tan-example (used to crash)

Percentage Accurate: 79.8% → 99.7%

Time: 11.1s

Alternatives: 14

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: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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, 0.2× speedup

Math

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

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 (- 1.0 (* (tan y) (tan z)))))
   (+ x (+ (/ (/ (sin y) (cos y)) t_0) (- (/ (tan z) t_0) (tan a))))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = 1.0 - (tan(y) * tan(z));
	return x + (((sin(y) / cos(y)) / t_0) + ((tan(z) / t_0) - 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
    real(8) :: t_0
    t_0 = 1.0d0 - (tan(y) * tan(z))
    code = x + (((sin(y) / cos(y)) / t_0) + ((tan(z) / t_0) - tan(a)))
end function

Java

assert x < y && y < z && z < a;
public static double code(double x, double y, double z, double a) {
	double t_0 = 1.0 - (Math.tan(y) * Math.tan(z));
	return x + (((Math.sin(y) / Math.cos(y)) / t_0) + ((Math.tan(z) / t_0) - Math.tan(a)));
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	t_0 = 1.0 - (math.tan(y) * math.tan(z))
	return x + (((math.sin(y) / math.cos(y)) / t_0) + ((math.tan(z) / t_0) - math.tan(a)))

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	t_0 = Float64(1.0 - Float64(tan(y) * tan(z)))
	return Float64(x + Float64(Float64(Float64(sin(y) / cos(y)) / t_0) + Float64(Float64(tan(z) / t_0) - tan(a))))
end

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	t_0 = 1.0 - (tan(y) * tan(z));
	tmp = x + (((sin(y) / cos(y)) / t_0) + ((tan(z) / t_0) - tan(a)));
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x + N[(N[(N[(N[Sin[y], $MachinePrecision] / N[Cos[y], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(N[Tan[z], $MachinePrecision] / t$95$0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.3%

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

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

   4. quot-tan N/A

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

   5. quot-tan N/A

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

   6. div-add N/A

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

   7. lower-+.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift--.f64 N/A

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

6. Applied rewrites 99.6%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. lower-/.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-cos.f64 99.6

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

8. Applied rewrites 99.6%

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

Alternative 2: 99.7% accurate, 0.3× speedup

Math

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

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 (- 1.0 (* (tan y) (tan z)))))
   (+ x (+ (/ (tan y) t_0) (- (/ (tan z) t_0) (tan a))))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double t_0 = 1.0 - (tan(y) * tan(z));
	return x + ((tan(y) / t_0) + ((tan(z) / t_0) - 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
    real(8) :: t_0
    t_0 = 1.0d0 - (tan(y) * tan(z))
    code = x + ((tan(y) / t_0) + ((tan(z) / t_0) - tan(a)))
end function

Java

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

Python

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

Julia

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

MATLAB

x, y, z, a = num2cell(sort([x, y, z, a])){:}
function tmp = code(x, y, z, a)
	t_0 = 1.0 - (tan(y) * tan(z));
	tmp = x + ((tan(y) / t_0) + ((tan(z) / t_0) - tan(a)));
end

Wolfram

NOTE: x, y, z, and a should be sorted in increasing order before calling this function.
code[x_, y_, z_, a_] := Block[{t$95$0 = N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x + N[(N[(N[Tan[y], $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(N[Tan[z], $MachinePrecision] / t$95$0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.3%

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

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

   4. quot-tan N/A

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

   5. quot-tan N/A

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

   6. div-add N/A

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

   7. lower-+.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift--.f64 N/A

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

6. Applied rewrites 99.6%

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

Alternative 3: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} t_0 := \tan z + \tan y\\ \mathbf{if}\;\tan a \leq -2 \cdot 10^{-11} \lor \neg \left(\tan a \leq 10^{-16}\right):\\ \;\;\;\;\left(\frac{\frac{t\_0}{1} - \tan a}{x} - -1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t\_0}{1 - \tan z \cdot \tan y}\\ \end{array} \end{array} \]

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 (or (<= (tan a) -2e-11) (not (<= (tan a) 1e-16)))
     (* (- (/ (- (/ t_0 1.0) (tan a)) x) -1.0) x)
     (+ x (/ t_0 (- 1.0 (* (tan z) (tan y))))))))

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 ((tan(a) <= -2e-11) || !(tan(a) <= 1e-16)) {
		tmp = ((((t_0 / 1.0) - tan(a)) / x) - -1.0) * x;
	} else {
		tmp = x + (t_0 / (1.0 - (tan(z) * tan(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) :: t_0
    real(8) :: tmp
    t_0 = tan(z) + tan(y)
    if ((tan(a) <= (-2d-11)) .or. (.not. (tan(a) <= 1d-16))) then
        tmp = ((((t_0 / 1.0d0) - tan(a)) / x) - (-1.0d0)) * x
    else
        tmp = x + (t_0 / (1.0d0 - (tan(z) * tan(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 t_0 = Math.tan(z) + Math.tan(y);
	double tmp;
	if ((Math.tan(a) <= -2e-11) || !(Math.tan(a) <= 1e-16)) {
		tmp = ((((t_0 / 1.0) - Math.tan(a)) / x) - -1.0) * x;
	} else {
		tmp = x + (t_0 / (1.0 - (Math.tan(z) * Math.tan(y))));
	}
	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 (math.tan(a) <= -2e-11) or not (math.tan(a) <= 1e-16):
		tmp = ((((t_0 / 1.0) - math.tan(a)) / x) - -1.0) * x
	else:
		tmp = x + (t_0 / (1.0 - (math.tan(z) * math.tan(y))))
	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 ((tan(a) <= -2e-11) || !(tan(a) <= 1e-16))
		tmp = Float64(Float64(Float64(Float64(Float64(t_0 / 1.0) - tan(a)) / x) - -1.0) * x);
	else
		tmp = Float64(x + Float64(t_0 / Float64(1.0 - Float64(tan(z) * tan(y)))));
	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 ((tan(a) <= -2e-11) || ~((tan(a) <= 1e-16)))
		tmp = ((((t_0 / 1.0) - tan(a)) / x) - -1.0) * x;
	else
		tmp = x + (t_0 / (1.0 - (tan(z) * tan(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_] := Block[{t$95$0 = N[(N[Tan[z], $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[N[Tan[a], $MachinePrecision], -2e-11], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 1e-16]], $MachinePrecision]], N[(N[(N[(N[(N[(t$95$0 / 1.0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - -1.0), $MachinePrecision] * x), $MachinePrecision], N[(x + N[(t$95$0 / N[(1.0 - N[(N[Tan[z], $MachinePrecision] * N[Tan[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -1.99999999999999988e-11 or 9.9999999999999998e-17 < (tan.f64 a)

   1. Initial program 72.3%

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

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

      4. quot-tan N/A

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

      5. quot-tan N/A

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

      6. div-add N/A

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

      7. lower-+.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around -inf

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

   7. Applied rewrites 72.1%

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

      1. lift-+.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-sum N/A

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

      4. lower-/.f64 N/A

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

      5. tan-quot N/A

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

      6. tan-quot N/A

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

      7. lower-+.f64 N/A

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

      8. tan-quot N/A

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

      9. lift-tan.f64 N/A

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

      10. tan-quot N/A

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

      11. lift-tan.f64 N/A

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

      12. lower--.f64 N/A

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

      13. tan-quot N/A

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

      14. tan-quot N/A

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

      15. lower-*.f64 N/A

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

      16. tan-quot N/A

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

      17. lift-tan.f64 N/A

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

      18. tan-quot N/A

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

      19. lift-tan.f64 99.1

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

   9. Applied rewrites 99.1%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 73.0%

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

   if -1.99999999999999988e-11 < (tan.f64 a) < 9.9999999999999998e-17

   1. Initial program 80.3%

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

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

      4. quot-tan N/A

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

      5. quot-tan N/A

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

      6. div-add N/A

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

      7. lower-+.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 80.3%

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

      1. lift-+.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-sum N/A

         \[\leadsto x + \frac{\tan z + \tan y}{\color{blue}{1 - \tan z \cdot \tan y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x + \frac{\tan z + \tan y}{\color{blue}{1 - \tan z \cdot \tan y}} \]

      5. tan-quot N/A

         \[\leadsto x + \frac{\frac{\sin z}{\cos z} + \tan y}{1 - \tan z \cdot \tan y} \]

      6. tan-quot N/A

         \[\leadsto x + \frac{\frac{\sin z}{\cos z} + \frac{\sin y}{\cos y}}{1 - \tan z \cdot \tan y} \]

      7. lower-+.f64 N/A

         \[\leadsto x + \frac{\frac{\sin z}{\cos z} + \frac{\sin y}{\cos y}}{\color{blue}{1} - \tan z \cdot \tan y} \]

      8. tan-quot N/A

         \[\leadsto x + \frac{\tan z + \frac{\sin y}{\cos y}}{1 - \tan z \cdot \tan y} \]

      9. lift-tan.f64 N/A

         \[\leadsto x + \frac{\tan z + \frac{\sin y}{\cos y}}{1 - \tan z \cdot \tan y} \]

      10. tan-quot N/A

          \[\leadsto x + \frac{\tan z + \tan y}{1 - \tan z \cdot \tan y} \]

      11. lift-tan.f64 N/A

          \[\leadsto x + \frac{\tan z + \tan y}{1 - \tan z \cdot \tan y} \]

      12. lower--.f64 N/A

          \[\leadsto x + \frac{\tan z + \tan y}{1 - \color{blue}{\tan z \cdot \tan y}} \]

      13. tan-quot N/A

          \[\leadsto x + \frac{\tan z + \tan y}{1 - \frac{\sin z}{\cos z} \cdot \tan \color{blue}{y}} \]

      14. tan-quot N/A

          \[\leadsto x + \frac{\tan z + \tan y}{1 - \frac{\sin z}{\cos z} \cdot \frac{\sin y}{\color{blue}{\cos y}}} \]

      15. lower-*.f64 N/A

          \[\leadsto x + \frac{\tan z + \tan y}{1 - \frac{\sin z}{\cos z} \cdot \color{blue}{\frac{\sin y}{\cos y}}} \]

      16. tan-quot N/A

          \[\leadsto x + \frac{\tan z + \tan y}{1 - \tan z \cdot \frac{\color{blue}{\sin y}}{\cos y}} \]

      17. lift-tan.f64 N/A

          \[\leadsto x + \frac{\tan z + \tan y}{1 - \tan z \cdot \frac{\color{blue}{\sin y}}{\cos y}} \]

      18. tan-quot N/A

          \[\leadsto x + \frac{\tan z + \tan y}{1 - \tan z \cdot \tan y} \]

      19. lift-tan.f64 99.8

          \[\leadsto x + \frac{\tan z + \tan y}{1 - \tan z \cdot \tan y} \]

   9. Applied rewrites 99.8%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -2 \cdot 10^{-11} \lor \neg \left(\tan a \leq 10^{-16}\right):\\ \;\;\;\;\left(\frac{\frac{\tan z + \tan y}{1} - \tan a}{x} - -1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\tan z + \tan y}{1 - \tan z \cdot \tan y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 76.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 99.6

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

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 5: 79.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

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

   4. quot-tan N/A

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

   5. quot-tan N/A

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

   6. div-add N/A

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

   7. lower-+.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around -inf

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

7. Applied rewrites 76.2%

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

   1. lift-+.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. tan-sum N/A

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

   4. lower-/.f64 N/A

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

   5. tan-quot N/A

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

   6. tan-quot N/A

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

   7. lower-+.f64 N/A

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

   8. tan-quot N/A

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

   9. lift-tan.f64 N/A

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

   10. tan-quot N/A

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

   11. lift-tan.f64 N/A

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

   12. lower--.f64 N/A

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

   13. tan-quot N/A

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

   14. tan-quot N/A

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

   15. lower-*.f64 N/A

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

   16. tan-quot N/A

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

   17. lift-tan.f64 N/A

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

   18. tan-quot N/A

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

   19. lift-tan.f64 99.3

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

9. Applied rewrites 99.3%

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

10. Taylor expanded in y around 0

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

12. Applied rewrites 77.1%

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

13. Final simplification 77.1%

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

Alternative 6: 79.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Python

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

MATLAB

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

Derivation

1. Initial program 76.3%

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

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 76.3

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

4. Applied rewrites 76.3%

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

Alternative 7: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;y + z \leq 0.2:\\ \;\;\;\;\left(\tan y + x\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \tan z\\ \end{array} \end{array} \]

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) 0.2) (- (+ (tan y) x) (tan a)) (+ x (tan z))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 0.2) {
		tmp = (tan(y) + x) - tan(a);
	} else {
		tmp = x + tan(z);
	}
	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) <= 0.2d0) then
        tmp = (tan(y) + x) - tan(a)
    else
        tmp = x + tan(z)
    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) <= 0.2) {
		tmp = (Math.tan(y) + x) - Math.tan(a);
	} else {
		tmp = x + Math.tan(z);
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	tmp = 0
	if (y + z) <= 0.2:
		tmp = (math.tan(y) + x) - math.tan(a)
	else:
		tmp = x + math.tan(z)
	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) <= 0.2)
		tmp = Float64(Float64(tan(y) + x) - tan(a));
	else
		tmp = Float64(x + tan(z));
	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) <= 0.2)
		tmp = (tan(y) + x) - tan(a);
	else
		tmp = x + tan(z);
	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], 0.2], N[(N[(N[Tan[y], $MachinePrecision] + x), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], N[(x + N[Tan[z], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < 0.20000000000000001

   1. Initial program 81.8%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      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 67.3

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

   5. Applied rewrites 67.3%

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

   if 0.20000000000000001 < (+.f64 y z)

   1. Initial program 66.4%

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

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

      4. quot-tan N/A

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

      5. quot-tan N/A

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

      6. div-add N/A

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

      7. lower-+.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 48.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto x + \tan z \]
   9. Step-by-step derivation

   10. Applied rewrites 36.8%

       \[\leadsto x + \tan z \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq 0.2:\\ \;\;\;\;\left(\tan y + x\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \tan z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-10}:\\ \;\;\;\;\left(\tan y + x\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

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 -1.85e-10) (- (+ (tan y) x) (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 <= -1.85e-10) {
		tmp = (tan(y) + x) - 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 <= (-1.85d-10)) then
        tmp = (tan(y) + x) - 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 <= -1.85e-10) {
		tmp = (Math.tan(y) + x) - 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 <= -1.85e-10:
		tmp = (math.tan(y) + x) - 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 (y <= -1.85e-10)
		tmp = Float64(Float64(tan(y) + x) - 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 <= -1.85e-10)
		tmp = (tan(y) + x) - 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[y, -1.85e-10], N[(N[(N[Tan[y], $MachinePrecision] + x), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.85000000000000007e-10

   1. Initial program 52.6%

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

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

   if -1.85000000000000007e-10 < y

   1. Initial program 84.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 74.0%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-10}:\\ \;\;\;\;\left(\tan y + x\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-10}:\\ \;\;\;\;\left(\tan y + x\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;\left(\tan z + x\right) - \tan a\\ \end{array} \end{array} \]

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 -1.85e-10) (- (+ (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 <= -1.85e-10) {
		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 <= (-1.85d-10)) 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 <= -1.85e-10) {
		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 <= -1.85e-10:
		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 (y <= -1.85e-10)
		tmp = Float64(Float64(tan(y) + 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 <= -1.85e-10)
		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[y, -1.85e-10], N[(N[(N[Tan[y], $MachinePrecision] + x), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], N[(N[(N[Tan[z], $MachinePrecision] + x), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.85000000000000007e-10

   1. Initial program 52.6%

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

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

   if -1.85000000000000007e-10 < y

   1. Initial program 84.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   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 74.0

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

   5. Applied rewrites 74.0%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-10}:\\ \;\;\;\;\left(\tan y + x\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;\left(\tan z + x\right) - \tan a\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.8% 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 76.3%

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

Alternative 11: 50.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y, z, a] = \mathsf{sort}([x, y, z, a])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-13}:\\ \;\;\;\;x + \tan y\\ \mathbf{else}:\\ \;\;\;\;x + \tan z\\ \end{array} \end{array} \]

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 -3.5e-13) (+ x (tan y)) (+ x (tan z))))

C

assert(x < y && y < z && z < a);
double code(double x, double y, double z, double a) {
	double tmp;
	if (y <= -3.5e-13) {
		tmp = x + tan(y);
	} else {
		tmp = x + tan(z);
	}
	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 <= (-3.5d-13)) then
        tmp = x + tan(y)
    else
        tmp = x + tan(z)
    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 <= -3.5e-13) {
		tmp = x + Math.tan(y);
	} else {
		tmp = x + Math.tan(z);
	}
	return tmp;
}

Python

[x, y, z, a] = sort([x, y, z, a])
def code(x, y, z, a):
	tmp = 0
	if y <= -3.5e-13:
		tmp = x + math.tan(y)
	else:
		tmp = x + math.tan(z)
	return tmp

Julia

x, y, z, a = sort([x, y, z, a])
function code(x, y, z, a)
	tmp = 0.0
	if (y <= -3.5e-13)
		tmp = Float64(x + tan(y));
	else
		tmp = Float64(x + tan(z));
	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 <= -3.5e-13)
		tmp = x + tan(y);
	else
		tmp = x + tan(z);
	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[y, -3.5e-13], N[(x + N[Tan[y], $MachinePrecision]), $MachinePrecision], N[(x + N[Tan[z], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.5000000000000002e-13

   1. Initial program 52.6%

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

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

      4. quot-tan N/A

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

      5. quot-tan N/A

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

      6. div-add N/A

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

      7. lower-+.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 34.5%

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

   8. Taylor expanded in y around inf

      \[\leadsto x + \tan y \]
   9. Step-by-step derivation

   10. Applied rewrites 35.3%

       \[\leadsto x + \tan y \]

   if -3.5000000000000002e-13 < y

   1. Initial program 84.7%

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

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

      4. quot-tan N/A

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

      5. quot-tan N/A

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

      6. div-add N/A

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

      7. lower-+.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 57.7%

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

   8. Taylor expanded in y around 0

      \[\leadsto x + \tan z \]
   9. Step-by-step derivation

   10. Applied rewrites 51.9%

       \[\leadsto x + \tan z \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-13}:\\ \;\;\;\;x + \tan y\\ \mathbf{else}:\\ \;\;\;\;x + \tan z\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.9% accurate, 2.0× speedup

Math

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

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 y))))

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 76.3%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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 51.6

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

5. Applied rewrites 51.6%

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

6. Final simplification 51.6%

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

Alternative 13: 40.9% 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 76.3%

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

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

   4. quot-tan N/A

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

   5. quot-tan N/A

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

   6. div-add N/A

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

   7. lower-+.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a around 0

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

7. Applied rewrites 51.6%

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

8. Taylor expanded in y around inf

   \[\leadsto x + \tan y \]
9. Step-by-step derivation

10. Applied rewrites 40.6%

    \[\leadsto x + \tan y \]

11. Final simplification 40.6%

    \[\leadsto x + \tan y \]
12. Add Preprocessing

Alternative 14: 31.7% 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 76.3%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 33.0%

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

6. Final simplification 33.0%

   \[\leadsto x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025037 
(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))))