tan-example (used to crash)

Percentage Accurate: 78.8% → 99.7%

Time: 35.6s

Alternatives: 17

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: 78.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. tan-sum N/A

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

   2. flip-+ N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. pow2 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-tan.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 N/A

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

   17. lower--.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. lift--.f64 N/A

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

   8. *-commutative N/A

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. distribute-lft-in N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

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

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

   17. lower-*.f64 N/A

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

   18. lower-neg.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (+
  x
  (-
   (/
    (- (pow (tan y) 2.0) (pow (tan z) 2.0))
    (* (- (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 + (((pow(tan(y), 2.0) - pow(tan(z), 2.0)) / ((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) ** 2.0d0) - (tan(z) ** 2.0d0)) / ((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.pow(Math.tan(y), 2.0) - Math.pow(Math.tan(z), 2.0)) / ((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.pow(math.tan(y), 2.0) - math.pow(math.tan(z), 2.0)) / ((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) ^ 2.0) - (tan(z) ^ 2.0)) / 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) ^ 2.0) - (tan(z) ^ 2.0)) / ((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[Power[N[Tan[y], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Tan[z], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / 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]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.3%

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

   1. tan-sum N/A

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

   2. flip-+ N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. pow2 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-tan.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-tan.f64 N/A

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

   17. lower--.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 87.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;x + \left(t\_0 - \frac{1}{\frac{1}{\tan a}}\right)\\ \mathbf{elif}\;\tan a \leq 5 \cdot 10^{-69}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{\mathsf{fma}\left(\tan z, -\tan y, 1\right)} - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t\_0 - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (<= (tan a) -0.02)
     (+ x (- t_0 (/ 1.0 (/ 1.0 (tan a)))))
     (if (<= (tan a) 5e-69)
       (+
        x
        (-
         (/ (+ (tan y) (tan z)) (fma (tan z) (- (tan y)) 1.0))
         (fma (* a a) (* a 0.3333333333333333) a)))
       (+ x (- t_0 (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if (tan(a) <= -0.02) {
		tmp = x + (t_0 - (1.0 / (1.0 / tan(a))));
	} else if (tan(a) <= 5e-69) {
		tmp = x + (((tan(y) + tan(z)) / fma(tan(z), -tan(y), 1.0)) - fma((a * a), (a * 0.3333333333333333), a));
	} else {
		tmp = x + (t_0 - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if (tan(a) <= -0.02)
		tmp = Float64(x + Float64(t_0 - Float64(1.0 / Float64(1.0 / tan(a)))));
	elseif (tan(a) <= 5e-69)
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / fma(tan(z), Float64(-tan(y)), 1.0)) - fma(Float64(a * a), Float64(a * 0.3333333333333333), a)));
	else
		tmp = Float64(x + Float64(t_0 - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.02], N[(x + N[(t$95$0 - N[(1.0 / N[(1.0 / N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 5e-69], N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[z], $MachinePrecision] * (-N[Tan[y], $MachinePrecision]) + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0200000000000000004

   1. Initial program 73.5%

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

      1. tan-quot N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. clear-num N/A

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

      5. tan-quot N/A

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

      6. lift-tan.f64 N/A

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

      7. lower-/.f64 73.5

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

   4. Applied rewrites 73.5%

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

   if -0.0200000000000000004 < (tan.f64 a) < 5.00000000000000033e-69

   1. Initial program 78.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot \frac{1}{3}\right) \cdot {a}^{2}} + a \cdot 1\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{{a}^{2} \cdot \left(a \cdot \frac{1}{3}\right)} + a \cdot 1\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left({a}^{2} \cdot \left(a \cdot \frac{1}{3}\right) + \color{blue}{a}\right)\right) \]

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)}\right) \]
   6. Step-by-step derivation

      1. tan-sum N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. lower-+.f64 99.7

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{1 + \left(\mathsf{neg}\left(\tan y \cdot \tan z\right)\right)}} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      12. +-commutative N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{\left(\mathsf{neg}\left(\tan y \cdot \tan z\right)\right) + 1}} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{\left(\mathsf{neg}\left(\color{blue}{\tan y \cdot \tan z}\right)\right) + 1} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      14. *-commutative N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{\left(\mathsf{neg}\left(\color{blue}{\tan z \cdot \tan y}\right)\right) + 1} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

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

          \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{\tan z \cdot \left(\mathsf{neg}\left(\tan y\right)\right)} + 1} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{\mathsf{fma}\left(\tan z, \mathsf{neg}\left(\tan y\right), 1\right)}} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      17. lower-neg.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if 5.00000000000000033e-69 < (tan.f64 a)

   1. Initial program 82.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.
4. 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 78.3%

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

   1. tan-sum N/A

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

   2. lower-/.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-tan.f64 N/A

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

   5. lower-tan.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. lower-tan.f64 99.7

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

4. Applied rewrites 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: 88.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0105:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\sin \left(y + z\right)}{x \cdot \cos \left(y + z\right)} - \frac{\sin a}{x \cdot \cos a}, x\right)\\ \mathbf{elif}\;a \leq 4.7 \cdot 10^{-66}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{\mathsf{fma}\left(\tan z, -\tan y, 1\right)} - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \frac{1}{\frac{1}{\tan a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -0.0105)
   (fma
    x
    (- (/ (sin (+ y z)) (* x (cos (+ y z)))) (/ (sin a) (* x (cos a))))
    x)
   (if (<= a 4.7e-66)
     (+
      x
      (-
       (/ (+ (tan y) (tan z)) (fma (tan z) (- (tan y)) 1.0))
       (fma (* a a) (* a 0.3333333333333333) a)))
     (+ x (- (tan (+ y z)) (/ 1.0 (/ 1.0 (tan a))))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -0.0105) {
		tmp = fma(x, ((sin((y + z)) / (x * cos((y + z)))) - (sin(a) / (x * cos(a)))), x);
	} else if (a <= 4.7e-66) {
		tmp = x + (((tan(y) + tan(z)) / fma(tan(z), -tan(y), 1.0)) - fma((a * a), (a * 0.3333333333333333), a));
	} else {
		tmp = x + (tan((y + z)) - (1.0 / (1.0 / tan(a))));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -0.0105)
		tmp = fma(x, Float64(Float64(sin(Float64(y + z)) / Float64(x * cos(Float64(y + z)))) - Float64(sin(a) / Float64(x * cos(a)))), x);
	elseif (a <= 4.7e-66)
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / fma(tan(z), Float64(-tan(y)), 1.0)) - fma(Float64(a * a), Float64(a * 0.3333333333333333), a)));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) - Float64(1.0 / Float64(1.0 / tan(a)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -0.0105], N[(x * N[(N[(N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision] / N[(x * N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] / N[(x * N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 4.7e-66], N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[z], $MachinePrecision] * (-N[Tan[y], $MachinePrecision]) + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(1.0 / N[(1.0 / N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0105000000000000007

   1. Initial program 73.1%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l/ N/A

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

      6. associate-/l/ N/A

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

      7. div-sub N/A

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

      8. lower-fma.f64 N/A

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

   5. Applied rewrites 73.1%

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

   if -0.0105000000000000007 < a < 4.6999999999999999e-66

   1. Initial program 78.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot \frac{1}{3}\right) \cdot {a}^{2}} + a \cdot 1\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{{a}^{2} \cdot \left(a \cdot \frac{1}{3}\right)} + a \cdot 1\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left({a}^{2} \cdot \left(a \cdot \frac{1}{3}\right) + \color{blue}{a}\right)\right) \]

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)}\right) \]
   6. Step-by-step derivation

      1. tan-sum N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. lower-+.f64 99.7

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{1 + \left(\mathsf{neg}\left(\tan y \cdot \tan z\right)\right)}} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      12. +-commutative N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{\left(\mathsf{neg}\left(\tan y \cdot \tan z\right)\right) + 1}} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{\left(\mathsf{neg}\left(\color{blue}{\tan y \cdot \tan z}\right)\right) + 1} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      14. *-commutative N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{\left(\mathsf{neg}\left(\color{blue}{\tan z \cdot \tan y}\right)\right) + 1} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

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

          \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{\tan z \cdot \left(\mathsf{neg}\left(\tan y\right)\right)} + 1} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{\color{blue}{\mathsf{fma}\left(\tan z, \mathsf{neg}\left(\tan y\right), 1\right)}} - \mathsf{fma}\left(a \cdot a, a \cdot \frac{1}{3}, a\right)\right) \]

      17. lower-neg.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if 4.6999999999999999e-66 < a

   1. Initial program 82.8%

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

      1. tan-quot N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. clear-num N/A

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

      5. tan-quot N/A

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

      6. lift-tan.f64 N/A

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

      7. lower-/.f64 82.8

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

   4. Applied rewrites 82.8%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0105:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\sin \left(y + z\right)}{x \cdot \cos \left(y + z\right)} - \frac{\sin a}{x \cdot \cos a}, x\right)\\ \mathbf{elif}\;a \leq 4.7 \cdot 10^{-66}:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{\mathsf{fma}\left(\tan z, -\tan y, 1\right)} - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \frac{1}{\frac{1}{\tan a}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

Alternative 7: 50.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{1}{x}}\\ \mathbf{if}\;a \leq -1.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.6:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(a \cdot a, 0.05396825396825397, 0.13333333333333333\right), 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ 1.0 x))))
   (if (<= a -1.6)
     t_0
     (if (<= a 1.6)
       (+
        x
        (-
         (tan (+ y z))
         (fma
          (fma
           a
           (* a (fma (* a a) 0.05396825396825397 0.13333333333333333))
           0.3333333333333333)
          (* a (* a a))
          a)))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = 1.0 / (1.0 / x);
	double tmp;
	if (a <= -1.6) {
		tmp = t_0;
	} else if (a <= 1.6) {
		tmp = x + (tan((y + z)) - fma(fma(a, (a * fma((a * a), 0.05396825396825397, 0.13333333333333333)), 0.3333333333333333), (a * (a * a)), a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(1.0 / Float64(1.0 / x))
	tmp = 0.0
	if (a <= -1.6)
		tmp = t_0;
	elseif (a <= 1.6)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(fma(a, Float64(a * fma(Float64(a * a), 0.05396825396825397, 0.13333333333333333)), 0.3333333333333333), Float64(a * Float64(a * a)), a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.6], t$95$0, If[LessEqual[a, 1.6], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(a * N[(a * N[(N[(a * a), $MachinePrecision] * 0.05396825396825397 + 0.13333333333333333), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.6000000000000001 or 1.6000000000000001 < a

   1. Initial program 76.8%

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

      1. lift-+.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}} \]

      6. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}}} \]

      8. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}}}} \]

   4. Applied rewrites 76.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 21.8

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   7. Applied rewrites 21.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   if -1.6000000000000001 < a < 1.6000000000000001

   1. Initial program 80.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left({a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right) + 1\right)}\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(\left({a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right) \cdot a + 1 \cdot a\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{a \cdot \left({a}^{2} \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)\right)} + 1 \cdot a\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot {a}^{2}\right) \cdot \left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right)} + 1 \cdot a\right)\right) \]

      5. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right) \cdot \left(a \cdot {a}^{2}\right)} + 1 \cdot a\right)\right) \]

      6. *-lft-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\left(\frac{1}{3} + {a}^{2} \cdot \left(\frac{2}{15} + \frac{17}{315} \cdot {a}^{2}\right)\right) \cdot \left(a \cdot {a}^{2}\right) + \color{blue}{a}\right)\right) \]

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 79.8%

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

Alternative 8: 50.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{1}{x}}\\ \mathbf{if}\;a \leq -1.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.6:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot 0.13333333333333333, 0.3333333333333333\right), a \cdot \left(a \cdot a\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ 1.0 x))))
   (if (<= a -1.6)
     t_0
     (if (<= a 1.6)
       (+
        x
        (-
         (tan (+ y z))
         (fma
          (fma a (* a 0.13333333333333333) 0.3333333333333333)
          (* a (* a a))
          a)))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = 1.0 / (1.0 / x);
	double tmp;
	if (a <= -1.6) {
		tmp = t_0;
	} else if (a <= 1.6) {
		tmp = x + (tan((y + z)) - fma(fma(a, (a * 0.13333333333333333), 0.3333333333333333), (a * (a * a)), a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(1.0 / Float64(1.0 / x))
	tmp = 0.0
	if (a <= -1.6)
		tmp = t_0;
	elseif (a <= 1.6)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(fma(a, Float64(a * 0.13333333333333333), 0.3333333333333333), Float64(a * Float64(a * a)), a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.6], t$95$0, If[LessEqual[a, 1.6], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(a * N[(a * 0.13333333333333333), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.6000000000000001 or 1.6000000000000001 < a

   1. Initial program 76.8%

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

      1. lift-+.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}} \]

      6. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}}} \]

      8. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}}}} \]

   4. Applied rewrites 76.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 21.8

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   7. Applied rewrites 21.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   if -1.6000000000000001 < a < 1.6000000000000001

   1. Initial program 80.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left({a}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right) + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left({a}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right)\right) + a \cdot 1\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot {a}^{2}\right) \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right)} + a \cdot 1\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right) \cdot \left(a \cdot {a}^{2}\right)} + a \cdot 1\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\left(\frac{1}{3} + \frac{2}{15} \cdot {a}^{2}\right) \cdot \left(a \cdot {a}^{2}\right) + \color{blue}{a}\right)\right) \]

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

          \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot \frac{2}{15}, \frac{1}{3}\right), a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

      15. lower-*.f64 79.7

          \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(a, a \cdot 0.13333333333333333, 0.3333333333333333\right), a \cdot \color{blue}{\left(a \cdot a\right)}, a\right)\right) \]

   5. Applied rewrites 79.7%

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

Alternative 9: 50.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{1}{x}}\\ \mathbf{if}\;a \leq -1.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.6:\\ \;\;\;\;\tan \left(y + z\right) - \left(\mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ 1.0 x))))
   (if (<= a -1.5)
     t_0
     (if (<= a 1.6)
       (- (tan (+ y z)) (- (fma (* a a) (* a 0.3333333333333333) a) x))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = 1.0 / (1.0 / x);
	double tmp;
	if (a <= -1.5) {
		tmp = t_0;
	} else if (a <= 1.6) {
		tmp = tan((y + z)) - (fma((a * a), (a * 0.3333333333333333), a) - x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(1.0 / Float64(1.0 / x))
	tmp = 0.0
	if (a <= -1.5)
		tmp = t_0;
	elseif (a <= 1.6)
		tmp = Float64(tan(Float64(y + z)) - Float64(fma(Float64(a * a), Float64(a * 0.3333333333333333), a) - x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.5], t$95$0, If[LessEqual[a, 1.6], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(N[(a * a), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.5 or 1.6000000000000001 < a

   1. Initial program 76.8%

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

      1. lift-+.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}} \]

      6. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}}} \]

      8. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}}}} \]

   4. Applied rewrites 76.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 21.8

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   7. Applied rewrites 21.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   if -1.5 < a < 1.6000000000000001

   1. Initial program 80.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot \frac{1}{3}\right) \cdot {a}^{2}} + a \cdot 1\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{{a}^{2} \cdot \left(a \cdot \frac{1}{3}\right)} + a \cdot 1\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left({a}^{2} \cdot \left(a \cdot \frac{1}{3}\right) + \color{blue}{a}\right)\right) \]

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)}\right) \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto x + \left(\tan \color{blue}{\left(y + z\right)} - \left(\left(a \cdot a\right) \cdot \left(a \cdot \frac{1}{3}\right) + a\right)\right) \]

      2. lift-tan.f64 N/A

         \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \left(\left(a \cdot a\right) \cdot \left(a \cdot \frac{1}{3}\right) + a\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot a\right)} \cdot \left(a \cdot \frac{1}{3}\right) + a\right)\right) \]

      4. lift-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot \frac{1}{3}\right)} + a\right)\right) \]

      5. lift-fma.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. lift--.f64 N/A

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

      9. associate-+l- N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 79.6

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

   7. Applied rewrites 79.6%

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

Alternative 10: 50.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{1}{x}}\\ \mathbf{if}\;a \leq -1.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.6:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ 1.0 x))))
   (if (<= a -1.5)
     t_0
     (if (<= a 1.6)
       (+ x (- (tan (+ y z)) (fma (* a a) (* a 0.3333333333333333) a)))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = 1.0 / (1.0 / x);
	double tmp;
	if (a <= -1.5) {
		tmp = t_0;
	} else if (a <= 1.6) {
		tmp = x + (tan((y + z)) - fma((a * a), (a * 0.3333333333333333), a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(1.0 / Float64(1.0 / x))
	tmp = 0.0
	if (a <= -1.5)
		tmp = t_0;
	elseif (a <= 1.6)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - fma(Float64(a * a), Float64(a * 0.3333333333333333), a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.5], t$95$0, If[LessEqual[a, 1.6], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.5 or 1.6000000000000001 < a

   1. Initial program 76.8%

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

      1. lift-+.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}} \]

      6. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}}} \]

      8. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}}}} \]

   4. Applied rewrites 76.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 21.8

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   7. Applied rewrites 21.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   if -1.5 < a < 1.6000000000000001

   1. Initial program 80.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot \frac{1}{3}\right) \cdot {a}^{2}} + a \cdot 1\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{{a}^{2} \cdot \left(a \cdot \frac{1}{3}\right)} + a \cdot 1\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left({a}^{2} \cdot \left(a \cdot \frac{1}{3}\right) + \color{blue}{a}\right)\right) \]

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

Alternative 11: 50.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{1}{x}}\\ \mathbf{if}\;a \leq -6.5 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.6:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - 0.3333333333333333 \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ 1.0 x))))
   (if (<= a -6.5e-21)
     t_0
     (if (<= a 1.6)
       (+ x (- (tan (+ y z)) (* 0.3333333333333333 (* a (* a a)))))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = 1.0 / (1.0 / x);
	double tmp;
	if (a <= -6.5e-21) {
		tmp = t_0;
	} else if (a <= 1.6) {
		tmp = x + (tan((y + z)) - (0.3333333333333333 * (a * (a * 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 = 1.0d0 / (1.0d0 / x)
    if (a <= (-6.5d-21)) then
        tmp = t_0
    else if (a <= 1.6d0) then
        tmp = x + (tan((y + z)) - (0.3333333333333333d0 * (a * (a * 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 = 1.0 / (1.0 / x);
	double tmp;
	if (a <= -6.5e-21) {
		tmp = t_0;
	} else if (a <= 1.6) {
		tmp = x + (Math.tan((y + z)) - (0.3333333333333333 * (a * (a * a))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = 1.0 / (1.0 / x)
	tmp = 0
	if a <= -6.5e-21:
		tmp = t_0
	elif a <= 1.6:
		tmp = x + (math.tan((y + z)) - (0.3333333333333333 * (a * (a * a))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(1.0 / Float64(1.0 / x))
	tmp = 0.0
	if (a <= -6.5e-21)
		tmp = t_0;
	elseif (a <= 1.6)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - Float64(0.3333333333333333 * Float64(a * Float64(a * a)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = 1.0 / (1.0 / x);
	tmp = 0.0;
	if (a <= -6.5e-21)
		tmp = t_0;
	elseif (a <= 1.6)
		tmp = x + (tan((y + z)) - (0.3333333333333333 * (a * (a * a))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.5e-21], t$95$0, If[LessEqual[a, 1.6], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(0.3333333333333333 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.49999999999999987e-21 or 1.6000000000000001 < a

   1. Initial program 76.2%

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

      1. lift-+.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}} \]

      6. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}}} \]

      8. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}}}} \]

   4. Applied rewrites 75.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 22.2

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   7. Applied rewrites 22.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   if -6.49999999999999987e-21 < a < 1.6000000000000001

   1. Initial program 81.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot \frac{1}{3}\right) \cdot {a}^{2}} + a \cdot 1\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{{a}^{2} \cdot \left(a \cdot \frac{1}{3}\right)} + a \cdot 1\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left({a}^{2} \cdot \left(a \cdot \frac{1}{3}\right) + \color{blue}{a}\right)\right) \]

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 80.4

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

   5. Applied rewrites 80.4%

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

   6. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\frac{1}{3} \cdot {a}^{3}}\right) \]

      2. cube-mult N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \frac{1}{3} \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)}\right) \]

      3. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \frac{1}{3} \cdot \left(a \cdot \color{blue}{{a}^{2}}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \frac{1}{3} \cdot \color{blue}{\left(a \cdot {a}^{2}\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \frac{1}{3} \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \]

      6. lower-*.f64 80.1

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

   8. Applied rewrites 80.1%

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

Alternative 12: 41.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{1}{x}}\\ \mathbf{if}\;a \leq -1.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.6:\\ \;\;\;\;\tan y + \left(x - \mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ 1.0 x))))
   (if (<= a -1.5)
     t_0
     (if (<= a 1.6)
       (+ (tan y) (- x (fma a (* a (* a 0.3333333333333333)) a)))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = 1.0 / (1.0 / x);
	double tmp;
	if (a <= -1.5) {
		tmp = t_0;
	} else if (a <= 1.6) {
		tmp = tan(y) + (x - fma(a, (a * (a * 0.3333333333333333)), a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(1.0 / Float64(1.0 / x))
	tmp = 0.0
	if (a <= -1.5)
		tmp = t_0;
	elseif (a <= 1.6)
		tmp = Float64(tan(y) + Float64(x - fma(a, Float64(a * Float64(a * 0.3333333333333333)), a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.5], t$95$0, If[LessEqual[a, 1.6], N[(N[Tan[y], $MachinePrecision] + N[(x - N[(a * N[(a * N[(a * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.5 or 1.6000000000000001 < a

   1. Initial program 76.8%

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

      1. lift-+.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift--.f64 N/A

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

      5. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}} \]

      6. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}}} \]

      8. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}}}} \]

   4. Applied rewrites 76.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 21.8

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   7. Applied rewrites 21.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   if -1.5 < a < 1.6000000000000001

   1. Initial program 80.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

      3. associate-*r* N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot \frac{1}{3}\right) \cdot {a}^{2}} + a \cdot 1\right)\right) \]

      4. *-commutative N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{{a}^{2} \cdot \left(a \cdot \frac{1}{3}\right)} + a \cdot 1\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto x + \left(\tan \left(y + z\right) - \left({a}^{2} \cdot \left(a \cdot \frac{1}{3}\right) + \color{blue}{a}\right)\right) \]

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x + \frac{\sin y}{\cos y}\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right)} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(x + \frac{\sin y}{\cos y}\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} + x\right)} - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} + x\right)} - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sin y}{\cos y}} + x\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sin y}}{\cos y} + x\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto \left(\frac{\sin y}{\color{blue}{\cos y}} + x\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\frac{\sin y}{\cos y} + x\right) - \color{blue}{\left(\frac{1}{3} \cdot {a}^{3} + a\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\frac{\sin y}{\cos y} + x\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {a}^{3}, a\right)} \]

      9. cube-mult N/A

         \[\leadsto \left(\frac{\sin y}{\cos y} + x\right) - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot \left(a \cdot a\right)}, a\right) \]

      10. unpow2 N/A

          \[\leadsto \left(\frac{\sin y}{\cos y} + x\right) - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{{a}^{2}}, a\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\frac{\sin y}{\cos y} + x\right) - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot {a}^{2}}, a\right) \]

      12. unpow2 N/A

          \[\leadsto \left(\frac{\sin y}{\cos y} + x\right) - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right) \]

      13. lower-*.f64 62.9

          \[\leadsto \left(\frac{\sin y}{\cos y} + x\right) - \mathsf{fma}\left(0.3333333333333333, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right) \]

   8. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} + x\right) - \mathsf{fma}\left(0.3333333333333333, a \cdot \left(a \cdot a\right), a\right)} \]
   9. Step-by-step derivation

      1. tan-quot N/A

         \[\leadsto \left(\color{blue}{\tan y} + x\right) - \left(\frac{1}{3} \cdot \left(a \cdot \left(a \cdot a\right)\right) + a\right) \]

      2. lift-tan.f64 N/A

         \[\leadsto \left(\color{blue}{\tan y} + x\right) - \left(\frac{1}{3} \cdot \left(a \cdot \left(a \cdot a\right)\right) + a\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\tan y + x\right) - \left(\frac{1}{3} \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) + a\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\tan y + x\right) - \left(\frac{1}{3} \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} + a\right) \]

      5. lift-fma.f64 N/A

         \[\leadsto \left(\tan y + x\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{3}, a \cdot \left(a \cdot a\right), a\right)} \]

      6. associate--l+ N/A

         \[\leadsto \color{blue}{\tan y + \left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \left(a \cdot a\right), a\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \left(a \cdot a\right), a\right)\right) + \tan y} \]

      8. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \left(a \cdot a\right), a\right)\right) + \tan y} \]

   10. Applied rewrites 62.9%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.5:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{elif}\;a \leq 1.6:\\ \;\;\;\;\tan y + \left(x - \mathsf{fma}\left(a, a \cdot \left(a \cdot 0.3333333333333333\right), a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 31.8% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{x}} \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (/ 1.0 (/ 1.0 x)))

C

double code(double x, double y, double z, double a) {
	return 1.0 / (1.0 / 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 = 1.0d0 / (1.0d0 / x)
end function

Java

public static double code(double x, double y, double z, double a) {
	return 1.0 / (1.0 / x);
}

Python

def code(x, y, z, a):
	return 1.0 / (1.0 / x)

Julia

function code(x, y, z, a)
	return Float64(1.0 / Float64(1.0 / x))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = 1.0 / (1.0 / x);
end

Wolfram

code[x_, y_, z_, a_] := N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.3%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. lift--.f64 N/A

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

   5. flip3-+ N/A

      \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}} \]

   6. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}}} \]

   8. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {\left(\tan \left(y + z\right) - \tan a\right)}^{3}}{x \cdot x + \left(\left(\tan \left(y + z\right) - \tan a\right) \cdot \left(\tan \left(y + z\right) - \tan a\right) - x \cdot \left(\tan \left(y + z\right) - \tan a\right)\right)}}}} \]

4. Applied rewrites 78.1%

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

5. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
6. Step-by-step derivation

   1. lower-/.f64 32.7

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

7. Applied rewrites 32.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
8. Add Preprocessing

Alternative 14: 22.3% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ x - \mathsf{fma}\left(0.3333333333333333, a \cdot \left(a \cdot a\right), a\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (- x (fma 0.3333333333333333 (* a (* a a)) a)))

C

double code(double x, double y, double z, double a) {
	return x - fma(0.3333333333333333, (a * (a * a)), a);
}

Julia

function code(x, y, z, a)
	return Float64(x - fma(0.3333333333333333, Float64(a * Float64(a * a)), a))
end

Wolfram

code[x_, y_, z_, a_] := N[(x - N[(0.3333333333333333 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.3%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

   2. distribute-lft-in N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot \frac{1}{3}\right) \cdot {a}^{2}} + a \cdot 1\right)\right) \]

   4. *-commutative N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{{a}^{2} \cdot \left(a \cdot \frac{1}{3}\right)} + a \cdot 1\right)\right) \]

   5. *-rgt-identity N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \left({a}^{2} \cdot \left(a \cdot \frac{1}{3}\right) + \color{blue}{a}\right)\right) \]

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 38.1

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

5. Applied rewrites 38.1%

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

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\left(x + \frac{\sin y}{\cos y}\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right)} \]
7. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(x + \frac{\sin y}{\cos y}\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} + x\right)} - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} + x\right)} - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

   4. lower-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{\sin y}{\cos y}} + x\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

   5. lower-sin.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{\sin y}}{\cos y} + x\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

   6. lower-cos.f64 N/A

      \[\leadsto \left(\frac{\sin y}{\color{blue}{\cos y}} + x\right) - \left(a + \frac{1}{3} \cdot {a}^{3}\right) \]

   7. +-commutative N/A

      \[\leadsto \left(\frac{\sin y}{\cos y} + x\right) - \color{blue}{\left(\frac{1}{3} \cdot {a}^{3} + a\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\frac{\sin y}{\cos y} + x\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {a}^{3}, a\right)} \]

   9. cube-mult N/A

      \[\leadsto \left(\frac{\sin y}{\cos y} + x\right) - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot \left(a \cdot a\right)}, a\right) \]

   10. unpow2 N/A

       \[\leadsto \left(\frac{\sin y}{\cos y} + x\right) - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{{a}^{2}}, a\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\frac{\sin y}{\cos y} + x\right) - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot {a}^{2}}, a\right) \]

   12. unpow2 N/A

       \[\leadsto \left(\frac{\sin y}{\cos y} + x\right) - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right) \]

   13. lower-*.f64 30.4

       \[\leadsto \left(\frac{\sin y}{\cos y} + x\right) - \mathsf{fma}\left(0.3333333333333333, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right) \]

8. Applied rewrites 30.4%

   \[\leadsto \color{blue}{\left(\frac{\sin y}{\cos y} + x\right) - \mathsf{fma}\left(0.3333333333333333, a \cdot \left(a \cdot a\right), a\right)} \]

9. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)} \]
10. Step-by-step derivation

    1. lower--.f64 N/A

       \[\leadsto \color{blue}{x - \left(a + \frac{1}{3} \cdot {a}^{3}\right)} \]

    2. +-commutative N/A

       \[\leadsto x - \color{blue}{\left(\frac{1}{3} \cdot {a}^{3} + a\right)} \]

    3. lower-fma.f64 N/A

       \[\leadsto x - \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {a}^{3}, a\right)} \]

    4. cube-mult N/A

       \[\leadsto x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot \left(a \cdot a\right)}, a\right) \]

    5. unpow2 N/A

       \[\leadsto x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{{a}^{2}}, a\right) \]

    6. lower-*.f64 N/A

       \[\leadsto x - \mathsf{fma}\left(\frac{1}{3}, \color{blue}{a \cdot {a}^{2}}, a\right) \]

    7. unpow2 N/A

       \[\leadsto x - \mathsf{fma}\left(\frac{1}{3}, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right) \]

    8. lower-*.f64 22.5

       \[\leadsto x - \mathsf{fma}\left(0.3333333333333333, a \cdot \color{blue}{\left(a \cdot a\right)}, a\right) \]

11. Applied rewrites 22.5%

    \[\leadsto \color{blue}{x - \mathsf{fma}\left(0.3333333333333333, a \cdot \left(a \cdot a\right), a\right)} \]
12. Add Preprocessing

Alternative 15: 22.1% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ x + \left(a \cdot \left(a \cdot a\right)\right) \cdot -0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ x (* (* a (* a a)) -0.3333333333333333)))

C

double code(double x, double y, double z, double a) {
	return x + ((a * (a * a)) * -0.3333333333333333);
}

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 + ((a * (a * a)) * (-0.3333333333333333d0))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + ((a * (a * a)) * -0.3333333333333333);
}

Python

def code(x, y, z, a):
	return x + ((a * (a * a)) * -0.3333333333333333)

Julia

function code(x, y, z, a)
	return Float64(x + Float64(Float64(a * Float64(a * a)) * -0.3333333333333333))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + ((a * (a * a)) * -0.3333333333333333);
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.3%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

   2. distribute-lft-in N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot \frac{1}{3}\right) \cdot {a}^{2}} + a \cdot 1\right)\right) \]

   4. *-commutative N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{{a}^{2} \cdot \left(a \cdot \frac{1}{3}\right)} + a \cdot 1\right)\right) \]

   5. *-rgt-identity N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \left({a}^{2} \cdot \left(a \cdot \frac{1}{3}\right) + \color{blue}{a}\right)\right) \]

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 38.1

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

5. Applied rewrites 38.1%

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

6. Taylor expanded in a around inf

   \[\leadsto x + \color{blue}{\frac{-1}{3} \cdot {a}^{3}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto x + \color{blue}{\frac{-1}{3} \cdot {a}^{3}} \]

   2. cube-mult N/A

      \[\leadsto x + \frac{-1}{3} \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} \]

   3. unpow2 N/A

      \[\leadsto x + \frac{-1}{3} \cdot \left(a \cdot \color{blue}{{a}^{2}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto x + \frac{-1}{3} \cdot \color{blue}{\left(a \cdot {a}^{2}\right)} \]

   5. unpow2 N/A

      \[\leadsto x + \frac{-1}{3} \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   6. lower-*.f64 22.4

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

8. Applied rewrites 22.4%

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

9. Final simplification 22.4%

   \[\leadsto x + \left(a \cdot \left(a \cdot a\right)\right) \cdot -0.3333333333333333 \]
10. Add Preprocessing

Alternative 16: 2.8% accurate, 13.1× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(\left(a \cdot a\right) \cdot -0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (* a (* (* a a) -0.3333333333333333)))

C

double code(double x, double y, double z, double a) {
	return a * ((a * a) * -0.3333333333333333);
}

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 = a * ((a * a) * (-0.3333333333333333d0))
end function

Java

public static double code(double x, double y, double z, double a) {
	return a * ((a * a) * -0.3333333333333333);
}

Python

def code(x, y, z, a):
	return a * ((a * a) * -0.3333333333333333)

Julia

function code(x, y, z, a)
	return Float64(a * Float64(Float64(a * a) * -0.3333333333333333))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = a * ((a * a) * -0.3333333333333333);
end

Wolfram

code[x_, y_, z_, a_] := N[(a * N[(N[(a * a), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.3%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

   2. distribute-lft-in N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot \frac{1}{3}\right) \cdot {a}^{2}} + a \cdot 1\right)\right) \]

   4. *-commutative N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{{a}^{2} \cdot \left(a \cdot \frac{1}{3}\right)} + a \cdot 1\right)\right) \]

   5. *-rgt-identity N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \left({a}^{2} \cdot \left(a \cdot \frac{1}{3}\right) + \color{blue}{a}\right)\right) \]

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 38.1

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

5. Applied rewrites 38.1%

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

6. Taylor expanded in a around inf

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot {a}^{3}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot {a}^{3}} \]

   2. cube-mult N/A

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} \]

   3. unpow2 N/A

      \[\leadsto \frac{-1}{3} \cdot \left(a \cdot \color{blue}{{a}^{2}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\left(a \cdot {a}^{2}\right)} \]

   5. unpow2 N/A

      \[\leadsto \frac{-1}{3} \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   6. lower-*.f64 3.0

      \[\leadsto -0.3333333333333333 \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

8. Applied rewrites 3.0%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot a\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{-1}{3} \cdot \left(\color{blue}{\left(a \cdot a\right)} \cdot a\right) \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{3} \cdot \left(a \cdot a\right)\right) \cdot a} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{3} \cdot \left(a \cdot a\right)\right) \cdot a} \]

   5. lower-*.f64 3.0

      \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot \left(a \cdot a\right)\right)} \cdot a \]

10. Applied rewrites 3.0%

    \[\leadsto \color{blue}{\left(-0.3333333333333333 \cdot \left(a \cdot a\right)\right) \cdot a} \]

11. Final simplification 3.0%

    \[\leadsto a \cdot \left(\left(a \cdot a\right) \cdot -0.3333333333333333\right) \]
12. Add Preprocessing

Alternative 17: 2.8% accurate, 13.1× speedup

Math

\[\begin{array}{l} \\ \left(a \cdot \left(a \cdot a\right)\right) \cdot -0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (* (* a (* a a)) -0.3333333333333333))

C

double code(double x, double y, double z, double a) {
	return (a * (a * a)) * -0.3333333333333333;
}

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 = (a * (a * a)) * (-0.3333333333333333d0)
end function

Java

public static double code(double x, double y, double z, double a) {
	return (a * (a * a)) * -0.3333333333333333;
}

Python

def code(x, y, z, a):
	return (a * (a * a)) * -0.3333333333333333

Julia

function code(x, y, z, a)
	return Float64(Float64(a * Float64(a * a)) * -0.3333333333333333)
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = (a * (a * a)) * -0.3333333333333333;
end

Wolfram

code[x_, y_, z_, a_] := N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 78.3%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - a \cdot \color{blue}{\left(\frac{1}{3} \cdot {a}^{2} + 1\right)}\right) \]

   2. distribute-lft-in N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\left(a \cdot \left(\frac{1}{3} \cdot {a}^{2}\right) + a \cdot 1\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{\left(a \cdot \frac{1}{3}\right) \cdot {a}^{2}} + a \cdot 1\right)\right) \]

   4. *-commutative N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \left(\color{blue}{{a}^{2} \cdot \left(a \cdot \frac{1}{3}\right)} + a \cdot 1\right)\right) \]

   5. *-rgt-identity N/A

      \[\leadsto x + \left(\tan \left(y + z\right) - \left({a}^{2} \cdot \left(a \cdot \frac{1}{3}\right) + \color{blue}{a}\right)\right) \]

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 38.1

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

5. Applied rewrites 38.1%

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

6. Taylor expanded in a around inf

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot {a}^{3}} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot {a}^{3}} \]

   2. cube-mult N/A

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)} \]

   3. unpow2 N/A

      \[\leadsto \frac{-1}{3} \cdot \left(a \cdot \color{blue}{{a}^{2}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{-1}{3} \cdot \color{blue}{\left(a \cdot {a}^{2}\right)} \]

   5. unpow2 N/A

      \[\leadsto \frac{-1}{3} \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   6. lower-*.f64 3.0

      \[\leadsto -0.3333333333333333 \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

8. Applied rewrites 3.0%

   \[\leadsto \color{blue}{-0.3333333333333333 \cdot \left(a \cdot \left(a \cdot a\right)\right)} \]

9. Final simplification 3.0%

   \[\leadsto \left(a \cdot \left(a \cdot a\right)\right) \cdot -0.3333333333333333 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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))))