tan-example (used to crash)

Percentage Accurate: 79.2% → 99.7%

Time: 33.9s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    t_0 = tan(y) * tan(z)
    code = x + (((tan(y) + tan(z)) / ((1.0d0 / (1.0d0 + t_0)) + (((tan(z) ** 2.0d0) * (tan(y) ** 2.0d0)) / ((-1.0d0) - t_0)))) - tan(a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 99.8%

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

4. Applied egg-rr 99.8%

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

   1. flip-- N/A

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

   2. metadata-eval N/A

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

   3. div-sub N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

6. Applied egg-rr 99.8%

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

   1. pow-prod-down N/A

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

   2. pow2 N/A

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

   3. *-commutative N/A

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

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

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

   5. pow2 N/A

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

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

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

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

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

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

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

   9. tan-lowering-tan.f64 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 87.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \left(t\_0 - \tan a\right)\\ \mathbf{if}\;\tan a \leq -0.05:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-39}:\\ \;\;\;\;x + \left(\frac{t\_0}{\mathsf{fma}\left(\tan y, 0 - \tan z, 1\right)} - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ x (- t_0 (tan a)))))
   (if (<= (tan a) -0.05)
     t_1
     (if (<= (tan a) 2e-39)
       (+ x (- (/ t_0 (fma (tan y) (- 0.0 (tan z)) 1.0)) a))
       t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = x + (t_0 - tan(a));
	double tmp;
	if (tan(a) <= -0.05) {
		tmp = t_1;
	} else if (tan(a) <= 2e-39) {
		tmp = x + ((t_0 / fma(tan(y), (0.0 - tan(z)), 1.0)) - a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(x + Float64(t_0 - tan(a)))
	tmp = 0.0
	if (tan(a) <= -0.05)
		tmp = t_1;
	elseif (tan(a) <= 2e-39)
		tmp = Float64(x + Float64(Float64(t_0 / fma(tan(y), Float64(0.0 - tan(z)), 1.0)) - a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.05], t$95$1, If[LessEqual[N[Tan[a], $MachinePrecision], 2e-39], N[(x + N[(N[(t$95$0 / N[(N[Tan[y], $MachinePrecision] * N[(0.0 - N[Tan[z], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.050000000000000003 or 1.99999999999999986e-39 < (tan.f64 a)

   1. Initial program 85.5%

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

      1. tan-sum N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. tan-lowering-tan.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 85.8%

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

   if -0.050000000000000003 < (tan.f64 a) < 1.99999999999999986e-39

   1. Initial program 81.1%

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

      1. tan-sum N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. tan-lowering-tan.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. fma-define N/A

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

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

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

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

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

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

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

      8. tan-lowering-tan.f64 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in a around 0

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

   9. Simplified 99.8%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.05:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-39}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{\mathsf{fma}\left(\tan y, 0 - \tan z, 1\right)} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 99.8%

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

4. Applied egg-rr 99.8%

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

Alternative 4: 88.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ t_1 := x + \left(t\_0 - \tan a\right)\\ \mathbf{if}\;a \leq -5 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{t\_0}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))) (t_1 (+ x (- t_0 (tan a)))))
   (if (<= a -5e-10)
     t_1
     (if (<= a 4.8e-39) (+ x (/ t_0 (- 1.0 (* (tan y) (tan z))))) t_1))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double t_1 = x + (t_0 - tan(a));
	double tmp;
	if (a <= -5e-10) {
		tmp = t_1;
	} else if (a <= 4.8e-39) {
		tmp = x + (t_0 / (1.0 - (tan(y) * tan(z))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = tan(y) + tan(z)
    t_1 = x + (t_0 - tan(a))
    if (a <= (-5d-10)) then
        tmp = t_1
    else if (a <= 4.8d-39) then
        tmp = x + (t_0 / (1.0d0 - (tan(y) * tan(z))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan(y) + Math.tan(z);
	double t_1 = x + (t_0 - Math.tan(a));
	double tmp;
	if (a <= -5e-10) {
		tmp = t_1;
	} else if (a <= 4.8e-39) {
		tmp = x + (t_0 / (1.0 - (Math.tan(y) * Math.tan(z))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan(y) + math.tan(z)
	t_1 = x + (t_0 - math.tan(a))
	tmp = 0
	if a <= -5e-10:
		tmp = t_1
	elif a <= 4.8e-39:
		tmp = x + (t_0 / (1.0 - (math.tan(y) * math.tan(z))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	t_1 = Float64(x + Float64(t_0 - tan(a)))
	tmp = 0.0
	if (a <= -5e-10)
		tmp = t_1;
	elseif (a <= 4.8e-39)
		tmp = Float64(x + Float64(t_0 / Float64(1.0 - Float64(tan(y) * tan(z)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan(y) + tan(z);
	t_1 = x + (t_0 - tan(a));
	tmp = 0.0;
	if (a <= -5e-10)
		tmp = t_1;
	elseif (a <= 4.8e-39)
		tmp = x + (t_0 / (1.0 - (tan(y) * tan(z))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -5e-10], t$95$1, If[LessEqual[a, 4.8e-39], N[(x + N[(t$95$0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.00000000000000031e-10 or 4.80000000000000031e-39 < a

   1. Initial program 85.5%

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

      1. tan-sum N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. tan-lowering-tan.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 85.8%

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

   if -5.00000000000000031e-10 < a < 4.80000000000000031e-39

   1. Initial program 81.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 80.8%

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

   5. Simplified 80.8%

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

      1. quot-tan N/A

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

      2. tan-sum N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. tan-lowering-tan.f64 99.3%

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

   7. Applied egg-rr 99.3%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-10}:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) - \tan a\right)\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% 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 83.2%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in y around 0

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

7. Simplified 83.5%

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

8. Final simplification 83.5%

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

Alternative 6: 58.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq 10^{-35}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) 1e-35) (+ x (- (tan y) (tan a))) (+ x (tan (+ y z)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 1e-35) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + tan((y + z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y + z) <= 1d-35) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + tan((y + z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 1e-35) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + Math.tan((y + z));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= 1e-35:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + math.tan((y + z))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= 1e-35)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + tan(Float64(y + z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= 1e-35)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + tan((y + z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], 1e-35], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < 1.00000000000000001e-35

   1. Initial program 87.9%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 74.5%

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

   if 1.00000000000000001e-35 < (+.f64 y z)

   1. Initial program 76.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 49.1%

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

   5. Simplified 49.1%

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

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

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

      2. quot-tan N/A

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

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

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

      4. +-lowering-+.f64 49.1%

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

   7. Applied egg-rr 49.1%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq 10^{-35}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.0045:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 0.0045) (+ x (- (tan y) (tan a))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 0.0045) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (tan(z) - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= 0.0045d0) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (tan(z) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 0.0045) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 0.0045:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 0.0045)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 0.0045)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 0.0045], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.00449999999999999966

   1. Initial program 89.5%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 79.3%

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

   if 0.00449999999999999966 < z

   1. Initial program 61.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 61.0%

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

Alternative 8: 79.2% 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 83.2%

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

Alternative 9: 59.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -3.4 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-5}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -3.4e-11) t_0 (if (<= a 3.7e-5) (+ x (- (tan (+ y z)) a)) t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -3.4e-11) {
		tmp = t_0;
	} else if (a <= 3.7e-5) {
		tmp = x + (tan((y + z)) - a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - tan(a)
    if (a <= (-3.4d-11)) then
        tmp = t_0
    else if (a <= 3.7d-5) then
        tmp = x + (tan((y + z)) - a)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = x - Math.tan(a);
	double tmp;
	if (a <= -3.4e-11) {
		tmp = t_0;
	} else if (a <= 3.7e-5) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = x - math.tan(a)
	tmp = 0
	if a <= -3.4e-11:
		tmp = t_0
	elif a <= 3.7e-5:
		tmp = x + (math.tan((y + z)) - a)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -3.4e-11)
		tmp = t_0;
	elseif (a <= 3.7e-5)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = x - tan(a);
	tmp = 0.0;
	if (a <= -3.4e-11)
		tmp = t_0;
	elseif (a <= 3.7e-5)
		tmp = x + (tan((y + z)) - a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.4e-11], t$95$0, If[LessEqual[a, 3.7e-5], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.3999999999999999e-11 or 3.69999999999999981e-5 < a

   1. Initial program 84.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. cos-lowering-cos.f64 56.8%

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

   5. Simplified 56.8%

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

   6. Taylor expanded in z around 0

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

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

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

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

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

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

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

      4. cos-lowering-cos.f64 39.6%

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

   8. Simplified 39.6%

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

      1. tan-quot N/A

         \[\leadsto x - \tan a \]

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

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

      3. tan-lowering-tan.f64 39.7%

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

   10. Applied egg-rr 39.7%

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

   if -3.3999999999999999e-11 < a < 3.69999999999999981e-5

   1. Initial program 82.2%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 82.2%

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

Alternative 10: 59.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -3.4 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -3.4e-11) t_0 (if (<= a 1.45e-5) (+ x (tan (+ y z))) t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -3.4e-11) {
		tmp = t_0;
	} else if (a <= 1.45e-5) {
		tmp = x + tan((y + z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - tan(a)
    if (a <= (-3.4d-11)) then
        tmp = t_0
    else if (a <= 1.45d-5) then
        tmp = x + tan((y + z))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = x - Math.tan(a);
	double tmp;
	if (a <= -3.4e-11) {
		tmp = t_0;
	} else if (a <= 1.45e-5) {
		tmp = x + Math.tan((y + z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = x - math.tan(a)
	tmp = 0
	if a <= -3.4e-11:
		tmp = t_0
	elif a <= 1.45e-5:
		tmp = x + math.tan((y + z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -3.4e-11)
		tmp = t_0;
	elseif (a <= 1.45e-5)
		tmp = Float64(x + tan(Float64(y + z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = x - tan(a);
	tmp = 0.0;
	if (a <= -3.4e-11)
		tmp = t_0;
	elseif (a <= 1.45e-5)
		tmp = x + tan((y + z));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.4e-11], t$95$0, If[LessEqual[a, 1.45e-5], N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.3999999999999999e-11 or 1.45e-5 < a

   1. Initial program 84.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. cos-lowering-cos.f64 56.8%

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

   5. Simplified 56.8%

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

   6. Taylor expanded in z around 0

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

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

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

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

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

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

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

      4. cos-lowering-cos.f64 39.6%

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

   8. Simplified 39.6%

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

      1. tan-quot N/A

         \[\leadsto x - \tan a \]

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

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

      3. tan-lowering-tan.f64 39.7%

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

   10. Applied egg-rr 39.7%

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

   if -3.3999999999999999e-11 < a < 1.45e-5

   1. Initial program 82.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 81.9%

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

   5. Simplified 81.9%

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

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

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

      2. quot-tan N/A

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

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

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

      4. +-lowering-+.f64 81.9%

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

   7. Applied egg-rr 81.9%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{-11}:\\ \;\;\;\;x - \tan a\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-5}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x - \tan a\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. cos-lowering-cos.f64 57.5%

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

5. Simplified 57.5%

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

6. Taylor expanded in z around 0

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

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

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

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

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

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

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

   4. cos-lowering-cos.f64 41.5%

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

8. Simplified 41.5%

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

   1. tan-quot N/A

      \[\leadsto x - \tan a \]

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

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

   3. tan-lowering-tan.f64 41.5%

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

10. Applied egg-rr 41.5%

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

Alternative 12: 31.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, a_] := N[(x + N[Sin[y], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.2%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

   7. +-lowering-+.f64 54.8%

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

5. Simplified 54.8%

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

6. Taylor expanded in y around 0

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

8. Simplified 41.6%

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

9. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x + \sin y} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \sin y + \color{blue}{x} \]

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

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

    3. sin-lowering-sin.f64 33.3%

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

11. Simplified 33.3%

    \[\leadsto \color{blue}{\sin y + x} \]

12. Final simplification 33.3%

    \[\leadsto x + \sin y \]
13. Add Preprocessing

Alternative 13: 31.5% 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 83.2%

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

3. Taylor expanded in x around inf

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

5. Simplified 33.2%

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

Reproduce

herbie shell --seed 2024148 
(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))))