tan-example (used to crash)

Percentage Accurate: 79.2% → 99.7%

Time: 7.0s

Alternatives: 17

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

(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

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

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

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

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

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

MATLAB

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

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[Tan[y], $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[Tan[z], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.2%

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

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

3. Applied rewrites 99.7%

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

Alternative 2: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\tan a \leq 4 \cdot 10^{-10}:\\ \;\;\;\;x + \left(\frac{\tan z + \tan y}{1 - \tan z \cdot \tan y} - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (- (tan (+ y z)) (tan a)))))
   (if (<= (tan a) -0.02)
     t_0
     (if (<= (tan a) 4e-10)
       (+ x (- (/ (+ (tan z) (tan y)) (- 1.0 (* (tan z) (tan y)))) a))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x + (tan((y + z)) - tan(a));
	double tmp;
	if (tan(a) <= -0.02) {
		tmp = t_0;
	} else if (tan(a) <= 4e-10) {
		tmp = x + (((tan(z) + tan(y)) / (1.0 - (tan(z) * tan(y)))) - a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (tan((y + z)) - tan(a))
    if (tan(a) <= (-0.02d0)) then
        tmp = t_0
    else if (tan(a) <= 4d-10) then
        tmp = x + (((tan(z) + tan(y)) / (1.0d0 - (tan(z) * tan(y)))) - a)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = x + (Math.tan((y + z)) - Math.tan(a));
	double tmp;
	if (Math.tan(a) <= -0.02) {
		tmp = t_0;
	} else if (Math.tan(a) <= 4e-10) {
		tmp = x + (((Math.tan(z) + Math.tan(y)) / (1.0 - (Math.tan(z) * Math.tan(y)))) - a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = x + (math.tan((y + z)) - math.tan(a))
	tmp = 0
	if math.tan(a) <= -0.02:
		tmp = t_0
	elif math.tan(a) <= 4e-10:
		tmp = x + (((math.tan(z) + math.tan(y)) / (1.0 - (math.tan(z) * math.tan(y)))) - a)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
	tmp = 0.0
	if (tan(a) <= -0.02)
		tmp = t_0;
	elseif (tan(a) <= 4e-10)
		tmp = Float64(x + Float64(Float64(Float64(tan(z) + tan(y)) / Float64(1.0 - Float64(tan(z) * tan(y)))) - a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = x + (tan((y + z)) - tan(a));
	tmp = 0.0;
	if (tan(a) <= -0.02)
		tmp = t_0;
	elseif (tan(a) <= 4e-10)
		tmp = x + (((tan(z) + tan(y)) / (1.0 - (tan(z) * tan(y)))) - a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.02], t$95$0, If[LessEqual[N[Tan[a], $MachinePrecision], 4e-10], 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] - a), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0200000000000000004 or 4.00000000000000015e-10 < (tan.f64 a)

   1. Initial program 80.2%

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

   if -0.0200000000000000004 < (tan.f64 a) < 4.00000000000000015e-10

   1. Initial program 78.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. 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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 99.0%

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

Alternative 3: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. 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 99.7

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

3. Applied rewrites 99.7%

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

Alternative 4: 90.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(\frac{\frac{\sin z}{\cos z} + \tan y}{1} - \tan a\right)\\ \mathbf{if}\;a \leq -0.04:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.00042:\\ \;\;\;\;x + \left(\frac{\tan z + \tan y}{1 - \tan z \cdot \tan y} - \mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, 0.13333333333333333, 0.3333333333333333\right), a \cdot a, 1\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (- (/ (+ (/ (sin z) (cos z)) (tan y)) 1.0) (tan a)))))
   (if (<= a -0.04)
     t_0
     (if (<= a 0.00042)
       (+
        x
        (-
         (/ (+ (tan z) (tan y)) (- 1.0 (* (tan z) (tan y))))
         (*
          (fma
           (fma (* a a) 0.13333333333333333 0.3333333333333333)
           (* a a)
           1.0)
          a)))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x + ((((sin(z) / cos(z)) + tan(y)) / 1.0) - tan(a));
	double tmp;
	if (a <= -0.04) {
		tmp = t_0;
	} else if (a <= 0.00042) {
		tmp = x + (((tan(z) + tan(y)) / (1.0 - (tan(z) * tan(y)))) - (fma(fma((a * a), 0.13333333333333333, 0.3333333333333333), (a * a), 1.0) * a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x + Float64(Float64(Float64(Float64(sin(z) / cos(z)) + tan(y)) / 1.0) - tan(a)))
	tmp = 0.0
	if (a <= -0.04)
		tmp = t_0;
	elseif (a <= 0.00042)
		tmp = Float64(x + Float64(Float64(Float64(tan(z) + tan(y)) / Float64(1.0 - Float64(tan(z) * tan(y)))) - Float64(fma(fma(Float64(a * a), 0.13333333333333333, 0.3333333333333333), Float64(a * a), 1.0) * a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[(N[(N[(N[Sin[z], $MachinePrecision] / N[Cos[z], $MachinePrecision]), $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.04], t$95$0, If[LessEqual[a, 0.00042], 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[(N[(N[(N[(a * a), $MachinePrecision] * 0.13333333333333333 + 0.3333333333333333), $MachinePrecision] * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0400000000000000008 or 4.2000000000000002e-4 < a

   1. Initial program 80.3%

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

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

   3. Applied rewrites 99.6%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 80.7%

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

   if -0.0400000000000000008 < a < 4.2000000000000002e-4

   1. Initial program 78.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. 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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

Alternative 5: 90.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(\frac{\frac{\sin z}{\cos z} + \tan y}{1} - \tan a\right)\\ \mathbf{if}\;a \leq -0.0112:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.00042:\\ \;\;\;\;x + \left(\frac{\tan z + \tan y}{1 - \tan z \cdot \tan y} - \mathsf{fma}\left(a \cdot a, 0.3333333333333333, 1\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (- (/ (+ (/ (sin z) (cos z)) (tan y)) 1.0) (tan a)))))
   (if (<= a -0.0112)
     t_0
     (if (<= a 0.00042)
       (+
        x
        (-
         (/ (+ (tan z) (tan y)) (- 1.0 (* (tan z) (tan y))))
         (* (fma (* a a) 0.3333333333333333 1.0) a)))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x + ((((sin(z) / cos(z)) + tan(y)) / 1.0) - tan(a));
	double tmp;
	if (a <= -0.0112) {
		tmp = t_0;
	} else if (a <= 0.00042) {
		tmp = x + (((tan(z) + tan(y)) / (1.0 - (tan(z) * tan(y)))) - (fma((a * a), 0.3333333333333333, 1.0) * a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x + Float64(Float64(Float64(Float64(sin(z) / cos(z)) + tan(y)) / 1.0) - tan(a)))
	tmp = 0.0
	if (a <= -0.0112)
		tmp = t_0;
	elseif (a <= 0.00042)
		tmp = Float64(x + Float64(Float64(Float64(tan(z) + tan(y)) / Float64(1.0 - Float64(tan(z) * tan(y)))) - Float64(fma(Float64(a * a), 0.3333333333333333, 1.0) * a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[(N[(N[(N[Sin[z], $MachinePrecision] / N[Cos[z], $MachinePrecision]), $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.0112], t$95$0, If[LessEqual[a, 0.00042], 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[(N[(N[(a * a), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0111999999999999999 or 4.2000000000000002e-4 < a

   1. Initial program 80.4%

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

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

   3. Applied rewrites 99.6%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 80.7%

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

   if -0.0111999999999999999 < a < 4.2000000000000002e-4

   1. Initial program 78.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. 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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

Alternative 6: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(\frac{\frac{\sin z}{\cos z} + \tan y}{1} - \tan a\right)\\ \mathbf{if}\;a \leq -0.0054:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.000215:\\ \;\;\;\;x + \left(\frac{\tan z + \tan y}{1 - \tan z \cdot \tan y} - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ x (- (/ (+ (/ (sin z) (cos z)) (tan y)) 1.0) (tan a)))))
   (if (<= a -0.0054)
     t_0
     (if (<= a 0.000215)
       (+ x (- (/ (+ (tan z) (tan y)) (- 1.0 (* (tan z) (tan y)))) a))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x + ((((sin(z) / cos(z)) + tan(y)) / 1.0) - tan(a));
	double tmp;
	if (a <= -0.0054) {
		tmp = t_0;
	} else if (a <= 0.000215) {
		tmp = x + (((tan(z) + tan(y)) / (1.0 - (tan(z) * tan(y)))) - a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + ((((sin(z) / cos(z)) + tan(y)) / 1.0d0) - tan(a))
    if (a <= (-0.0054d0)) then
        tmp = t_0
    else if (a <= 0.000215d0) then
        tmp = x + (((tan(z) + tan(y)) / (1.0d0 - (tan(z) * tan(y)))) - a)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = x + ((((Math.sin(z) / Math.cos(z)) + Math.tan(y)) / 1.0) - Math.tan(a));
	double tmp;
	if (a <= -0.0054) {
		tmp = t_0;
	} else if (a <= 0.000215) {
		tmp = x + (((Math.tan(z) + Math.tan(y)) / (1.0 - (Math.tan(z) * Math.tan(y)))) - a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = x + ((((math.sin(z) / math.cos(z)) + math.tan(y)) / 1.0) - math.tan(a))
	tmp = 0
	if a <= -0.0054:
		tmp = t_0
	elif a <= 0.000215:
		tmp = x + (((math.tan(z) + math.tan(y)) / (1.0 - (math.tan(z) * math.tan(y)))) - a)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(x + Float64(Float64(Float64(Float64(sin(z) / cos(z)) + tan(y)) / 1.0) - tan(a)))
	tmp = 0.0
	if (a <= -0.0054)
		tmp = t_0;
	elseif (a <= 0.000215)
		tmp = Float64(x + Float64(Float64(Float64(tan(z) + tan(y)) / Float64(1.0 - Float64(tan(z) * tan(y)))) - a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = x + ((((sin(z) / cos(z)) + tan(y)) / 1.0) - tan(a));
	tmp = 0.0;
	if (a <= -0.0054)
		tmp = t_0;
	elseif (a <= 0.000215)
		tmp = x + (((tan(z) + tan(y)) / (1.0 - (tan(z) * tan(y)))) - a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x + N[(N[(N[(N[(N[Sin[z], $MachinePrecision] / N[Cos[z], $MachinePrecision]), $MachinePrecision] + N[Tan[y], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.0054], t$95$0, If[LessEqual[a, 0.000215], 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] - a), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0054000000000000003 or 2.14999999999999995e-4 < a

   1. Initial program 80.3%

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

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

   3. Applied rewrites 99.6%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 80.7%

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

   if -0.0054000000000000003 < a < 2.14999999999999995e-4

   1. Initial program 78.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. 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 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 99.6%

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

Alternative 7: 60.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq 4 \cdot 10^{-102}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) 4e-102) (+ x (- (tan y) (tan a))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 4e-102) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (tan(z) - tan(a));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((y + z) <= 4d-102) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (tan(z) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 4e-102) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= 4e-102:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= 4e-102)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= 4e-102)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], 4e-102], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < 3.99999999999999973e-102

   1. Initial program 81.0%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 63.2%

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

   if 3.99999999999999973e-102 < (+.f64 y z)

   1. Initial program 77.1%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 56.4%

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

Alternative 8: 60.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq 4 \cdot 10^{-102}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan z + \left(x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) 4e-102) (+ x (- (tan y) (tan a))) (+ (tan z) (- x (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 4e-102) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = tan(z) + (x - tan(a));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((y + z) <= 4d-102) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = tan(z) + (x - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 4e-102) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = Math.tan(z) + (x - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= 4e-102:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = math.tan(z) + (x - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= 4e-102)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(tan(z) + Float64(x - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= 4e-102)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = tan(z) + (x - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], 4e-102], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Tan[z], $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < 3.99999999999999973e-102

   1. Initial program 81.0%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 63.2%

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

   if 3.99999999999999973e-102 < (+.f64 y z)

   1. Initial program 77.1%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x + \frac{\sin z}{\cos z}\right) - \frac{\sin a}{\cos a}} \]
   3. 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 56.3

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

   4. Applied rewrites 56.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. tan-quot N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-tan.f64 N/A

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

      11. lower--.f64 N/A

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

      12. tan-quot N/A

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

      13. lift-tan.f64 56.3

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

   6. Applied rewrites 56.3%

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

Alternative 9: 60.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq 4 \cdot 10^{-102}:\\ \;\;\;\;\left(\tan y + x\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;\tan z + \left(x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) 4e-102) (- (+ (tan y) x) (tan a)) (+ (tan z) (- x (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 4e-102) {
		tmp = (tan(y) + x) - tan(a);
	} else {
		tmp = tan(z) + (x - tan(a));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((y + z) <= 4d-102) then
        tmp = (tan(y) + x) - tan(a)
    else
        tmp = tan(z) + (x - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 4e-102) {
		tmp = (Math.tan(y) + x) - Math.tan(a);
	} else {
		tmp = Math.tan(z) + (x - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= 4e-102:
		tmp = (math.tan(y) + x) - math.tan(a)
	else:
		tmp = math.tan(z) + (x - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= 4e-102)
		tmp = Float64(Float64(tan(y) + x) - tan(a));
	else
		tmp = Float64(tan(z) + Float64(x - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= 4e-102)
		tmp = (tan(y) + x) - tan(a);
	else
		tmp = tan(z) + (x - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], 4e-102], N[(N[(N[Tan[y], $MachinePrecision] + x), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], N[(N[Tan[z], $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < 3.99999999999999973e-102

   1. Initial program 81.0%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. quot-tan N/A

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

      5. lower-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. lift-tan.f64 63.1

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

   4. Applied rewrites 63.1%

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

   if 3.99999999999999973e-102 < (+.f64 y z)

   1. Initial program 77.1%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x + \frac{\sin z}{\cos z}\right) - \frac{\sin a}{\cos a}} \]
   3. 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 56.3

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

   4. Applied rewrites 56.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. tan-quot N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-tan.f64 N/A

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

      11. lower--.f64 N/A

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

      12. tan-quot N/A

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

      13. lift-tan.f64 56.3

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

   6. Applied rewrites 56.3%

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

Alternative 10: 60.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -0.5:\\ \;\;\;\;x + \tan \left(z + y\right)\\ \mathbf{else}:\\ \;\;\;\;\tan z + \left(x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -0.5) (+ x (tan (+ z y))) (+ (tan z) (- x (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -0.5) {
		tmp = x + tan((z + y));
	} else {
		tmp = tan(z) + (x - tan(a));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((y + z) <= (-0.5d0)) then
        tmp = x + tan((z + y))
    else
        tmp = tan(z) + (x - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -0.5) {
		tmp = x + Math.tan((z + y));
	} else {
		tmp = Math.tan(z) + (x - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -0.5:
		tmp = x + math.tan((z + y))
	else:
		tmp = math.tan(z) + (x - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -0.5)
		tmp = Float64(x + tan(Float64(z + y)));
	else
		tmp = Float64(tan(z) + Float64(x - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -0.5)
		tmp = x + tan((z + y));
	else
		tmp = tan(z) + (x - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -0.5], N[(x + N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Tan[z], $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -0.5

   1. Initial program 72.3%

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

   2. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} \]
   3. 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 46.4

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

   4. Applied rewrites 46.4%

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

   if -0.5 < (+.f64 y z)

   1. Initial program 83.6%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x + \frac{\sin z}{\cos z}\right) - \frac{\sin a}{\cos a}} \]
   3. 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 68.4

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

   4. Applied rewrites 68.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. tan-quot N/A

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

      7. associate--l+ N/A

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

      8. lower-+.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-tan.f64 N/A

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

      11. lower--.f64 N/A

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

      12. tan-quot N/A

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

      13. lift-tan.f64 68.4

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

   6. Applied rewrites 68.4%

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

Alternative 11: 79.2% 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]

Derivation

1. Initial program 79.2%

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

Alternative 12: 59.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -0.0052:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.15:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \mathsf{fma}\left(0.3333333333333333, a \cdot a, 1\right) \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -0.0052)
     t_0
     (if (<= a 0.15)
       (+ x (- (tan (+ y z)) (* (fma 0.3333333333333333 (* a a) 1.0) a)))
       t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -0.0052) {
		tmp = t_0;
	} else if (a <= 0.15) {
		tmp = x + (tan((y + z)) - (fma(0.3333333333333333, (a * a), 1.0) * a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -0.0052)
		tmp = t_0;
	elseif (a <= 0.15)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - Float64(fma(0.3333333333333333, Float64(a * a), 1.0) * a)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.0052], t$95$0, If[LessEqual[a, 0.15], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[(N[(0.3333333333333333 * N[(a * a), $MachinePrecision] + 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0051999999999999998 or 0.149999999999999994 < a

   1. Initial program 80.3%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x + \frac{\sin z}{\cos z}\right) - \frac{\sin a}{\cos a}} \]
   3. 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 61.5

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

   4. Applied rewrites 61.5%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 41.9%

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

   if -0.0051999999999999998 < a < 0.149999999999999994

   1. Initial program 78.2%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 78.1

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

   4. Applied rewrites 78.1%

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

Alternative 13: 54.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -0.5:\\ \;\;\;\;x + \tan \left(z + y\right)\\ \mathbf{elif}\;y + z \leq 0.0004:\\ \;\;\;\;\left(z + x\right) - \tan a\\ \mathbf{else}:\\ \;\;\;\;\tan z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -0.5)
   (+ x (tan (+ z y)))
   (if (<= (+ y z) 0.0004) (- (+ z x) (tan a)) (+ (tan z) x))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -0.5) {
		tmp = x + tan((z + y));
	} else if ((y + z) <= 0.0004) {
		tmp = (z + x) - tan(a);
	} else {
		tmp = tan(z) + x;
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((y + z) <= (-0.5d0)) then
        tmp = x + tan((z + y))
    else if ((y + z) <= 0.0004d0) then
        tmp = (z + x) - tan(a)
    else
        tmp = tan(z) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -0.5) {
		tmp = x + Math.tan((z + y));
	} else if ((y + z) <= 0.0004) {
		tmp = (z + x) - Math.tan(a);
	} else {
		tmp = Math.tan(z) + x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -0.5:
		tmp = x + math.tan((z + y))
	elif (y + z) <= 0.0004:
		tmp = (z + x) - math.tan(a)
	else:
		tmp = math.tan(z) + x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -0.5)
		tmp = Float64(x + tan(Float64(z + y)));
	elseif (Float64(y + z) <= 0.0004)
		tmp = Float64(Float64(z + x) - tan(a));
	else
		tmp = Float64(tan(z) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -0.5)
		tmp = x + tan((z + y));
	elseif ((y + z) <= 0.0004)
		tmp = (z + x) - tan(a);
	else
		tmp = tan(z) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -0.5], N[(x + N[Tan[N[(z + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y + z), $MachinePrecision], 0.0004], N[(N[(z + x), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], N[(N[Tan[z], $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 y z) < -0.5

   1. Initial program 72.3%

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

   2. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} \]
   3. 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 46.4

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

   4. Applied rewrites 46.4%

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

   if -0.5 < (+.f64 y z) < 4.00000000000000019e-4

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x + \frac{\sin z}{\cos z}\right) - \frac{\sin a}{\cos a}} \]
   3. 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 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in z around 0

      \[\leadsto \left(z + x\right) - \tan a \]
   6. Step-by-step derivation

   7. Applied rewrites 98.3%

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

   if 4.00000000000000019e-4 < (+.f64 y z)

   1. Initial program 73.1%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x + \frac{\sin z}{\cos z}\right) - \frac{\sin a}{\cos a}} \]
   3. 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 49.0

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

   4. Applied rewrites 49.0%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\sin z}{\cos z} + x \]

      2. tan-quot N/A

         \[\leadsto \tan z + x \]

      3. lift-tan.f64 N/A

         \[\leadsto \tan z + x \]

      4. lift-+.f64 34.9

         \[\leadsto \tan z + x \]

   7. Applied rewrites 34.9%

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

Alternative 14: 59.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \tan a\\ \mathbf{if}\;a \leq -0.00095:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 0.033:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))))
   (if (<= a -0.00095) t_0 (if (<= a 0.033) (+ x (- (tan (+ y z)) a)) t_0))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double tmp;
	if (a <= -0.00095) {
		tmp = t_0;
	} else if (a <= 0.033) {
		tmp = x + (tan((y + z)) - a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - tan(a)
    if (a <= (-0.00095d0)) then
        tmp = t_0
    else if (a <= 0.033d0) then
        tmp = x + (tan((y + z)) - a)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = x - Math.tan(a);
	double tmp;
	if (a <= -0.00095) {
		tmp = t_0;
	} else if (a <= 0.033) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = x - math.tan(a)
	tmp = 0
	if a <= -0.00095:
		tmp = t_0
	elif a <= 0.033:
		tmp = x + (math.tan((y + z)) - a)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	tmp = 0.0
	if (a <= -0.00095)
		tmp = t_0;
	elseif (a <= 0.033)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = x - tan(a);
	tmp = 0.0;
	if (a <= -0.00095)
		tmp = t_0;
	elseif (a <= 0.033)
		tmp = x + (tan((y + z)) - a);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -0.00095], t$95$0, If[LessEqual[a, 0.033], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -9.49999999999999998e-4 or 0.033000000000000002 < a

   1. Initial program 80.3%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x + \frac{\sin z}{\cos z}\right) - \frac{\sin a}{\cos a}} \]
   3. 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 61.5

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

   4. Applied rewrites 61.5%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 41.9%

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

   if -9.49999999999999998e-4 < a < 0.033000000000000002

   1. Initial program 78.2%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 78.0%

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

Alternative 15: 45.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq 10^{-20}:\\ \;\;\;\;x - \tan a\\ \mathbf{else}:\\ \;\;\;\;\tan z + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) 1e-20) (- x (tan a)) (+ (tan z) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 1e-20) {
		tmp = x - tan(a);
	} else {
		tmp = tan(z) + x;
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((y + z) <= 1d-20) then
        tmp = x - tan(a)
    else
        tmp = tan(z) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 1e-20) {
		tmp = x - Math.tan(a);
	} else {
		tmp = Math.tan(z) + x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= 1e-20:
		tmp = x - math.tan(a)
	else:
		tmp = math.tan(z) + x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= 1e-20)
		tmp = Float64(x - tan(a));
	else
		tmp = Float64(tan(z) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= 1e-20)
		tmp = x - tan(a);
	else
		tmp = tan(z) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], 1e-20], N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision], N[(N[Tan[z], $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < 9.99999999999999945e-21

   1. Initial program 82.6%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x + \frac{\sin z}{\cos z}\right) - \frac{\sin a}{\cos a}} \]
   3. 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 67.2

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

   4. Applied rewrites 67.2%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 51.5%

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

   if 9.99999999999999945e-21 < (+.f64 y z)

   1. Initial program 74.0%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x + \frac{\sin z}{\cos z}\right) - \frac{\sin a}{\cos a}} \]
   3. 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 50.4

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

   4. Applied rewrites 50.4%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\sin z}{\cos z} + x \]

      2. tan-quot N/A

         \[\leadsto \tan z + x \]

      3. lift-tan.f64 N/A

         \[\leadsto \tan z + x \]

      4. lift-+.f64 35.5

         \[\leadsto \tan z + x \]

   7. Applied rewrites 35.5%

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

Alternative 16: 40.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \tan z + x \end{array} \]

FPCore

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

C

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

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 = tan(z) + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(x + \frac{\sin z}{\cos z}\right) - \frac{\sin a}{\cos a}} \]
3. 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 60.7

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

4. Applied rewrites 60.7%

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

5. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{\sin z}{\cos z} + x \]

   2. tan-quot N/A

      \[\leadsto \tan z + x \]

   3. lift-tan.f64 N/A

      \[\leadsto \tan z + x \]

   4. lift-+.f64 40.6

      \[\leadsto \tan z + x \]

7. Applied rewrites 40.6%

   \[\leadsto \tan z + \color{blue}{x} \]
8. Add Preprocessing

Alternative 17: 31.3% accurate, 210.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

function code(x, y, z, a)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, a_] := x

Derivation

1. Initial program 79.2%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 31.3%

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

Reproduce

herbie shell --seed 2025105 
(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))))