= -0.00038)
		tmp = x + (t_0 - tan(a));
	elseif (a <= 0.00065)
		tmp = x + ((t_0 / (1.0 - (tan(y) * tan(z)))) - a);
	else
		tmp = x + (((1.0 / cos((y + z))) * sin((y + z))) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.00038], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.00065], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(1.0 / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.8000000000000002e-4

   1. Initial program 78.7%

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

      1. div-inv N/A

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around 0

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

   9. Simplified 79.5%

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

   if -3.8000000000000002e-4 < a < 6.4999999999999997e-4

   1. Initial program 79.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 79.1%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a}\right) \]
   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. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. tan-lowering-tan.f64 99.7%

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

   7. Applied egg-rr 99.7%

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

   if 6.4999999999999997e-4 < a

   1. Initial program 85.2%

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

      1. tan-quot N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

      9. +-lowering-+.f64 85.2%

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

   4. Applied egg-rr 85.2%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00038:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 0.00065:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{1}{\cos \left(y + z\right)} \cdot \sin \left(y + z\right) - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(\tan y - \tan a\right)\\ \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\tan a \leq 10^{-8}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + a \cdot \left(-1 - \left(a \cdot a\right) \cdot \left(0.3333333333333333 + \left(a \cdot a\right) \cdot 0.13333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (- (tan y) (tan a)))))
   (if (<= (tan a) -0.02)
     t_0
     (if (<= (tan a) 1e-8)
       (+
        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(y) - tan(a));
	double tmp;
	if (tan(a) <= -0.02) {
		tmp = t_0;
	} else if (tan(a) <= 1e-8) {
		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(y) - tan(a))
    if (tan(a) <= (-0.02d0)) then
        tmp = t_0
    else if (tan(a) <= 1d-8) 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(y) - Math.tan(a));
	double tmp;
	if (Math.tan(a) <= -0.02) {
		tmp = t_0;
	} else if (Math.tan(a) <= 1e-8) {
		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(y) - math.tan(a))
	tmp = 0
	if math.tan(a) <= -0.02:
		tmp = t_0
	elif math.tan(a) <= 1e-8:
		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 + Float64(tan(y) - tan(a)))
	tmp = 0.0
	if (tan(a) <= -0.02)
		tmp = t_0;
	elseif (tan(a) <= 1e-8)
		tmp = Float64(x + Float64(tan(Float64(y + z)) + Float64(a * Float64(-1.0 - Float64(Float64(a * a) * Float64(0.3333333333333333 + Float64(Float64(a * a) * 0.13333333333333333)))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = x + (tan(y) - tan(a));
	tmp = 0.0;
	if (tan(a) <= -0.02)
		tmp = t_0;
	elseif (tan(a) <= 1e-8)
		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[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.02], t$95$0, If[LessEqual[N[Tan[a], $MachinePrecision], 1e-8], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] + N[(a * N[(-1.0 - N[(N[(a * a), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(a * a), $MachinePrecision] * 0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0200000000000000004 or 1e-8 < (tan.f64 a)

   1. Initial program 81.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 58.5%

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

   if -0.0200000000000000004 < (tan.f64 a) < 1e-8

   1. Initial program 79.8%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 79.8%

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

   5. Simplified 79.8%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 10^{-8}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + a \cdot \left(-1 - \left(a \cdot a\right) \cdot \left(0.3333333333333333 + \left(a \cdot a\right) \cdot 0.13333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

   1. div-inv N/A

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

   2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in y around 0

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

9. Simplified 80.7%

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

10. Final simplification 80.7%

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

Alternative 9: 60.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -5 \cdot 10^{-12}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -5e-12) (+ x (- (tan y) (tan a))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -5e-12) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (tan(z) - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y + z) <= (-5d-12)) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (tan(z) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -5e-12) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -5e-12:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -5e-12)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -5e-12)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -5e-12], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -4.9999999999999997e-12

   1. Initial program 76.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 47.9%

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

   if -4.9999999999999997e-12 < (+.f64 y z)

   1. Initial program 83.3%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 67.6%

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

Alternative 10: 79.6% 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 80.4%

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

Alternative 11: 51.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2000000:\\ \;\;\;\;\frac{1}{\frac{1}{x + \tan y}}\\ \mathbf{elif}\;y + z \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{1}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.3333333333333333 + 0.13333333333333333 \cdot \left(y \cdot y\right)\right)\right) + \left(x - \tan a\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + a \cdot \left(-1 - \left(a \cdot a\right) \cdot \left(0.3333333333333333 + \left(a \cdot a\right) \cdot 0.13333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -2000000.0)
   (/ 1.0 (/ 1.0 (+ x (tan y))))
   (if (<= (+ y z) 2e-7)
     (/
      1.0
      (/
       1.0
       (+
        (*
         y
         (+
          1.0
          (* (* y y) (+ 0.3333333333333333 (* 0.13333333333333333 (* y y))))))
        (- x (tan a)))))
     (+
      x
      (+
       (tan (+ y z))
       (*
        a
        (-
         -1.0
         (*
          (* a a)
          (+ 0.3333333333333333 (* (* a a) 0.13333333333333333))))))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2000000.0) {
		tmp = 1.0 / (1.0 / (x + tan(y)));
	} else if ((y + z) <= 2e-7) {
		tmp = 1.0 / (1.0 / ((y * (1.0 + ((y * y) * (0.3333333333333333 + (0.13333333333333333 * (y * y)))))) + (x - tan(a))));
	} else {
		tmp = x + (tan((y + z)) + (a * (-1.0 - ((a * a) * (0.3333333333333333 + ((a * a) * 0.13333333333333333))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y + z) <= (-2000000.0d0)) then
        tmp = 1.0d0 / (1.0d0 / (x + tan(y)))
    else if ((y + z) <= 2d-7) then
        tmp = 1.0d0 / (1.0d0 / ((y * (1.0d0 + ((y * y) * (0.3333333333333333d0 + (0.13333333333333333d0 * (y * y)))))) + (x - tan(a))))
    else
        tmp = x + (tan((y + z)) + (a * ((-1.0d0) - ((a * a) * (0.3333333333333333d0 + ((a * a) * 0.13333333333333333d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2000000.0) {
		tmp = 1.0 / (1.0 / (x + Math.tan(y)));
	} else if ((y + z) <= 2e-7) {
		tmp = 1.0 / (1.0 / ((y * (1.0 + ((y * y) * (0.3333333333333333 + (0.13333333333333333 * (y * y)))))) + (x - Math.tan(a))));
	} else {
		tmp = x + (Math.tan((y + z)) + (a * (-1.0 - ((a * a) * (0.3333333333333333 + ((a * a) * 0.13333333333333333))))));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -2000000.0:
		tmp = 1.0 / (1.0 / (x + math.tan(y)))
	elif (y + z) <= 2e-7:
		tmp = 1.0 / (1.0 / ((y * (1.0 + ((y * y) * (0.3333333333333333 + (0.13333333333333333 * (y * y)))))) + (x - math.tan(a))))
	else:
		tmp = x + (math.tan((y + z)) + (a * (-1.0 - ((a * a) * (0.3333333333333333 + ((a * a) * 0.13333333333333333))))))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -2000000.0)
		tmp = Float64(1.0 / Float64(1.0 / Float64(x + tan(y))));
	elseif (Float64(y + z) <= 2e-7)
		tmp = Float64(1.0 / Float64(1.0 / Float64(Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.3333333333333333 + Float64(0.13333333333333333 * Float64(y * y)))))) + Float64(x - tan(a)))));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) + Float64(a * Float64(-1.0 - Float64(Float64(a * a) * Float64(0.3333333333333333 + Float64(Float64(a * a) * 0.13333333333333333)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -2000000.0)
		tmp = 1.0 / (1.0 / (x + tan(y)));
	elseif ((y + z) <= 2e-7)
		tmp = 1.0 / (1.0 / ((y * (1.0 + ((y * y) * (0.3333333333333333 + (0.13333333333333333 * (y * y)))))) + (x - tan(a))));
	else
		tmp = x + (tan((y + z)) + (a * (-1.0 - ((a * a) * (0.3333333333333333 + ((a * a) * 0.13333333333333333))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -2000000.0], N[(1.0 / N[(1.0 / N[(x + N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 2e-7], N[(1.0 / N[(1.0 / N[(N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.3333333333333333 + N[(0.13333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] + N[(a * N[(-1.0 - N[(N[(a * a), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(a * a), $MachinePrecision] * 0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -2e6

   1. Initial program 75.2%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-+ N/A

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

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

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

      7. associate-+r- N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      14. tan-lowering-tan.f64 75.1%

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

   4. Applied egg-rr 75.1%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 46.5%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 37.4%

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

   if -2e6 < (+.f64 y z) < 1.9999999999999999e-7

   1. Initial program 99.9%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-+ N/A

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

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

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

      7. associate-+r- N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      14. tan-lowering-tan.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 98.7%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 98.7%

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

   10. Simplified 98.7%

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.3333333333333333 + \left(y \cdot y\right) \cdot 0.13333333333333333\right)\right)} + \left(x - \tan a\right)}} \]

   if 1.9999999999999999e-7 < (+.f64 y z)

   1. Initial program 73.6%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 33.7%

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

   5. Simplified 33.7%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -2000000:\\ \;\;\;\;\frac{1}{\frac{1}{x + \tan y}}\\ \mathbf{elif}\;y + z \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{1}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.3333333333333333 + 0.13333333333333333 \cdot \left(y \cdot y\right)\right)\right) + \left(x - \tan a\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + a \cdot \left(-1 - \left(a \cdot a\right) \cdot \left(0.3333333333333333 + \left(a \cdot a\right) \cdot 0.13333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2000000:\\ \;\;\;\;\frac{1}{\frac{1}{x + \tan y}}\\ \mathbf{elif}\;y + z \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{1}{\left(x - \tan a\right) + y \cdot \left(1 + 0.3333333333333333 \cdot \left(y \cdot y\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + a \cdot \left(-1 - \left(a \cdot a\right) \cdot \left(0.3333333333333333 + \left(a \cdot a\right) \cdot 0.13333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -2000000.0)
   (/ 1.0 (/ 1.0 (+ x (tan y))))
   (if (<= (+ y z) 2e-7)
     (/
      1.0
      (/ 1.0 (+ (- x (tan a)) (* y (+ 1.0 (* 0.3333333333333333 (* y y)))))))
     (+
      x
      (+
       (tan (+ y z))
       (*
        a
        (-
         -1.0
         (*
          (* a a)
          (+ 0.3333333333333333 (* (* a a) 0.13333333333333333))))))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2000000.0) {
		tmp = 1.0 / (1.0 / (x + tan(y)));
	} else if ((y + z) <= 2e-7) {
		tmp = 1.0 / (1.0 / ((x - tan(a)) + (y * (1.0 + (0.3333333333333333 * (y * y))))));
	} else {
		tmp = x + (tan((y + z)) + (a * (-1.0 - ((a * a) * (0.3333333333333333 + ((a * a) * 0.13333333333333333))))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y + z) <= (-2000000.0d0)) then
        tmp = 1.0d0 / (1.0d0 / (x + tan(y)))
    else if ((y + z) <= 2d-7) then
        tmp = 1.0d0 / (1.0d0 / ((x - tan(a)) + (y * (1.0d0 + (0.3333333333333333d0 * (y * y))))))
    else
        tmp = x + (tan((y + z)) + (a * ((-1.0d0) - ((a * a) * (0.3333333333333333d0 + ((a * a) * 0.13333333333333333d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2000000.0) {
		tmp = 1.0 / (1.0 / (x + Math.tan(y)));
	} else if ((y + z) <= 2e-7) {
		tmp = 1.0 / (1.0 / ((x - Math.tan(a)) + (y * (1.0 + (0.3333333333333333 * (y * y))))));
	} else {
		tmp = x + (Math.tan((y + z)) + (a * (-1.0 - ((a * a) * (0.3333333333333333 + ((a * a) * 0.13333333333333333))))));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -2000000.0:
		tmp = 1.0 / (1.0 / (x + math.tan(y)))
	elif (y + z) <= 2e-7:
		tmp = 1.0 / (1.0 / ((x - math.tan(a)) + (y * (1.0 + (0.3333333333333333 * (y * y))))))
	else:
		tmp = x + (math.tan((y + z)) + (a * (-1.0 - ((a * a) * (0.3333333333333333 + ((a * a) * 0.13333333333333333))))))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -2000000.0)
		tmp = Float64(1.0 / Float64(1.0 / Float64(x + tan(y))));
	elseif (Float64(y + z) <= 2e-7)
		tmp = Float64(1.0 / Float64(1.0 / Float64(Float64(x - tan(a)) + Float64(y * Float64(1.0 + Float64(0.3333333333333333 * Float64(y * y)))))));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) + Float64(a * Float64(-1.0 - Float64(Float64(a * a) * Float64(0.3333333333333333 + Float64(Float64(a * a) * 0.13333333333333333)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -2000000.0)
		tmp = 1.0 / (1.0 / (x + tan(y)));
	elseif ((y + z) <= 2e-7)
		tmp = 1.0 / (1.0 / ((x - tan(a)) + (y * (1.0 + (0.3333333333333333 * (y * y))))));
	else
		tmp = x + (tan((y + z)) + (a * (-1.0 - ((a * a) * (0.3333333333333333 + ((a * a) * 0.13333333333333333))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -2000000.0], N[(1.0 / N[(1.0 / N[(x + N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 2e-7], N[(1.0 / N[(1.0 / N[(N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision] + N[(y * N[(1.0 + N[(0.3333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] + N[(a * N[(-1.0 - N[(N[(a * a), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(a * a), $MachinePrecision] * 0.13333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -2e6

   1. Initial program 75.2%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-+ N/A

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

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

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

      7. associate-+r- N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      14. tan-lowering-tan.f64 75.1%

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

   4. Applied egg-rr 75.1%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 46.5%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 37.4%

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

   if -2e6 < (+.f64 y z) < 1.9999999999999999e-7

   1. Initial program 99.9%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-+ N/A

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

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

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

      7. associate-+r- N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      14. tan-lowering-tan.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 98.7%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 98.4%

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

   10. Simplified 98.4%

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

   if 1.9999999999999999e-7 < (+.f64 y z)

   1. Initial program 73.6%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 33.7%

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

   5. Simplified 33.7%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -2000000:\\ \;\;\;\;\frac{1}{\frac{1}{x + \tan y}}\\ \mathbf{elif}\;y + z \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{1}{\left(x - \tan a\right) + y \cdot \left(1 + 0.3333333333333333 \cdot \left(y \cdot y\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) + a \cdot \left(-1 - \left(a \cdot a\right) \cdot \left(0.3333333333333333 + \left(a \cdot a\right) \cdot 0.13333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2000000:\\ \;\;\;\;\frac{1}{\frac{1}{x + \tan y}}\\ \mathbf{elif}\;y + z \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{1}{\left(x - \tan a\right) + y \cdot \left(1 + 0.3333333333333333 \cdot \left(y \cdot y\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(z \cdot \left(1 + \frac{y}{z}\right)\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -2000000.0)
   (/ 1.0 (/ 1.0 (+ x (tan y))))
   (if (<= (+ y z) 2e-7)
     (/
      1.0
      (/ 1.0 (+ (- x (tan a)) (* y (+ 1.0 (* 0.3333333333333333 (* y y)))))))
     (+ x (- (tan (* z (+ 1.0 (/ y z)))) a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2000000.0) {
		tmp = 1.0 / (1.0 / (x + tan(y)));
	} else if ((y + z) <= 2e-7) {
		tmp = 1.0 / (1.0 / ((x - tan(a)) + (y * (1.0 + (0.3333333333333333 * (y * y))))));
	} else {
		tmp = x + (tan((z * (1.0 + (y / z)))) - a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y + z) <= (-2000000.0d0)) then
        tmp = 1.0d0 / (1.0d0 / (x + tan(y)))
    else if ((y + z) <= 2d-7) then
        tmp = 1.0d0 / (1.0d0 / ((x - tan(a)) + (y * (1.0d0 + (0.3333333333333333d0 * (y * y))))))
    else
        tmp = x + (tan((z * (1.0d0 + (y / z)))) - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2000000.0) {
		tmp = 1.0 / (1.0 / (x + Math.tan(y)));
	} else if ((y + z) <= 2e-7) {
		tmp = 1.0 / (1.0 / ((x - Math.tan(a)) + (y * (1.0 + (0.3333333333333333 * (y * y))))));
	} else {
		tmp = x + (Math.tan((z * (1.0 + (y / z)))) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -2000000.0:
		tmp = 1.0 / (1.0 / (x + math.tan(y)))
	elif (y + z) <= 2e-7:
		tmp = 1.0 / (1.0 / ((x - math.tan(a)) + (y * (1.0 + (0.3333333333333333 * (y * y))))))
	else:
		tmp = x + (math.tan((z * (1.0 + (y / z)))) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -2000000.0)
		tmp = Float64(1.0 / Float64(1.0 / Float64(x + tan(y))));
	elseif (Float64(y + z) <= 2e-7)
		tmp = Float64(1.0 / Float64(1.0 / Float64(Float64(x - tan(a)) + Float64(y * Float64(1.0 + Float64(0.3333333333333333 * Float64(y * y)))))));
	else
		tmp = Float64(x + Float64(tan(Float64(z * Float64(1.0 + Float64(y / z)))) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -2000000.0)
		tmp = 1.0 / (1.0 / (x + tan(y)));
	elseif ((y + z) <= 2e-7)
		tmp = 1.0 / (1.0 / ((x - tan(a)) + (y * (1.0 + (0.3333333333333333 * (y * y))))));
	else
		tmp = x + (tan((z * (1.0 + (y / z)))) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -2000000.0], N[(1.0 / N[(1.0 / N[(x + N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 2e-7], N[(1.0 / N[(1.0 / N[(N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision] + N[(y * N[(1.0 + N[(0.3333333333333333 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(z * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -2e6

   1. Initial program 75.2%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-+ N/A

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

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

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

      7. associate-+r- N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      14. tan-lowering-tan.f64 75.1%

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

   4. Applied egg-rr 75.1%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 46.5%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 37.4%

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

   if -2e6 < (+.f64 y z) < 1.9999999999999999e-7

   1. Initial program 99.9%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-+ N/A

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

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

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

      7. associate-+r- N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      14. tan-lowering-tan.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 98.7%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 98.4%

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

   10. Simplified 98.4%

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

   if 1.9999999999999999e-7 < (+.f64 y z)

   1. Initial program 73.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 33.7%

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

   6. Taylor expanded in z around inf

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

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

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

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

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

      3. /-lowering-/.f64 31.0%

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

   8. Simplified 31.0%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -2000000:\\ \;\;\;\;\frac{1}{\frac{1}{x + \tan y}}\\ \mathbf{elif}\;y + z \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{1}{\left(x - \tan a\right) + y \cdot \left(1 + 0.3333333333333333 \cdot \left(y \cdot y\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(z \cdot \left(1 + \frac{y}{z}\right)\right) - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2000000:\\ \;\;\;\;\frac{1}{\frac{1}{x + \tan y}}\\ \mathbf{elif}\;y + z \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{1}{y + \left(x - \tan a\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(z \cdot \left(1 + \frac{y}{z}\right)\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -2000000.0)
   (/ 1.0 (/ 1.0 (+ x (tan y))))
   (if (<= (+ y z) 2e-7)
     (/ 1.0 (/ 1.0 (+ y (- x (tan a)))))
     (+ x (- (tan (* z (+ 1.0 (/ y z)))) a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2000000.0) {
		tmp = 1.0 / (1.0 / (x + tan(y)));
	} else if ((y + z) <= 2e-7) {
		tmp = 1.0 / (1.0 / (y + (x - tan(a))));
	} else {
		tmp = x + (tan((z * (1.0 + (y / z)))) - a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y + z) <= (-2000000.0d0)) then
        tmp = 1.0d0 / (1.0d0 / (x + tan(y)))
    else if ((y + z) <= 2d-7) then
        tmp = 1.0d0 / (1.0d0 / (y + (x - tan(a))))
    else
        tmp = x + (tan((z * (1.0d0 + (y / z)))) - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2000000.0) {
		tmp = 1.0 / (1.0 / (x + Math.tan(y)));
	} else if ((y + z) <= 2e-7) {
		tmp = 1.0 / (1.0 / (y + (x - Math.tan(a))));
	} else {
		tmp = x + (Math.tan((z * (1.0 + (y / z)))) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -2000000.0:
		tmp = 1.0 / (1.0 / (x + math.tan(y)))
	elif (y + z) <= 2e-7:
		tmp = 1.0 / (1.0 / (y + (x - math.tan(a))))
	else:
		tmp = x + (math.tan((z * (1.0 + (y / z)))) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -2000000.0)
		tmp = Float64(1.0 / Float64(1.0 / Float64(x + tan(y))));
	elseif (Float64(y + z) <= 2e-7)
		tmp = Float64(1.0 / Float64(1.0 / Float64(y + Float64(x - tan(a)))));
	else
		tmp = Float64(x + Float64(tan(Float64(z * Float64(1.0 + Float64(y / z)))) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -2000000.0)
		tmp = 1.0 / (1.0 / (x + tan(y)));
	elseif ((y + z) <= 2e-7)
		tmp = 1.0 / (1.0 / (y + (x - tan(a))));
	else
		tmp = x + (tan((z * (1.0 + (y / z)))) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -2000000.0], N[(1.0 / N[(1.0 / N[(x + N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 2e-7], N[(1.0 / N[(1.0 / N[(y + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(z * N[(1.0 + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -2e6

   1. Initial program 75.2%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-+ N/A

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

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

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

      7. associate-+r- N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      14. tan-lowering-tan.f64 75.1%

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

   4. Applied egg-rr 75.1%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 46.5%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 37.4%

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

   if -2e6 < (+.f64 y z) < 1.9999999999999999e-7

   1. Initial program 99.9%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-+ N/A

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

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

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

      7. associate-+r- N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      14. tan-lowering-tan.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 98.7%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 98.1%

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

   if 1.9999999999999999e-7 < (+.f64 y z)

   1. Initial program 73.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 33.7%

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

   6. Taylor expanded in z around inf

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

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

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

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

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

      3. /-lowering-/.f64 31.0%

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

   8. Simplified 31.0%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -2000000:\\ \;\;\;\;\frac{1}{\frac{1}{x + \tan y}}\\ \mathbf{elif}\;y + z \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{1}{y + \left(x - \tan a\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(z \cdot \left(1 + \frac{y}{z}\right)\right) - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2000000:\\ \;\;\;\;\frac{1}{\frac{1}{x + \tan y}}\\ \mathbf{elif}\;y + z \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{1}{y + \left(x - \tan a\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -2000000.0)
   (/ 1.0 (/ 1.0 (+ x (tan y))))
   (if (<= (+ y z) 2e-7)
     (/ 1.0 (/ 1.0 (+ y (- x (tan a)))))
     (+ x (- (tan (+ y z)) a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2000000.0) {
		tmp = 1.0 / (1.0 / (x + tan(y)));
	} else if ((y + z) <= 2e-7) {
		tmp = 1.0 / (1.0 / (y + (x - tan(a))));
	} else {
		tmp = x + (tan((y + z)) - a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y + z) <= (-2000000.0d0)) then
        tmp = 1.0d0 / (1.0d0 / (x + tan(y)))
    else if ((y + z) <= 2d-7) then
        tmp = 1.0d0 / (1.0d0 / (y + (x - tan(a))))
    else
        tmp = x + (tan((y + z)) - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -2000000.0) {
		tmp = 1.0 / (1.0 / (x + Math.tan(y)));
	} else if ((y + z) <= 2e-7) {
		tmp = 1.0 / (1.0 / (y + (x - Math.tan(a))));
	} else {
		tmp = x + (Math.tan((y + z)) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -2000000.0:
		tmp = 1.0 / (1.0 / (x + math.tan(y)))
	elif (y + z) <= 2e-7:
		tmp = 1.0 / (1.0 / (y + (x - math.tan(a))))
	else:
		tmp = x + (math.tan((y + z)) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -2000000.0)
		tmp = Float64(1.0 / Float64(1.0 / Float64(x + tan(y))));
	elseif (Float64(y + z) <= 2e-7)
		tmp = Float64(1.0 / Float64(1.0 / Float64(y + Float64(x - tan(a)))));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -2000000.0)
		tmp = 1.0 / (1.0 / (x + tan(y)));
	elseif ((y + z) <= 2e-7)
		tmp = 1.0 / (1.0 / (y + (x - tan(a))));
	else
		tmp = x + (tan((y + z)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -2000000.0], N[(1.0 / N[(1.0 / N[(x + N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 2e-7], N[(1.0 / N[(1.0 / N[(y + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -2e6

   1. Initial program 75.2%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-+ N/A

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

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

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

      7. associate-+r- N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      14. tan-lowering-tan.f64 75.1%

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

   4. Applied egg-rr 75.1%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 46.5%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 37.4%

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

   if -2e6 < (+.f64 y z) < 1.9999999999999999e-7

   1. Initial program 99.9%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-+ N/A

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

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

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

      7. associate-+r- N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      14. tan-lowering-tan.f64 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 98.7%

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

   8. Taylor expanded in y around 0

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

   10. Simplified 98.1%

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

   if 1.9999999999999999e-7 < (+.f64 y z)

   1. Initial program 73.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 33.7%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -2000000:\\ \;\;\;\;\frac{1}{\frac{1}{x + \tan y}}\\ \mathbf{elif}\;y + z \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{1}{y + \left(x - \tan a\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.56:\\ \;\;\;\;\frac{1}{\frac{1}{x + \tan y}}\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{-5}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.56)
   (/ 1.0 (/ 1.0 (+ x (tan y))))
   (if (<= a 7.2e-5) (+ x (- (tan (+ y z)) a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.56) {
		tmp = 1.0 / (1.0 / (x + tan(y)));
	} else if (a <= 7.2e-5) {
		tmp = x + (tan((y + z)) - a);
	} else {
		tmp = x;
	}
	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 (a <= (-1.56d0)) then
        tmp = 1.0d0 / (1.0d0 / (x + tan(y)))
    else if (a <= 7.2d-5) then
        tmp = x + (tan((y + z)) - a)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.56) {
		tmp = 1.0 / (1.0 / (x + Math.tan(y)));
	} else if (a <= 7.2e-5) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -1.56:
		tmp = 1.0 / (1.0 / (x + math.tan(y)))
	elif a <= 7.2e-5:
		tmp = x + (math.tan((y + z)) - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.56)
		tmp = Float64(1.0 / Float64(1.0 / Float64(x + tan(y))));
	elseif (a <= 7.2e-5)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -1.56)
		tmp = 1.0 / (1.0 / (x + tan(y)));
	elseif (a <= 7.2e-5)
		tmp = x + (tan((y + z)) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.56], N[(1.0 / N[(1.0 / N[(x + N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 7.2e-5], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if a < -1.5600000000000001

   1. Initial program 78.1%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-+ N/A

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

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

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

      7. associate-+r- N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

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

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

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

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

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

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

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

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

      14. tan-lowering-tan.f64 78.0%

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

   4. Applied egg-rr 78.0%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 59.1%

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

   8. Taylor expanded in x around inf

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

   10. Simplified 22.8%

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

   if -1.5600000000000001 < a < 7.20000000000000018e-5

   1. Initial program 79.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 79.3%

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

   if 7.20000000000000018e-5 < a

   1. Initial program 84.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 22.3%

      \[\leadsto \color{blue}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.56:\\ \;\;\;\;\frac{1}{\frac{1}{x + \tan y}}\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{-5}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 50.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.95:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{-5}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.95) x (if (<= a 7.2e-5) (+ x (- (tan (+ y z)) a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.95) {
		tmp = x;
	} else if (a <= 7.2e-5) {
		tmp = x + (tan((y + z)) - a);
	} else {
		tmp = x;
	}
	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 (a <= (-1.95d0)) then
        tmp = x
    else if (a <= 7.2d-5) then
        tmp = x + (tan((y + z)) - a)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.95) {
		tmp = x;
	} else if (a <= 7.2e-5) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -1.95:
		tmp = x
	elif a <= 7.2e-5:
		tmp = x + (math.tan((y + z)) - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.95)
		tmp = x;
	elseif (a <= 7.2e-5)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -1.95)
		tmp = x;
	elseif (a <= 7.2e-5)
		tmp = x + (tan((y + z)) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.95], x, If[LessEqual[a, 7.2e-5], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -1.94999999999999996 or 7.20000000000000018e-5 < a

   1. Initial program 81.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 21.6%

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

   if -1.94999999999999996 < a < 7.20000000000000018e-5

   1. Initial program 79.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 79.3%

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

Alternative 18: 40.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-31}:\\ \;\;\;\;x + \left(\tan y - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.5e-15) x (if (<= a 4.4e-31) (+ x (- (tan y) a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.5e-15) {
		tmp = x;
	} else if (a <= 4.4e-31) {
		tmp = x + (tan(y) - a);
	} else {
		tmp = x;
	}
	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 (a <= (-1.5d-15)) then
        tmp = x
    else if (a <= 4.4d-31) then
        tmp = x + (tan(y) - a)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.5e-15) {
		tmp = x;
	} else if (a <= 4.4e-31) {
		tmp = x + (Math.tan(y) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -1.5e-15:
		tmp = x
	elif a <= 4.4e-31:
		tmp = x + (math.tan(y) - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.5e-15)
		tmp = x;
	elseif (a <= 4.4e-31)
		tmp = Float64(x + Float64(tan(y) - a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -1.5e-15)
		tmp = x;
	elseif (a <= 4.4e-31)
		tmp = x + (tan(y) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.5e-15], x, If[LessEqual[a, 4.4e-31], N[(x + N[(N[Tan[y], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -1.5e-15 or 4.40000000000000019e-31 < a

   1. Initial program 80.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 22.4%

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

   if -1.5e-15 < a < 4.40000000000000019e-31

   1. Initial program 80.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 80.6%

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

   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 59.3%

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

Alternative 19: 36.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.6 \cdot 10^{-7}:\\ \;\;\;\;x + \left(\tan y - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 1.6e-7) (+ x (- (tan y) a)) (+ x (- (tan z) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 1.6e-7) {
		tmp = x + (tan(y) - a);
	} else {
		tmp = x + (tan(z) - 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 <= 1.6d-7) then
        tmp = x + (tan(y) - a)
    else
        tmp = x + (tan(z) - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 1.6e-7) {
		tmp = x + (Math.tan(y) - a);
	} else {
		tmp = x + (Math.tan(z) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 1.6e-7:
		tmp = x + (math.tan(y) - a)
	else:
		tmp = x + (math.tan(z) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 1.6e-7)
		tmp = Float64(x + Float64(tan(y) - a));
	else
		tmp = Float64(x + Float64(tan(z) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 1.6e-7)
		tmp = x + (tan(y) - a);
	else
		tmp = x + (tan(z) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 1.6e-7], N[(x + N[(N[Tan[y], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.6e-7

   1. Initial program 88.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 45.7%

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

   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 38.1%

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

   if 1.6e-7 < z

   1. Initial program 59.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 28.6%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 28.6%

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

Alternative 20: 31.4% 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 80.4%

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

3. Taylor expanded in x around inf

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

5. Simplified 30.4%

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

Reproduce

herbie shell --seed 2024160 
(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))))