tan-example (used to crash)

Percentage Accurate: 79.9% → 99.7%

Time: 16.8s

Alternatives: 9

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: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+
  x
  (-
   (/ (+ (/ (sin y) (cos y)) (tan z)) (fma (- (tan z)) (tan y) 1.0))
   (tan a))))

C

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

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(x + Float64(Float64(Float64(Float64(sin(y) / cos(y)) + tan(z)) / fma(Float64(-tan(z)), tan(y), 1.0)) - tan(a)))
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(x + N[(N[(N[(N[(N[Sin[y], $MachinePrecision] / N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[Tan[z], $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 76.9%

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

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

   5. div-add N/A

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

   6. lower-+.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-tan.f64 N/A

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

   13. lower-tan.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower--.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lower-tan.f64 N/A

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

   20. lower-tan.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. div-add-rev N/A

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

   6. lift-tan.f64 N/A

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

   7. lift-tan.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. lift-tan.f64 N/A

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

   11. lower-+.f64 99.7

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

   12. lift--.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. fp-cancel-sub-sign-inv N/A

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

   15. +-commutative N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-neg.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-cos.f64 99.7

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

8. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+ x (- (/ (+ (tan y) (tan z)) (fma (- (tan z)) (tan y) 1.0)) (tan a))))

C

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

Julia

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

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := 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[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

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

   5. div-add N/A

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

   6. lower-+.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-tan.f64 N/A

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

   13. lower-tan.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower--.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lower-tan.f64 N/A

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

   20. lower-tan.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. div-add-rev N/A

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

   6. lift-tan.f64 N/A

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

   7. lift-tan.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. lift-tan.f64 N/A

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

   11. lower-+.f64 99.7

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

   12. lift--.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. fp-cancel-sub-sign-inv N/A

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

   15. +-commutative N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-neg.f64 99.7

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

6. Applied rewrites 99.7%

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

Alternative 3: 80.3% accurate, 0.7× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (+ x (- (/ (+ (tan y) (tan z)) 1.0) (tan a))))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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(a))
end function

Java

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

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	return x + (((math.tan(y) + math.tan(z)) / 1.0) - math.tan(a))

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / 1.0) - tan(a)))
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = x + (((tan(y) + tan(z)) / 1.0) - tan(a));
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

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

   5. div-add N/A

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

   6. lower-+.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-tan.f64 N/A

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

   13. lower-tan.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-tan.f64 N/A

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

   16. lower--.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lower-tan.f64 N/A

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

   20. lower-tan.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. div-add-rev N/A

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

   6. lift-tan.f64 N/A

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

   7. lift-tan.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. lift-tan.f64 N/A

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

   11. lower-+.f64 99.7

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

   12. lift--.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. fp-cancel-sub-sign-inv N/A

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

   15. +-commutative N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-neg.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in y around 0

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

9. Applied rewrites 77.5%

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

Alternative 4: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;y + z \leq 10^{-35}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(z \cdot \left(\frac{y}{z} - -1\right)\right) - \left(-x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) 1e-35)
   (+ x (- (tan y) (tan a)))
   (- (tan (* z (- (/ y z) -1.0))) (- x))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 1e-35) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = tan((z * ((y / z) - -1.0))) - -x;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y + z) <= 1d-35) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = tan((z * ((y / z) - (-1.0d0)))) - -x
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 1e-35) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = Math.tan((z * ((y / z) - -1.0))) - -x;
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	tmp = 0
	if (y + z) <= 1e-35:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = math.tan((z * ((y / z) - -1.0))) - -x
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= 1e-35)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(tan(Float64(z * Float64(Float64(y / z) - -1.0))) - Float64(-x));
	end
	return tmp
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= 1e-35)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = tan((z * ((y / z) - -1.0))) - -x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], 1e-35], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Tan[N[(z * N[(N[(y / z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 79.9%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 66.3

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

   5. Applied rewrites 66.3%

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

   7. Applied rewrites 66.3%

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

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

   1. Initial program 72.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 72.7

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

   4. Applied rewrites 72.7%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.4

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

   7. Applied rewrites 48.4%

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

   8. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 37.4

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

   10. Applied rewrites 37.4%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq 10^{-35}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(z \cdot \left(\frac{y}{z} - -1\right)\right) - \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 6.4 \cdot 10^{-8}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan z - \tan a\right) + x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (if (<= z 6.4e-8) (+ x (- (tan y) (tan a))) (+ (- (tan z) (tan a)) x)))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 6.4e-8) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = (tan(z) - tan(a)) + x;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= 6.4d-8) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = (tan(z) - tan(a)) + x
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 6.4e-8) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = (Math.tan(z) - Math.tan(a)) + x;
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	tmp = 0
	if z <= 6.4e-8:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = (math.tan(z) - math.tan(a)) + x
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	tmp = 0.0
	if (z <= 6.4e-8)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(Float64(tan(z) - tan(a)) + x);
	end
	return tmp
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 6.4e-8)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = (tan(z) - tan(a)) + x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := If[LessEqual[z, 6.4e-8], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.4000000000000004e-8

   1. Initial program 83.9%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 73.0%

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

   if 6.4000000000000004e-8 < z

   1. Initial program 59.3%

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

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

      5. div-add N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-tan.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-tan.f64 N/A

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

      13. lower-tan.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-tan.f64 N/A

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

      16. lower--.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-tan.f64 N/A

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

      20. lower-tan.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. div-add-rev N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. lift-tan.f64 N/A

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

      11. lower-+.f64 99.7

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

      12. lift--.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. fp-cancel-sub-sign-inv N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-neg.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 59.2

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

   9. Applied rewrites 59.2%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 59.2

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

   11. Applied rewrites 59.2%

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

Alternative 6: 79.9% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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

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

Python

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

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 76.9%

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

Alternative 7: 59.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;y + z \leq -1:\\ \;\;\;\;\tan \left(z + y\right) - \left(-x\right)\\ \mathbf{elif}\;y + z \leq 10^{-35}:\\ \;\;\;\;\left(y + x\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;\tan \left(z \cdot \left(\frac{y}{z} - -1\right)\right) - \left(-x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -1.0)
   (- (tan (+ z y)) (- x))
   (if (<= (+ y z) 1e-35)
     (- (+ y x) (tan a))
     (- (tan (* z (- (/ y z) -1.0))) (- x)))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -1.0) {
		tmp = tan((z + y)) - -x;
	} else if ((y + z) <= 1e-35) {
		tmp = (y + x) - tan(a);
	} else {
		tmp = tan((z * ((y / z) - -1.0))) - -x;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y + z) <= (-1.0d0)) then
        tmp = tan((z + y)) - -x
    else if ((y + z) <= 1d-35) then
        tmp = (y + x) - tan(a)
    else
        tmp = tan((z * ((y / z) - (-1.0d0)))) - -x
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -1.0) {
		tmp = Math.tan((z + y)) - -x;
	} else if ((y + z) <= 1e-35) {
		tmp = (y + x) - Math.tan(a);
	} else {
		tmp = Math.tan((z * ((y / z) - -1.0))) - -x;
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -1.0:
		tmp = math.tan((z + y)) - -x
	elif (y + z) <= 1e-35:
		tmp = (y + x) - math.tan(a)
	else:
		tmp = math.tan((z * ((y / z) - -1.0))) - -x
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -1.0)
		tmp = Float64(tan(Float64(z + y)) - Float64(-x));
	elseif (Float64(y + z) <= 1e-35)
		tmp = Float64(Float64(y + x) - tan(a));
	else
		tmp = Float64(tan(Float64(z * Float64(Float64(y / z) - -1.0))) - Float64(-x));
	end
	return tmp
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -1.0)
		tmp = tan((z + y)) - -x;
	elseif ((y + z) <= 1e-35)
		tmp = (y + x) - tan(a);
	else
		tmp = tan((z * ((y / z) - -1.0))) - -x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -1.0], N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 1e-35], N[(N[(y + x), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], N[(N[Tan[N[(z * N[(N[(y / z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -1

   1. Initial program 68.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 68.7

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

   4. Applied rewrites 68.7%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 41.0

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

   7. Applied rewrites 41.0%

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

   if -1 < (+.f64 y z) < 1.00000000000000001e-35

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.8%

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

   10. Applied rewrites 99.8%

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

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

   1. Initial program 72.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 72.7

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

   4. Applied rewrites 72.7%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.4

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

   7. Applied rewrites 48.4%

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

   8. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 37.4

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

   10. Applied rewrites 37.4%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -1:\\ \;\;\;\;\tan \left(z + y\right) - \left(-x\right)\\ \mathbf{elif}\;y + z \leq 10^{-35}:\\ \;\;\;\;\left(y + x\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;\tan \left(z \cdot \left(\frac{y}{z} - -1\right)\right) - \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;y + z \leq -1 \lor \neg \left(y + z \leq 10^{-35}\right):\\ \;\;\;\;\tan \left(z + y\right) - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \tan a\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (if (or (<= (+ y z) -1.0) (not (<= (+ y z) 1e-35)))
   (- (tan (+ z y)) (- x))
   (- (+ y x) (tan a))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double tmp;
	if (((y + z) <= -1.0) || !((y + z) <= 1e-35)) {
		tmp = tan((z + y)) - -x;
	} else {
		tmp = (y + x) - tan(a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((y + z) <= (-1.0d0)) .or. (.not. ((y + z) <= 1d-35))) then
        tmp = tan((z + y)) - -x
    else
        tmp = (y + x) - tan(a)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	double tmp;
	if (((y + z) <= -1.0) || !((y + z) <= 1e-35)) {
		tmp = Math.tan((z + y)) - -x;
	} else {
		tmp = (y + x) - Math.tan(a);
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	tmp = 0
	if ((y + z) <= -1.0) or not ((y + z) <= 1e-35):
		tmp = math.tan((z + y)) - -x
	else:
		tmp = (y + x) - math.tan(a)
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	tmp = 0.0
	if ((Float64(y + z) <= -1.0) || !(Float64(y + z) <= 1e-35))
		tmp = Float64(tan(Float64(z + y)) - Float64(-x));
	else
		tmp = Float64(Float64(y + x) - tan(a));
	end
	return tmp
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (((y + z) <= -1.0) || ~(((y + z) <= 1e-35)))
		tmp = tan((z + y)) - -x;
	else
		tmp = (y + x) - tan(a);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := If[Or[LessEqual[N[(y + z), $MachinePrecision], -1.0], N[Not[LessEqual[N[(y + z), $MachinePrecision], 1e-35]], $MachinePrecision]], N[(N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision] - (-x)), $MachinePrecision], N[(N[(y + x), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -1 or 1.00000000000000001e-35 < (+.f64 y z)

   1. Initial program 71.0%

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

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower--.f64 70.9

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

   4. Applied rewrites 70.9%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 45.0

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

   7. Applied rewrites 45.0%

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

   if -1 < (+.f64 y z) < 1.00000000000000001e-35

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-cos.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.8%

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

   10. Applied rewrites 99.8%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -1 \lor \neg \left(y + z \leq 10^{-35}\right):\\ \;\;\;\;\tan \left(z + y\right) - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \tan a\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 31.7% accurate, 2.0× speedup

Math

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

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a) :precision binary64 (- (+ y x) (tan a)))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
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 = (y + x) - tan(a)
end function

Java

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

Python

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

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return Float64(Float64(y + x) - tan(a))
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = (y + x) - tan(a);
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := N[(N[(y + x), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

3. Taylor expanded in z around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 59.1

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

5. Applied rewrites 59.1%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 30.2%

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

10. Applied rewrites 30.2%

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

Reproduce

herbie shell --seed 2024340 
(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))))