tan-example (used to crash)

Percentage Accurate: 79.3% → 99.7%

Time: 31.2s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. lift-tan.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-tan.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-tan.f64 N/A

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

   11. lower-tan.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-quot N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. lower-cos.f64 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. tan-quot N/A

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

   7. lift-tan.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-tan.f64 N/A

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

   10. tan-quot N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. clear-num N/A

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

   14. un-div-inv N/A

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

   15. lower-/.f64 N/A

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

   16. clear-num N/A

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

   17. lift-sin.f64 N/A

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

   18. lift-cos.f64 N/A

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

   19. tan-quot N/A

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

   20. lift-tan.f64 N/A

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

   21. lower-/.f64 99.6

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

8. Applied rewrites 99.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 99.6

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. lift-tan.f64 N/A

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

   9. tan-quot N/A

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

   10. clear-num N/A

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

   11. lower-/.f64 N/A

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

   12. div-inv N/A

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

   13. clear-num N/A

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

   14. tan-quot N/A

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

   15. lift-tan.f64 N/A

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

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

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

   17. lower-fma.f64 N/A

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

   18. lower-neg.f64 99.7

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

10. Applied rewrites 99.7%

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

Alternative 2: 89.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan z + \tan y\\ \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;x + \left(\tan \left(z + y\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 0.0001:\\ \;\;\;\;x + \left(\frac{t\_0}{1 - \tan z \cdot \tan y} - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{t\_0}{1 - \frac{\tan z}{\frac{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0021164021164021165, -0.022222222222222223\right), -0.3333333333333333\right), 1\right)}{y}}} - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan z) (tan y))))
   (if (<= (tan a) -0.02)
     (+ x (- (tan (+ z y)) (tan a)))
     (if (<= (tan a) 0.0001)
       (+
        x
        (-
         (/ t_0 (- 1.0 (* (tan z) (tan y))))
         (fma (* a a) (* a 0.3333333333333333) a)))
       (+
        x
        (-
         (/
          t_0
          (-
           1.0
           (/
            (tan z)
            (/
             (fma
              (* y y)
              (fma
               y
               (* y (fma (* y y) -0.0021164021164021165 -0.022222222222222223))
               -0.3333333333333333)
              1.0)
             y))))
         (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(z) + tan(y);
	double tmp;
	if (tan(a) <= -0.02) {
		tmp = x + (tan((z + y)) - tan(a));
	} else if (tan(a) <= 0.0001) {
		tmp = x + ((t_0 / (1.0 - (tan(z) * tan(y)))) - fma((a * a), (a * 0.3333333333333333), a));
	} else {
		tmp = x + ((t_0 / (1.0 - (tan(z) / (fma((y * y), fma(y, (y * fma((y * y), -0.0021164021164021165, -0.022222222222222223)), -0.3333333333333333), 1.0) / y)))) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(z) + tan(y))
	tmp = 0.0
	if (tan(a) <= -0.02)
		tmp = Float64(x + Float64(tan(Float64(z + y)) - tan(a)));
	elseif (tan(a) <= 0.0001)
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(z) * tan(y)))) - fma(Float64(a * a), Float64(a * 0.3333333333333333), a)));
	else
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(z) / Float64(fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), -0.0021164021164021165, -0.022222222222222223)), -0.3333333333333333), 1.0) / y)))) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.02], N[(x + N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 0.0001], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[z], $MachinePrecision] * N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[z], $MachinePrecision] / N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * -0.0021164021164021165 + -0.022222222222222223), $MachinePrecision]), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 76.1%

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

   if -0.0200000000000000004 < (tan.f64 a) < 1.00000000000000005e-4

   1. Initial program 81.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 81.9

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

   5. Applied rewrites 81.9%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. lift-*.f64 N/A

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

      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. lift-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if 1.00000000000000005e-4 < (tan.f64 a)

   1. Initial program 76.0%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-tan.f64 N/A

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

      7. lower-tan.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-tan.f64 N/A

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

      11. lower-tan.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-quot N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-cos.f64 99.5

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

   6. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. tan-quot N/A

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

      7. lift-tan.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-tan.f64 N/A

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

      10. tan-quot N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

      15. lower-/.f64 N/A

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

      16. clear-num N/A

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

      17. lift-sin.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. tan-quot N/A

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

      20. lift-tan.f64 N/A

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

      21. lower-/.f64 99.5

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in y around 0

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

       1. lower-/.f64 N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan z}{\color{blue}{\frac{1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{-2}{945} \cdot {y}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{y}}}} - \tan a\right) \]

   11. Applied rewrites 77.0%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;x + \left(\tan \left(z + y\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 0.0001:\\ \;\;\;\;x + \left(\frac{\tan z + \tan y}{1 - \tan z \cdot \tan y} - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{\tan z + \tan y}{1 - \frac{\tan z}{\frac{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0021164021164021165, -0.022222222222222223\right), -0.3333333333333333\right), 1\right)}{y}}} - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan z + \tan y\\ \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;x + \left(\tan \left(z + y\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 0.0001:\\ \;\;\;\;x + \left(\frac{t\_0}{1 - \tan z \cdot \tan y} - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{t\_0}{1 - \frac{\tan z}{\frac{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.022222222222222223, -0.3333333333333333\right), 1\right)}{y}}} - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan z) (tan y))))
   (if (<= (tan a) -0.02)
     (+ x (- (tan (+ z y)) (tan a)))
     (if (<= (tan a) 0.0001)
       (+
        x
        (-
         (/ t_0 (- 1.0 (* (tan z) (tan y))))
         (fma (* a a) (* a 0.3333333333333333) a)))
       (+
        x
        (-
         (/
          t_0
          (-
           1.0
           (/
            (tan z)
            (/
             (fma
              (* y y)
              (fma (* y y) -0.022222222222222223 -0.3333333333333333)
              1.0)
             y))))
         (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(z) + tan(y);
	double tmp;
	if (tan(a) <= -0.02) {
		tmp = x + (tan((z + y)) - tan(a));
	} else if (tan(a) <= 0.0001) {
		tmp = x + ((t_0 / (1.0 - (tan(z) * tan(y)))) - fma((a * a), (a * 0.3333333333333333), a));
	} else {
		tmp = x + ((t_0 / (1.0 - (tan(z) / (fma((y * y), fma((y * y), -0.022222222222222223, -0.3333333333333333), 1.0) / y)))) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(z) + tan(y))
	tmp = 0.0
	if (tan(a) <= -0.02)
		tmp = Float64(x + Float64(tan(Float64(z + y)) - tan(a)));
	elseif (tan(a) <= 0.0001)
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(z) * tan(y)))) - fma(Float64(a * a), Float64(a * 0.3333333333333333), a)));
	else
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(z) / Float64(fma(Float64(y * y), fma(Float64(y * y), -0.022222222222222223, -0.3333333333333333), 1.0) / y)))) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.02], N[(x + N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 0.0001], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[z], $MachinePrecision] * N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[z], $MachinePrecision] / N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * -0.022222222222222223 + -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 76.1%

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

   if -0.0200000000000000004 < (tan.f64 a) < 1.00000000000000005e-4

   1. Initial program 81.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 81.9

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

   5. Applied rewrites 81.9%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. lift-*.f64 N/A

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

      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. lift-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if 1.00000000000000005e-4 < (tan.f64 a)

   1. Initial program 76.0%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-tan.f64 N/A

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

      7. lower-tan.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-tan.f64 N/A

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

      11. lower-tan.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-quot N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-cos.f64 99.5

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

   6. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. tan-quot N/A

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

      7. lift-tan.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-tan.f64 N/A

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

      10. tan-quot N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

      15. lower-/.f64 N/A

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

      16. clear-num N/A

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

      17. lift-sin.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. tan-quot N/A

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

      20. lift-tan.f64 N/A

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

      21. lower-/.f64 99.5

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in y around 0

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

       1. lower-/.f64 N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan z}{\color{blue}{\frac{1 + {y}^{2} \cdot \left(\frac{-1}{45} \cdot {y}^{2} - \frac{1}{3}\right)}{y}}}} - \tan a\right) \]

       2. +-commutative N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan z}{\frac{\color{blue}{{y}^{2} \cdot \left(\frac{-1}{45} \cdot {y}^{2} - \frac{1}{3}\right) + 1}}{y}}} - \tan a\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan z}{\frac{\color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{45} \cdot {y}^{2} - \frac{1}{3}, 1\right)}}{y}}} - \tan a\right) \]

       4. unpow2 N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan z}{\frac{\mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{45} \cdot {y}^{2} - \frac{1}{3}, 1\right)}{y}}} - \tan a\right) \]

       5. lower-*.f64 N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan z}{\frac{\mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{45} \cdot {y}^{2} - \frac{1}{3}, 1\right)}{y}}} - \tan a\right) \]

       6. sub-neg N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan z}{\frac{\mathsf{fma}\left(y \cdot y, \color{blue}{\frac{-1}{45} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, 1\right)}{y}}} - \tan a\right) \]

       7. *-commutative N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan z}{\frac{\mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{2} \cdot \frac{-1}{45}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), 1\right)}{y}}} - \tan a\right) \]

       8. metadata-eval N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan z}{\frac{\mathsf{fma}\left(y \cdot y, {y}^{2} \cdot \frac{-1}{45} + \color{blue}{\frac{-1}{3}}, 1\right)}{y}}} - \tan a\right) \]

       9. lower-fma.f64 N/A

          \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan z}{\frac{\mathsf{fma}\left(y \cdot y, \color{blue}{\mathsf{fma}\left({y}^{2}, \frac{-1}{45}, \frac{-1}{3}\right)}, 1\right)}{y}}} - \tan a\right) \]

       10. unpow2 N/A

           \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan z}{\frac{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{45}, \frac{-1}{3}\right), 1\right)}{y}}} - \tan a\right) \]

       11. lower-*.f64 77.0

           \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \frac{\tan z}{\frac{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(\color{blue}{y \cdot y}, -0.022222222222222223, -0.3333333333333333\right), 1\right)}{y}}} - \tan a\right) \]

   11. Applied rewrites 77.0%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;x + \left(\tan \left(z + y\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 0.0001:\\ \;\;\;\;x + \left(\frac{\tan z + \tan y}{1 - \tan z \cdot \tan y} - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{\tan z + \tan y}{1 - \frac{\tan z}{\frac{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, -0.022222222222222223, -0.3333333333333333\right), 1\right)}{y}}} - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	return x + (((tan(z) + tan(y)) / (1.0 - (tan(z) * tan(y)))) - 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(z) + tan(y)) / (1.0d0 - (tan(z) * tan(y)))) - tan(a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. lift-tan.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-tan.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-tan.f64 N/A

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

   11. lower-tan.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 5: 88.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan z + \tan y\\ \mathbf{if}\;a \leq -0.0132:\\ \;\;\;\;x + \left(t\_0 \cdot 1 - \tan a\right)\\ \mathbf{elif}\;a \leq 2300:\\ \;\;\;\;x + \left(\frac{t\_0}{1 - \tan z \cdot \tan y} - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(z + y\right) - \frac{\sin a}{\cos a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan z) (tan y))))
   (if (<= a -0.0132)
     (+ x (- (* t_0 1.0) (tan a)))
     (if (<= a 2300.0)
       (+
        x
        (-
         (/ t_0 (- 1.0 (* (tan z) (tan y))))
         (fma (* a a) (* a 0.3333333333333333) a)))
       (+ x (- (tan (+ z y)) (/ (sin a) (cos a))))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(z) + tan(y);
	double tmp;
	if (a <= -0.0132) {
		tmp = x + ((t_0 * 1.0) - tan(a));
	} else if (a <= 2300.0) {
		tmp = x + ((t_0 / (1.0 - (tan(z) * tan(y)))) - fma((a * a), (a * 0.3333333333333333), a));
	} else {
		tmp = x + (tan((z + y)) - (sin(a) / cos(a)));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(z) + tan(y))
	tmp = 0.0
	if (a <= -0.0132)
		tmp = Float64(x + Float64(Float64(t_0 * 1.0) - tan(a)));
	elseif (a <= 2300.0)
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(z) * tan(y)))) - fma(Float64(a * a), Float64(a * 0.3333333333333333), a)));
	else
		tmp = Float64(x + Float64(tan(Float64(z + y)) - Float64(sin(a) / cos(a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.0132], N[(x + N[(N[(t$95$0 * 1.0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2300.0], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[z], $MachinePrecision] * N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] * N[(a * 0.3333333333333333), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0132

   1. Initial program 72.9%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-tan.f64 N/A

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

      11. lower-tan.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-tan.f64 N/A

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

      14. lower-tan.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 73.7%

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

   if -0.0132 < a < 2300

   1. Initial program 81.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 81.3

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

   5. Applied rewrites 81.3%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lift-tan.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. lift-*.f64 N/A

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

      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. lift-/.f64 99.0

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

   7. Applied rewrites 99.0%

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

   if 2300 < a

   1. Initial program 78.9%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 78.9

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

   4. Applied rewrites 78.9%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0132:\\ \;\;\;\;x + \left(\left(\tan z + \tan y\right) \cdot 1 - \tan a\right)\\ \mathbf{elif}\;a \leq 2300:\\ \;\;\;\;x + \left(\frac{\tan z + \tan y}{1 - \tan z \cdot \tan y} - \mathsf{fma}\left(a \cdot a, a \cdot 0.3333333333333333, a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(z + y\right) - \frac{\sin a}{\cos a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan z + \tan y\\ \mathbf{if}\;a \leq -7.8 \cdot 10^{-14}:\\ \;\;\;\;x + \left(t\_0 \cdot 1 - \tan a\right)\\ \mathbf{elif}\;a \leq 2300:\\ \;\;\;\;x + \frac{t\_0}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(z + y\right) - \frac{\sin a}{\cos a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan z) (tan y))))
   (if (<= a -7.8e-14)
     (+ x (- (* t_0 1.0) (tan a)))
     (if (<= a 2300.0)
       (+ x (/ t_0 (fma (- (tan z)) (tan y) 1.0)))
       (+ x (- (tan (+ z y)) (/ (sin a) (cos a))))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(z) + tan(y);
	double tmp;
	if (a <= -7.8e-14) {
		tmp = x + ((t_0 * 1.0) - tan(a));
	} else if (a <= 2300.0) {
		tmp = x + (t_0 / fma(-tan(z), tan(y), 1.0));
	} else {
		tmp = x + (tan((z + y)) - (sin(a) / cos(a)));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(z) + tan(y))
	tmp = 0.0
	if (a <= -7.8e-14)
		tmp = Float64(x + Float64(Float64(t_0 * 1.0) - tan(a)));
	elseif (a <= 2300.0)
		tmp = Float64(x + Float64(t_0 / fma(Float64(-tan(z)), tan(y), 1.0)));
	else
		tmp = Float64(x + Float64(tan(Float64(z + y)) - Float64(sin(a) / cos(a))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -7.8e-14], N[(x + N[(N[(t$95$0 * 1.0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2300.0], N[(x + N[(t$95$0 / N[((-N[Tan[z], $MachinePrecision]) * N[Tan[y], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - N[(N[Sin[a], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -7.7999999999999996e-14

   1. Initial program 72.9%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-tan.f64 N/A

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

      11. lower-tan.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-tan.f64 N/A

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

      14. lower-tan.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 73.7%

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

   if -7.7999999999999996e-14 < a < 2300

   1. Initial program 81.2%

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

      1. lift-tan.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. tan-sum N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-tan.f64 N/A

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

      7. lower-tan.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-tan.f64 N/A

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

      11. 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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lift-tan.f64 N/A

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

      9. tan-sum N/A

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

      10. lift-+.f64 N/A

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

      11. lift-tan.f64 81.2

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

      12. unpow1 N/A

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

      13. sqr-pow N/A

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

      14. pow-prod-down N/A

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

      15. unpow2 N/A

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

      16. lift-pow.f64 N/A

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

      17. lower-pow.f64 N/A

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

   6. Applied rewrites 61.1%

      \[\leadsto x + \color{blue}{{\left({\left(\tan \left(y + z\right) - \tan a\right)}^{2}\right)}^{0.5}} \]

   7. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-+.f64 81.1

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

   9. Applied rewrites 81.1%

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

   11. Applied rewrites 98.6%

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

   if 2300 < a

   1. Initial program 78.9%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 78.9

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

   4. Applied rewrites 78.9%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{-14}:\\ \;\;\;\;x + \left(\left(\tan z + \tan y\right) \cdot 1 - \tan a\right)\\ \mathbf{elif}\;a \leq 2300:\\ \;\;\;\;x + \frac{\tan z + \tan y}{\mathsf{fma}\left(-\tan z, \tan y, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(z + y\right) - \frac{\sin a}{\cos a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. lift-tan.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. tan-sum N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-tan.f64 N/A

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

   11. lower-tan.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. lower-tan.f64 N/A

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

   14. lower-tan.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 79.1%

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

8. Final simplification 79.1%

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

Alternative 8: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Final simplification 78.9%

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

Alternative 9: 50.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

   1. lift-tan.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-tan.f64 N/A

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

   7. lower-tan.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-tan.f64 N/A

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

   11. lower-tan.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. lift-tan.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-tan.f64 N/A

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

   8. lift-tan.f64 N/A

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

   9. tan-sum N/A

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

   10. lift-+.f64 N/A

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

   11. lift-tan.f64 78.9

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

   12. unpow1 N/A

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

   13. sqr-pow N/A

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

   14. pow-prod-down N/A

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

   15. unpow2 N/A

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

   16. lift-pow.f64 N/A

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

   17. lower-pow.f64 N/A

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

6. Applied rewrites 54.6%

   \[\leadsto x + \color{blue}{{\left({\left(\tan \left(y + z\right) - \tan a\right)}^{2}\right)}^{0.5}} \]

7. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-+.f64 51.4

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

9. Applied rewrites 51.4%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lower-+.f64 51.4

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

11. Applied rewrites 51.4%

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

12. Final simplification 51.4%

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

Alternative 10: 31.6% 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 78.9%

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

   1. lift-+.f64 N/A

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

   2. flip3-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip3-+ N/A

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

   7. lift-+.f64 N/A

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

   8. lower-/.f64 78.7

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

4. Applied rewrites 78.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 31.6

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

7. Applied rewrites 31.6%

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

Reproduce

herbie shell --seed 2024221 
(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))))