tan-example (used to crash)

Percentage Accurate: 79.5% → 99.7%

Time: 8.4s

Alternatives: 17

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    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: 79.5% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    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

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

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 z) (tan y)) (- 1.0 (* (/ (sin z) (cos z)) (tan y)))) (tan a))))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    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 - ((sin(z) / cos(z)) * tan(y)))) - 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(z) + Math.tan(y)) / (1.0 - ((Math.sin(z) / Math.cos(z)) * Math.tan(y)))) - Math.tan(a));
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	return x + (((math.tan(z) + math.tan(y)) / (1.0 - ((math.sin(z) / math.cos(z)) * math.tan(y)))) - 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(z) + tan(y)) / Float64(1.0 - Float64(Float64(sin(z) / cos(z)) * tan(y)))) - tan(a)))
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = x + (((tan(z) + tan(y)) / (1.0 - ((sin(z) / cos(z)) * tan(y)))) - 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[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[(N[Sin[z], $MachinePrecision] / N[Cos[z], $MachinePrecision]), $MachinePrecision] * N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.2%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. +-commutative N/A

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

   4. tan-sum N/A

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

   5. lower-/.f64 N/A

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

   6. quot-tan N/A

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

   7. quot-tan N/A

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

   8. lower-+.f64 N/A

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

   9. quot-tan N/A

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

   10. lower-tan.f64 N/A

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

   11. quot-tan N/A

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

   12. lower-tan.f64 N/A

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

   13. lower--.f64 N/A

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

   14. quot-tan N/A

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

   15. quot-tan N/A

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

   16. lower-*.f64 N/A

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

   17. quot-tan N/A

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

   18. lower-tan.f64 N/A

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

   19. quot-tan N/A

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

   20. lower-tan.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-cos.f64 99.6

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

6. Applied rewrites 99.6%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

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

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 z) (tan y)) (- 1.0 (* (tan z) (tan y)))) (tan a))))

C

assert(x < y && y < z && z < a);
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

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    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

assert x < y && y < z && z < a;
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

[x, y, z, a] = sort([x, y, z, a])
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

x, y, z, a = sort([x, y, z, a])
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

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	tmp = x + (((tan(z) + tan(y)) / (1.0 - (tan(z) * tan(y)))) - 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[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 77.2%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. +-commutative N/A

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

   4. tan-sum N/A

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

   5. lower-/.f64 N/A

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

   6. quot-tan N/A

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

   7. quot-tan N/A

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

   8. lower-+.f64 N/A

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

   9. quot-tan N/A

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

   10. lower-tan.f64 N/A

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

   11. quot-tan N/A

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

   12. lower-tan.f64 N/A

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

   13. lower--.f64 N/A

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

   14. quot-tan N/A

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

   15. quot-tan N/A

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

   16. lower-*.f64 N/A

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

   17. quot-tan N/A

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

   18. lower-tan.f64 N/A

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

   19. quot-tan N/A

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

   20. lower-tan.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 3: 89.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -0.365:\\ \;\;\;\;x + \left(\frac{\tan z + \tan y}{1} - \tan a\right)\\ \mathbf{elif}\;a \leq 0.072:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.05396825396825397, a \cdot a, 0.13333333333333333\right), a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(\left(z + y\right) + \pi\right) - \tan a\right)\\ \end{array} \end{array} \]

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 (<= a -0.365)
   (+ x (- (/ (+ (tan z) (tan y)) 1.0) (tan a)))
   (if (<= a 0.072)
     (+
      x
      (-
       (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z))))
       (*
        (fma
         (fma
          (fma 0.05396825396825397 (* a a) 0.13333333333333333)
          (* a a)
          0.3333333333333333)
         (* a a)
         1.0)
        a)))
     (+ x (- (tan (+ (+ z y) PI)) (tan a))))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -0.365) {
		tmp = x + (((tan(z) + tan(y)) / 1.0) - tan(a));
	} else if (a <= 0.072) {
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - (fma(fma(fma(0.05396825396825397, (a * a), 0.13333333333333333), (a * a), 0.3333333333333333), (a * a), 1.0) * a));
	} else {
		tmp = x + (tan(((z + y) + ((double) M_PI))) - tan(a));
	}
	return tmp;
}

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	tmp = 0.0
	if (a <= -0.365)
		tmp = Float64(x + Float64(Float64(Float64(tan(z) + tan(y)) / 1.0) - tan(a)));
	elseif (a <= 0.072)
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - Float64(fma(fma(fma(0.05396825396825397, Float64(a * a), 0.13333333333333333), Float64(a * a), 0.3333333333333333), Float64(a * a), 1.0) * a)));
	else
		tmp = Float64(x + Float64(tan(Float64(Float64(z + y) + pi)) - tan(a)));
	end
	return 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[a, -0.365], N[(x + N[(N[(N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.072], N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(0.05396825396825397 * N[(a * a), $MachinePrecision] + 0.13333333333333333), $MachinePrecision] * N[(a * a), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(N[(z + y), $MachinePrecision] + Pi), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.36499999999999999

   1. Initial program 77.3%

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

      1. lift-+.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. +-commutative N/A

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

      4. tan-sum N/A

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

      5. lower-/.f64 N/A

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

      6. quot-tan N/A

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

      7. quot-tan N/A

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

      8. lower-+.f64 N/A

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

      9. quot-tan N/A

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

      10. lower-tan.f64 N/A

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

      11. quot-tan N/A

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

      12. lower-tan.f64 N/A

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

      13. lower--.f64 N/A

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

      14. quot-tan N/A

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

      15. quot-tan N/A

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

      16. lower-*.f64 N/A

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

      17. quot-tan N/A

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

      18. lower-tan.f64 N/A

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

      19. quot-tan N/A

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

      20. lower-tan.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 77.8%

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

   if -0.36499999999999999 < a < 0.0719999999999999946

   1. Initial program 75.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 75.6

          \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.05396825396825397, a \cdot a, 0.13333333333333333\right), a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a\right) \]

   5. Applied rewrites 75.6%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.05396825396825397, a \cdot a, 0.13333333333333333\right), a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a}\right) \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-sum N/A

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

      4. tan-quot N/A

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

      5. tan-quot N/A

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

      6. tan-quot N/A

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

      7. tan-quot N/A

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

      8. times-frac N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. tan-quot N/A

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

      12. lift-tan.f64 N/A

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

      13. tan-quot N/A

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

      14. lift-tan.f64 N/A

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

      15. lower--.f64 N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 99.0%

      \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.05396825396825397, a \cdot a, 0.13333333333333333\right), a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a\right) \]

   if 0.0719999999999999946 < a

   1. Initial program 80.0%

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

      1. lift-+.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-+PI-rev N/A

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

      4. lower-tan.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-PI.f64 80.0

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

   4. Applied rewrites 80.0%

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

Alternative 4: 89.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := \tan z + \tan y\\ \mathbf{if}\;a \leq -0.00088:\\ \;\;\;\;x + \left(\frac{t\_0}{1} - \tan a\right)\\ \mathbf{elif}\;a \leq 0.00186:\\ \;\;\;\;x + \left(\frac{t\_0}{1 - \tan z \cdot \tan y} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(\left(z + y\right) + \pi\right) - \tan a\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan z) (tan y))))
   (if (<= a -0.00088)
     (+ x (- (/ t_0 1.0) (tan a)))
     (if (<= a 0.00186)
       (+ x (- (/ t_0 (- 1.0 (* (tan z) (tan y)))) a))
       (+ x (- (tan (+ (+ z y) PI)) (tan a)))))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = tan(z) + tan(y);
	double tmp;
	if (a <= -0.00088) {
		tmp = x + ((t_0 / 1.0) - tan(a));
	} else if (a <= 0.00186) {
		tmp = x + ((t_0 / (1.0 - (tan(z) * tan(y)))) - a);
	} else {
		tmp = x + (tan(((z + y) + ((double) M_PI))) - tan(a));
	}
	return tmp;
}

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan(z) + Math.tan(y);
	double tmp;
	if (a <= -0.00088) {
		tmp = x + ((t_0 / 1.0) - Math.tan(a));
	} else if (a <= 0.00186) {
		tmp = x + ((t_0 / (1.0 - (Math.tan(z) * Math.tan(y)))) - a);
	} else {
		tmp = x + (Math.tan(((z + y) + Math.PI)) - Math.tan(a));
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	t_0 = math.tan(z) + math.tan(y)
	tmp = 0
	if a <= -0.00088:
		tmp = x + ((t_0 / 1.0) - math.tan(a))
	elif a <= 0.00186:
		tmp = x + ((t_0 / (1.0 - (math.tan(z) * math.tan(y)))) - a)
	else:
		tmp = x + (math.tan(((z + y) + math.pi)) - math.tan(a))
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(tan(z) + tan(y))
	tmp = 0.0
	if (a <= -0.00088)
		tmp = Float64(x + Float64(Float64(t_0 / 1.0) - tan(a)));
	elseif (a <= 0.00186)
		tmp = Float64(x + Float64(Float64(t_0 / Float64(1.0 - Float64(tan(z) * tan(y)))) - a));
	else
		tmp = Float64(x + Float64(tan(Float64(Float64(z + y) + pi)) - 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)
	t_0 = tan(z) + tan(y);
	tmp = 0.0;
	if (a <= -0.00088)
		tmp = x + ((t_0 / 1.0) - tan(a));
	elseif (a <= 0.00186)
		tmp = x + ((t_0 / (1.0 - (tan(z) * tan(y)))) - a);
	else
		tmp = x + (tan(((z + y) + pi)) - 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_] := Block[{t$95$0 = N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.00088], N[(x + N[(N[(t$95$0 / 1.0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.00186], N[(x + N[(N[(t$95$0 / N[(1.0 - N[(N[Tan[z], $MachinePrecision] * N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(N[(z + y), $MachinePrecision] + Pi), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -8.80000000000000031e-4

   1. Initial program 76.8%

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

      1. lift-+.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. +-commutative N/A

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

      4. tan-sum N/A

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

      5. lower-/.f64 N/A

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

      6. quot-tan N/A

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

      7. quot-tan N/A

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

      8. lower-+.f64 N/A

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

      9. quot-tan N/A

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

      10. lower-tan.f64 N/A

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

      11. quot-tan N/A

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

      12. lower-tan.f64 N/A

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

      13. lower--.f64 N/A

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

      14. quot-tan N/A

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

      15. quot-tan N/A

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

      16. lower-*.f64 N/A

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

      17. quot-tan N/A

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

      18. lower-tan.f64 N/A

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

      19. quot-tan N/A

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

      20. lower-tan.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 77.3%

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

   if -8.80000000000000031e-4 < a < 0.0018600000000000001

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. +-commutative N/A

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

      4. tan-sum N/A

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

      5. lower-/.f64 N/A

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

      6. quot-tan N/A

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

      7. quot-tan N/A

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

      8. lower-+.f64 N/A

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

      9. quot-tan N/A

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

      10. lower-tan.f64 N/A

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

      11. quot-tan N/A

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

      12. lower-tan.f64 N/A

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

      13. lower--.f64 N/A

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

      14. quot-tan N/A

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

      15. quot-tan N/A

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

      16. lower-*.f64 N/A

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

      17. quot-tan N/A

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

      18. lower-tan.f64 N/A

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

      19. quot-tan N/A

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

      20. lower-tan.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 99.6%

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

   if 0.0018600000000000001 < a

   1. Initial program 80.0%

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

      1. lift-+.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-+PI-rev N/A

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

      4. lower-tan.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-PI.f64 80.0

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

   4. Applied rewrites 80.0%

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

Alternative 5: 58.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.02 \lor \neg \left(\tan a \leq 0.1\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(z + y\right)\\ \end{array} \end{array} \]

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 (or (<= (tan a) -0.02) (not (<= (tan a) 0.1)))
   (- x (tan a))
   (+ x (tan (+ z y)))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double tmp;
	if ((tan(a) <= -0.02) || !(tan(a) <= 0.1)) {
		tmp = x - tan(a);
	} else {
		tmp = x + tan((z + y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((tan(a) <= (-0.02d0)) .or. (.not. (tan(a) <= 0.1d0))) then
        tmp = x - tan(a)
    else
        tmp = x + tan((z + y))
    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 ((Math.tan(a) <= -0.02) || !(Math.tan(a) <= 0.1)) {
		tmp = x - Math.tan(a);
	} else {
		tmp = x + Math.tan((z + y));
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	tmp = 0
	if (math.tan(a) <= -0.02) or not (math.tan(a) <= 0.1):
		tmp = x - math.tan(a)
	else:
		tmp = x + math.tan((z + y))
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	tmp = 0.0
	if ((tan(a) <= -0.02) || !(tan(a) <= 0.1))
		tmp = Float64(x - tan(a));
	else
		tmp = Float64(x + tan(Float64(z + y)));
	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 ((tan(a) <= -0.02) || ~((tan(a) <= 0.1)))
		tmp = x - tan(a);
	else
		tmp = x + tan((z + y));
	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[Tan[a], $MachinePrecision], -0.02], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 0.1]], $MachinePrecision]], N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision], N[(x + N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0200000000000000004 or 0.10000000000000001 < (tan.f64 a)

   1. Initial program 78.4%

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

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. quot-tan N/A

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

      5. lower-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. lift-tan.f64 63.0

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

   5. Applied rewrites 63.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 45.5%

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

   if -0.0200000000000000004 < (tan.f64 a) < 0.10000000000000001

   1. Initial program 76.2%

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

   3. Taylor expanded in a around 0

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

      1. tan-quot N/A

         \[\leadsto x + \tan \left(y + z\right) \]

      2. lift-tan.f64 N/A

         \[\leadsto x + \tan \left(y + z\right) \]

      3. +-commutative N/A

         \[\leadsto x + \tan \left(z + y\right) \]

      4. lower-+.f64 73.9

         \[\leadsto x + \tan \left(z + y\right) \]

   5. Applied rewrites 73.9%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.02 \lor \neg \left(\tan a \leq 0.1\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(z + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2 \cdot 10^{-12}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

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) -2e-12) (+ x (- (tan y) (tan a))) (+ x (- (tan z) (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) <= -2e-12) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (tan(z) - tan(a));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    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) <= (-2d-12)) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (tan(z) - 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) <= -2e-12) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan(z) - 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) <= -2e-12:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.tan(z) - 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) <= -2e-12)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(z) - 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) <= -2e-12)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan(z) - 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[LessEqual[N[(y + z), $MachinePrecision], -2e-12], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -1.99999999999999996e-12

   1. Initial program 75.2%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 53.4%

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

   if -1.99999999999999996e-12 < (+.f64 y z)

   1. Initial program 78.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 65.1%

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

Alternative 7: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2 \cdot 10^{-12}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan z + x\right) - \tan a\\ \end{array} \end{array} \]

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 (<= (+ y z) -2e-12) (+ x (- (tan y) (tan a))) (- (+ (tan z) 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) <= -2e-12) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = (tan(z) + x) - tan(a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    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) <= (-2d-12)) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = (tan(z) + 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) <= -2e-12) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = (Math.tan(z) + 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) <= -2e-12:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = (math.tan(z) + 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) <= -2e-12)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(Float64(tan(z) + 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) <= -2e-12)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = (tan(z) + 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[LessEqual[N[(y + z), $MachinePrecision], -2e-12], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Tan[z], $MachinePrecision] + x), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -1.99999999999999996e-12

   1. Initial program 75.2%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 53.4%

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

   if -1.99999999999999996e-12 < (+.f64 y z)

   1. Initial program 78.5%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. quot-tan N/A

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

      5. lower-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. lift-tan.f64 65.1

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

   5. Applied rewrites 65.1%

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

Alternative 8: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;y + z \leq -2 \cdot 10^{-12}:\\ \;\;\;\;\tan y + \left(x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan z + x\right) - \tan a\\ \end{array} \end{array} \]

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 (<= (+ y z) -2e-12) (+ (tan y) (- x (tan a))) (- (+ (tan z) 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) <= -2e-12) {
		tmp = tan(y) + (x - tan(a));
	} else {
		tmp = (tan(z) + x) - tan(a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    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) <= (-2d-12)) then
        tmp = tan(y) + (x - tan(a))
    else
        tmp = (tan(z) + 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) <= -2e-12) {
		tmp = Math.tan(y) + (x - Math.tan(a));
	} else {
		tmp = (Math.tan(z) + 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) <= -2e-12:
		tmp = math.tan(y) + (x - math.tan(a))
	else:
		tmp = (math.tan(z) + 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) <= -2e-12)
		tmp = Float64(tan(y) + Float64(x - tan(a)));
	else
		tmp = Float64(Float64(tan(z) + 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) <= -2e-12)
		tmp = tan(y) + (x - tan(a));
	else
		tmp = (tan(z) + 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[LessEqual[N[(y + z), $MachinePrecision], -2e-12], N[(N[Tan[y], $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Tan[z], $MachinePrecision] + x), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -1.99999999999999996e-12

   1. Initial program 75.2%

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

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. quot-tan N/A

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

      5. lower-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. lift-tan.f64 53.4

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

   5. Applied rewrites 53.4%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 53.4

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

   7. Applied rewrites 53.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. tan-quot N/A

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

      8. associate--l+ N/A

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

      9. lower-+.f64 N/A

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

      10. tan-quot N/A

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

      11. lift-tan.f64 N/A

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

      12. lower--.f64 N/A

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

      13. tan-quot N/A

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

      14. lift-tan.f64 53.4

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

   9. Applied rewrites 53.4%

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

   if -1.99999999999999996e-12 < (+.f64 y z)

   1. Initial program 78.5%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. quot-tan N/A

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

      5. lower-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. lift-tan.f64 65.1

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

   5. Applied rewrites 65.1%

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

Alternative 9: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq 0.0025:\\ \;\;\;\;\tan y + \left(x - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(z + y\right)\\ \end{array} \end{array} \]

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 (<= z 0.0025) (+ (tan y) (- x (tan a))) (+ x (tan (+ z y)))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 0.0025) {
		tmp = tan(y) + (x - tan(a));
	} else {
		tmp = x + tan((z + y));
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= 0.0025d0) then
        tmp = tan(y) + (x - tan(a))
    else
        tmp = x + tan((z + y))
    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 <= 0.0025) {
		tmp = Math.tan(y) + (x - Math.tan(a));
	} else {
		tmp = x + Math.tan((z + y));
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	tmp = 0
	if z <= 0.0025:
		tmp = math.tan(y) + (x - math.tan(a))
	else:
		tmp = x + math.tan((z + y))
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	tmp = 0.0
	if (z <= 0.0025)
		tmp = Float64(tan(y) + Float64(x - tan(a)));
	else
		tmp = Float64(x + tan(Float64(z + y)));
	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 <= 0.0025)
		tmp = tan(y) + (x - tan(a));
	else
		tmp = x + tan((z + y));
	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, 0.0025], N[(N[Tan[y], $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.00250000000000000005

   1. Initial program 86.4%

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

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. quot-tan N/A

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

      5. lower-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. lift-tan.f64 74.1

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

   5. Applied rewrites 74.1%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 74.0

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

   7. Applied rewrites 74.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. tan-quot N/A

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

      8. associate--l+ N/A

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

      9. lower-+.f64 N/A

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

      10. tan-quot N/A

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

      11. lift-tan.f64 N/A

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

      12. lower--.f64 N/A

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

      13. tan-quot N/A

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

      14. lift-tan.f64 74.1

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

   9. Applied rewrites 74.1%

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

   if 0.00250000000000000005 < z

   1. Initial program 52.7%

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

   3. Taylor expanded in a around 0

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

      1. tan-quot N/A

         \[\leadsto x + \tan \left(y + z\right) \]

      2. lift-tan.f64 N/A

         \[\leadsto x + \tan \left(y + z\right) \]

      3. +-commutative N/A

         \[\leadsto x + \tan \left(z + y\right) \]

      4. lower-+.f64 38.6

         \[\leadsto x + \tan \left(z + y\right) \]

   5. Applied rewrites 38.6%

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

Alternative 10: 79.5% 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.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    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 77.2%

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

Alternative 11: 59.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -0.35 \lor \neg \left(a \leq 2.8 \cdot 10^{-26}\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.05396825396825397, a \cdot a, 0.13333333333333333\right), a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a\right)\\ \end{array} \end{array} \]

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 (or (<= a -0.35) (not (<= a 2.8e-26)))
   (- x (tan a))
   (+
    x
    (-
     (tan (+ y z))
     (*
      (fma
       (fma
        (fma 0.05396825396825397 (* a a) 0.13333333333333333)
        (* a a)
        0.3333333333333333)
       (* a a)
       1.0)
      a)))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -0.35) || !(a <= 2.8e-26)) {
		tmp = x - tan(a);
	} else {
		tmp = x + (tan((y + z)) - (fma(fma(fma(0.05396825396825397, (a * a), 0.13333333333333333), (a * a), 0.3333333333333333), (a * a), 1.0) * a));
	}
	return tmp;
}

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -0.35) || !(a <= 2.8e-26))
		tmp = Float64(x - tan(a));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) - Float64(fma(fma(fma(0.05396825396825397, Float64(a * a), 0.13333333333333333), Float64(a * a), 0.3333333333333333), Float64(a * a), 1.0) * a)));
	end
	return 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[a, -0.35], N[Not[LessEqual[a, 2.8e-26]], $MachinePrecision]], N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(N[(N[(0.05396825396825397 * N[(a * a), $MachinePrecision] + 0.13333333333333333), $MachinePrecision] * N[(a * a), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.34999999999999998 or 2.8000000000000001e-26 < a

   1. Initial program 77.2%

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

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. quot-tan N/A

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

      5. lower-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. lift-tan.f64 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 44.2%

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

   if -0.34999999999999998 < a < 2.8000000000000001e-26

   1. Initial program 77.2%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 77.0

          \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.05396825396825397, a \cdot a, 0.13333333333333333\right), a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a\right) \]

   5. Applied rewrites 77.0%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.05396825396825397, a \cdot a, 0.13333333333333333\right), a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.35 \lor \neg \left(a \leq 2.8 \cdot 10^{-26}\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.05396825396825397, a \cdot a, 0.13333333333333333\right), a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -0.35 \lor \neg \left(a \leq 2.8 \cdot 10^{-26}\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a\right)\\ \end{array} \end{array} \]

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 (or (<= a -0.35) (not (<= a 2.8e-26)))
   (- x (tan a))
   (+
    x
    (-
     (tan (+ y z))
     (*
      (fma (fma 0.13333333333333333 (* a a) 0.3333333333333333) (* a a) 1.0)
      a)))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -0.35) || !(a <= 2.8e-26)) {
		tmp = x - tan(a);
	} else {
		tmp = x + (tan((y + z)) - (fma(fma(0.13333333333333333, (a * a), 0.3333333333333333), (a * a), 1.0) * a));
	}
	return tmp;
}

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -0.35) || !(a <= 2.8e-26))
		tmp = Float64(x - tan(a));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) - Float64(fma(fma(0.13333333333333333, Float64(a * a), 0.3333333333333333), Float64(a * a), 1.0) * a)));
	end
	return 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[a, -0.35], N[Not[LessEqual[a, 2.8e-26]], $MachinePrecision]], N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(N[(0.13333333333333333 * N[(a * a), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.34999999999999998 or 2.8000000000000001e-26 < a

   1. Initial program 77.2%

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

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. quot-tan N/A

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

      5. lower-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. lift-tan.f64 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 44.2%

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

   if -0.34999999999999998 < a < 2.8000000000000001e-26

   1. Initial program 77.2%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 76.9

          \[\leadsto x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a\right) \]

   5. Applied rewrites 76.9%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.35 \lor \neg \left(a \leq 2.8 \cdot 10^{-26}\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, a \cdot a, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -0.28 \lor \neg \left(a \leq 2.8 \cdot 10^{-26}\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(0.3333333333333333, a \cdot a, 1\right) \cdot a\right)\\ \end{array} \end{array} \]

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 (or (<= a -0.28) (not (<= a 2.8e-26)))
   (- x (tan a))
   (+ x (- (tan (+ y z)) (* (fma 0.3333333333333333 (* a a) 1.0) a)))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -0.28) || !(a <= 2.8e-26)) {
		tmp = x - tan(a);
	} else {
		tmp = x + (tan((y + z)) - (fma(0.3333333333333333, (a * a), 1.0) * a));
	}
	return tmp;
}

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -0.28) || !(a <= 2.8e-26))
		tmp = Float64(x - tan(a));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) - Float64(fma(0.3333333333333333, Float64(a * a), 1.0) * a)));
	end
	return 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[a, -0.28], N[Not[LessEqual[a, 2.8e-26]], $MachinePrecision]], N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(0.3333333333333333 * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.28000000000000003 or 2.8000000000000001e-26 < a

   1. Initial program 77.2%

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

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. quot-tan N/A

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

      5. lower-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. lift-tan.f64 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 44.2%

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

   if -0.28000000000000003 < a < 2.8000000000000001e-26

   1. Initial program 77.2%

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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 76.8

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

   5. Applied rewrites 76.8%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.28 \lor \neg \left(a \leq 2.8 \cdot 10^{-26}\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(0.3333333333333333, a \cdot a, 1\right) \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -0.28 \lor \neg \left(a \leq 2.8 \cdot 10^{-26}\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \end{array} \]

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 (or (<= a -0.28) (not (<= a 2.8e-26)))
   (- x (tan a))
   (+ x (- (tan (+ y z)) a))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double tmp;
	if ((a <= -0.28) || !(a <= 2.8e-26)) {
		tmp = x - tan(a);
	} else {
		tmp = x + (tan((y + z)) - a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-0.28d0)) .or. (.not. (a <= 2.8d-26))) then
        tmp = x - tan(a)
    else
        tmp = x + (tan((y + z)) - 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 ((a <= -0.28) || !(a <= 2.8e-26)) {
		tmp = x - Math.tan(a);
	} else {
		tmp = x + (Math.tan((y + z)) - a);
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	tmp = 0
	if (a <= -0.28) or not (a <= 2.8e-26):
		tmp = x - math.tan(a)
	else:
		tmp = x + (math.tan((y + z)) - a)
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	tmp = 0.0
	if ((a <= -0.28) || !(a <= 2.8e-26))
		tmp = Float64(x - tan(a));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) - 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 ((a <= -0.28) || ~((a <= 2.8e-26)))
		tmp = x - tan(a);
	else
		tmp = x + (tan((y + z)) - 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[a, -0.28], N[Not[LessEqual[a, 2.8e-26]], $MachinePrecision]], N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.28000000000000003 or 2.8000000000000001e-26 < a

   1. Initial program 77.2%

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

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. quot-tan N/A

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

      5. lower-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. lift-tan.f64 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 44.2%

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

   if -0.28000000000000003 < a < 2.8000000000000001e-26

   1. Initial program 77.2%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 76.8%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.28 \lor \neg \left(a \leq 2.8 \cdot 10^{-26}\right):\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.1% accurate, 1.9× speedup

Math

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

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 (<= (+ y z) -10.0) (+ x (tan 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) <= -10.0) {
		tmp = x + tan(y);
	} else {
		tmp = x - tan(a);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    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) <= (-10.0d0)) then
        tmp = x + tan(y)
    else
        tmp = 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) <= -10.0) {
		tmp = x + Math.tan(y);
	} else {
		tmp = 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) <= -10.0:
		tmp = x + math.tan(y)
	else:
		tmp = 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) <= -10.0)
		tmp = Float64(x + tan(y));
	else
		tmp = Float64(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) <= -10.0)
		tmp = x + tan(y);
	else
		tmp = 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[LessEqual[N[(y + z), $MachinePrecision], -10.0], N[(x + N[Tan[y], $MachinePrecision]), $MachinePrecision], N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -10

   1. Initial program 73.5%

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

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. quot-tan N/A

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

      5. lower-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. lift-tan.f64 51.0

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

   5. Applied rewrites 51.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. tan-quot N/A

         \[\leadsto x + \tan y \]

      3. lift-tan.f64 39.3

         \[\leadsto x + \tan y \]

   8. Applied rewrites 39.3%

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

   if -10 < (+.f64 y z)

   1. Initial program 79.4%

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

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. quot-tan N/A

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

      5. lower-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. lift-tan.f64 64.5

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

   5. Applied rewrites 64.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 51.0%

      \[\leadsto x - \tan \color{blue}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 40.6% accurate, 2.0× speedup

Math

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

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)))

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + tan(y)
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);
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 77.2%

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

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. quot-tan N/A

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

   5. lower-tan.f64 N/A

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

   6. tan-quot N/A

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

   7. lift-tan.f64 59.4

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

5. Applied rewrites 59.4%

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

6. Taylor expanded in a around 0

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

   1. lower-+.f64 N/A

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

   2. tan-quot N/A

      \[\leadsto x + \tan y \]

   3. lift-tan.f64 40.1

      \[\leadsto x + \tan y \]

8. Applied rewrites 40.1%

   \[\leadsto x + \color{blue}{\tan y} \]
9. Add Preprocessing

Alternative 17: 31.9% accurate, 210.0× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ x \end{array} \]

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)

C

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

Fortran

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	return x;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	return x

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	return x
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 29.6%

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

Reproduce

herbie shell --seed 2025082 
(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))))