Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3

Percentage Accurate: 97.2% → 97.2%

Time: 4.7s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / 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, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - ((y - z) / (((t - z) + 1.0d0) / a))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

Python

def code(x, y, z, t, a):
	return x - ((y - z) / (((t - z) + 1.0) / a))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / 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, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - ((y - z) / (((t - z) + 1.0d0) / a))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

Python

def code(x, y, z, t, a):
	return x - ((y - z) / (((t - z) + 1.0) / a))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(1 + t\right) - z}{a}\\ x - \left(\frac{y}{t\_1} - \frac{z}{t\_1}\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- (+ 1.0 t) z) a))) (- x (- (/ y t_1) (/ z t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((1.0 + t) - z) / a;
	return x - ((y / t_1) - (z / t_1));
}

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, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    t_1 = ((1.0d0 + t) - z) / a
    code = x - ((y / t_1) - (z / t_1))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((1.0 + t) - z) / a;
	return x - ((y / t_1) - (z / t_1));
}

Python

def code(x, y, z, t, a):
	t_1 = ((1.0 + t) - z) / a
	return x - ((y / t_1) - (z / t_1))

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(1.0 + t) - z) / a)
	return Float64(x - Float64(Float64(y / t_1) - Float64(z / t_1)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	t_1 = ((1.0 + t) - z) / a;
	tmp = x - ((y / t_1) - (z / t_1));
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(1.0 + t), $MachinePrecision] - z), $MachinePrecision] / a), $MachinePrecision]}, N[(x - N[(N[(y / t$95$1), $MachinePrecision] - N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 97.2%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
2. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift--.f64 N/A

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

   6. div-sub N/A

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

   7. lower--.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. +-commutative N/A

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

   10. associate--l+ N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-+.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. +-commutative N/A

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

   16. associate--l+ N/A

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

   17. lower-/.f64 N/A

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

   18. lower--.f64 N/A

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

   19. lower-+.f64 97.2

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

3. Applied rewrites 97.2%

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

Alternative 2: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / 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, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - ((y - z) / (((t - z) + 1.0d0) / a))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

Python

def code(x, y, z, t, a):
	return x - ((y - z) / (((t - z) + 1.0) / a))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.2%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
2. Add Preprocessing

Alternative 3: 94.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y - z}{\frac{t}{a} - \frac{z}{a}}\\ \mathbf{if}\;t \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+60}:\\ \;\;\;\;x - \left(y - z\right) \cdot \frac{a}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (- y z) (- (/ t a) (/ z a))))))
   (if (<= t -1.0)
     t_1
     (if (<= t 8.5e+60) (- x (* (- y z) (/ a (- 1.0 z)))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((y - z) / ((t / a) - (z / a)));
	double tmp;
	if (t <= -1.0) {
		tmp = t_1;
	} else if (t <= 8.5e+60) {
		tmp = x - ((y - z) * (a / (1.0 - z)));
	} else {
		tmp = t_1;
	}
	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, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x - ((y - z) / ((t / a) - (z / a)))
    if (t <= (-1.0d0)) then
        tmp = t_1
    else if (t <= 8.5d+60) then
        tmp = x - ((y - z) * (a / (1.0d0 - z)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((y - z) / ((t / a) - (z / a)));
	double tmp;
	if (t <= -1.0) {
		tmp = t_1;
	} else if (t <= 8.5e+60) {
		tmp = x - ((y - z) * (a / (1.0 - z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x - ((y - z) / ((t / a) - (z / a)))
	tmp = 0
	if t <= -1.0:
		tmp = t_1
	elif t <= 8.5e+60:
		tmp = x - ((y - z) * (a / (1.0 - z)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(y - z) / Float64(Float64(t / a) - Float64(z / a))))
	tmp = 0.0
	if (t <= -1.0)
		tmp = t_1;
	elseif (t <= 8.5e+60)
		tmp = Float64(x - Float64(Float64(y - z) * Float64(a / Float64(1.0 - z))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x - ((y - z) / ((t / a) - (z / a)));
	tmp = 0.0;
	if (t <= -1.0)
		tmp = t_1;
	elseif (t <= 8.5e+60)
		tmp = x - ((y - z) * (a / (1.0 - z)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(y - z), $MachinePrecision] / N[(N[(t / a), $MachinePrecision] - N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.0], t$95$1, If[LessEqual[t, 8.5e+60], N[(x - N[(N[(y - z), $MachinePrecision] * N[(a / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1 or 8.50000000000000064e60 < t

   1. Initial program 96.6%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. div-sub N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\frac{1 + t}{a} - \frac{z}{a}}} \]

      7. div-add-rev N/A

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

      8. lower--.f64 N/A

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

      9. div-add-rev N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\frac{1 + t}{a}} - \frac{z}{a}} \]

      10. lower-/.f64 N/A

          \[\leadsto x - \frac{y - z}{\color{blue}{\frac{1 + t}{a}} - \frac{z}{a}} \]

      11. lower-+.f64 N/A

          \[\leadsto x - \frac{y - z}{\frac{\color{blue}{1 + t}}{a} - \frac{z}{a}} \]

      12. lower-/.f64 92.6

          \[\leadsto x - \frac{y - z}{\frac{1 + t}{a} - \color{blue}{\frac{z}{a}}} \]

   3. Applied rewrites 92.6%

      \[\leadsto x - \frac{y - z}{\color{blue}{\frac{1 + t}{a} - \frac{z}{a}}} \]

   4. Taylor expanded in t around inf

      \[\leadsto x - \frac{y - z}{\frac{\color{blue}{t}}{a} - \frac{z}{a}} \]
   5. Step-by-step derivation

   6. Applied rewrites 92.4%

      \[\leadsto x - \frac{y - z}{\frac{\color{blue}{t}}{a} - \frac{z}{a}} \]

   if -1 < t < 8.50000000000000064e60

   1. Initial program 97.7%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower--.f64 85.2

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

   4. Applied rewrites 85.2%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 95.3

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

   6. Applied rewrites 95.3%

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

Alternative 4: 92.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{z}{t - z}, x\right)\\ \mathbf{if}\;z \leq -2.25 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+39}:\\ \;\;\;\;x - \frac{y - z}{\frac{t + 1}{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ z (- t z)) x)))
   (if (<= z -2.25e+17)
     t_1
     (if (<= z 3.9e+39) (- x (/ (- y z) (/ (+ t 1.0) a))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, (z / (t - z)), x);
	double tmp;
	if (z <= -2.25e+17) {
		tmp = t_1;
	} else if (z <= 3.9e+39) {
		tmp = x - ((y - z) / ((t + 1.0) / a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(z / Float64(t - z)), x)
	tmp = 0.0
	if (z <= -2.25e+17)
		tmp = t_1;
	elseif (z <= 3.9e+39)
		tmp = Float64(x - Float64(Float64(y - z) / Float64(Float64(t + 1.0) / a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(z / N[(t - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -2.25e+17], t$95$1, If[LessEqual[z, 3.9e+39], N[(x - N[(N[(y - z), $MachinePrecision] / N[(N[(t + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.25e17 or 3.9000000000000001e39 < z

   1. Initial program 94.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. associate--l+ N/A

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

      17. lower-/.f64 N/A

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

      18. lower--.f64 N/A

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

      19. lower-+.f64 94.9

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

   3. Applied rewrites 94.9%

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

   4. Taylor expanded in y around 0

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

      1. sub-div N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 86.7

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

   6. Applied rewrites 86.7%

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

   7. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{t - z}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 86.7%

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{t - z}, x\right) \]

   if -2.25e17 < z < 3.9000000000000001e39

   1. Initial program 99.1%

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

   2. Taylor expanded in z around 0

      \[\leadsto x - \frac{y - z}{\frac{\color{blue}{t} + 1}{a}} \]
   3. Step-by-step derivation

   4. Applied rewrites 96.4%

      \[\leadsto x - \frac{y - z}{\frac{\color{blue}{t} + 1}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 90.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y - z}{\frac{t}{a}}\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{+64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+43}:\\ \;\;\;\;x - \left(y - z\right) \cdot \frac{a}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (- y z) (/ t a)))))
   (if (<= t -5.5e+64)
     t_1
     (if (<= t 1.02e+43) (- x (* (- y z) (/ a (- 1.0 z)))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((y - z) / (t / a));
	double tmp;
	if (t <= -5.5e+64) {
		tmp = t_1;
	} else if (t <= 1.02e+43) {
		tmp = x - ((y - z) * (a / (1.0 - z)));
	} else {
		tmp = t_1;
	}
	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, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x - ((y - z) / (t / a))
    if (t <= (-5.5d+64)) then
        tmp = t_1
    else if (t <= 1.02d+43) then
        tmp = x - ((y - z) * (a / (1.0d0 - z)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((y - z) / (t / a));
	double tmp;
	if (t <= -5.5e+64) {
		tmp = t_1;
	} else if (t <= 1.02e+43) {
		tmp = x - ((y - z) * (a / (1.0 - z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x - ((y - z) / (t / a))
	tmp = 0
	if t <= -5.5e+64:
		tmp = t_1
	elif t <= 1.02e+43:
		tmp = x - ((y - z) * (a / (1.0 - z)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(y - z) / Float64(t / a)))
	tmp = 0.0
	if (t <= -5.5e+64)
		tmp = t_1;
	elseif (t <= 1.02e+43)
		tmp = Float64(x - Float64(Float64(y - z) * Float64(a / Float64(1.0 - z))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x - ((y - z) / (t / a));
	tmp = 0.0;
	if (t <= -5.5e+64)
		tmp = t_1;
	elseif (t <= 1.02e+43)
		tmp = x - ((y - z) * (a / (1.0 - z)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(y - z), $MachinePrecision] / N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.5e+64], t$95$1, If[LessEqual[t, 1.02e+43], N[(x - N[(N[(y - z), $MachinePrecision] * N[(a / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.4999999999999996e64 or 1.02e43 < t

   1. Initial program 97.0%

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

   2. Taylor expanded in t around inf

      \[\leadsto x - \frac{y - z}{\frac{\color{blue}{t}}{a}} \]
   3. Step-by-step derivation

   4. Applied rewrites 85.7%

      \[\leadsto x - \frac{y - z}{\frac{\color{blue}{t}}{a}} \]

   if -5.4999999999999996e64 < t < 1.02e43

   1. Initial program 97.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower--.f64 83.9

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

   4. Applied rewrites 83.9%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 93.5

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

   6. Applied rewrites 93.5%

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

Alternative 6: 89.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - y}{\left(1 + t\right) - z} \cdot a\\ t_2 := \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{\left(t + 1\right) - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- z y) (- (+ 1.0 t) z)) a))
        (t_2 (/ (- y z) (/ (+ (- t z) 1.0) a))))
   (if (<= t_2 -5e+95)
     t_1
     (if (<= t_2 1e+139) (fma a (/ z (- (+ t 1.0) z)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - y) / ((1.0 + t) - z)) * a;
	double t_2 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_2 <= -5e+95) {
		tmp = t_1;
	} else if (t_2 <= 1e+139) {
		tmp = fma(a, (z / ((t + 1.0) - z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - y) / Float64(Float64(1.0 + t) - z)) * a)
	t_2 = Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a))
	tmp = 0.0
	if (t_2 <= -5e+95)
		tmp = t_1;
	elseif (t_2 <= 1e+139)
		tmp = fma(a, Float64(z / Float64(Float64(t + 1.0) - z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / N[(N[(1.0 + t), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+95], t$95$1, If[LessEqual[t$95$2, 1e+139], N[(a * N[(z / N[(N[(t + 1.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -5.00000000000000025e95 or 1.00000000000000003e139 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 99.8%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 85.6

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

   4. Applied rewrites 85.6%

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

   if -5.00000000000000025e95 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < 1.00000000000000003e139

   1. Initial program 96.2%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. associate--l+ N/A

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

      17. lower-/.f64 N/A

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

      18. lower--.f64 N/A

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

      19. lower-+.f64 96.2

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

   3. Applied rewrites 96.2%

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

   4. Taylor expanded in y around 0

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

      1. sub-div N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 85.0

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

   6. Applied rewrites 85.0%

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

Alternative 7: 88.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - y}{\frac{\left(1 + t\right) - z}{a}}\\ t_2 := \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{\left(t + 1\right) - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z y) (/ (- (+ 1.0 t) z) a)))
        (t_2 (/ (- y z) (/ (+ (- t z) 1.0) a))))
   (if (<= t_2 -5e+95)
     t_1
     (if (<= t_2 1e+139) (fma a (/ z (- (+ t 1.0) z)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - y) / (((1.0 + t) - z) / a);
	double t_2 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_2 <= -5e+95) {
		tmp = t_1;
	} else if (t_2 <= 1e+139) {
		tmp = fma(a, (z / ((t + 1.0) - z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - y) / Float64(Float64(Float64(1.0 + t) - z) / a))
	t_2 = Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a))
	tmp = 0.0
	if (t_2 <= -5e+95)
		tmp = t_1;
	elseif (t_2 <= 1e+139)
		tmp = fma(a, Float64(z / Float64(Float64(t + 1.0) - z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - y), $MachinePrecision] / N[(N[(N[(1.0 + t), $MachinePrecision] - z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+95], t$95$1, If[LessEqual[t$95$2, 1e+139], N[(a * N[(z / N[(N[(t + 1.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -5.00000000000000025e95 or 1.00000000000000003e139 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 99.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. div-sub N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\frac{1 + t}{a} - \frac{z}{a}}} \]

      7. div-add-rev N/A

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

      8. lower--.f64 N/A

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

      9. div-add-rev N/A

         \[\leadsto x - \frac{y - z}{\color{blue}{\frac{1 + t}{a}} - \frac{z}{a}} \]

      10. lower-/.f64 N/A

          \[\leadsto x - \frac{y - z}{\color{blue}{\frac{1 + t}{a}} - \frac{z}{a}} \]

      11. lower-+.f64 N/A

          \[\leadsto x - \frac{y - z}{\frac{\color{blue}{1 + t}}{a} - \frac{z}{a}} \]

      12. lower-/.f64 99.8

          \[\leadsto x - \frac{y - z}{\frac{1 + t}{a} - \color{blue}{\frac{z}{a}}} \]

   3. Applied rewrites 99.8%

      \[\leadsto x - \frac{y - z}{\color{blue}{\frac{1 + t}{a} - \frac{z}{a}}} \]

   4. Taylor expanded in x around 0

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

      1. sub-div N/A

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

      2. div-add-rev N/A

         \[\leadsto \frac{z - y}{\frac{1 + t}{a} - \frac{\color{blue}{z}}{a}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{z - y}{\color{blue}{\frac{1 + t}{a} - \frac{z}{a}}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{z - y}{\color{blue}{\frac{1 + t}{a}} - \frac{z}{a}} \]

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift-+.f64 85.5

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

   6. Applied rewrites 85.5%

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

   if -5.00000000000000025e95 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < 1.00000000000000003e139

   1. Initial program 96.2%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. associate--l+ N/A

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

      17. lower-/.f64 N/A

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

      18. lower--.f64 N/A

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

      19. lower-+.f64 96.2

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

   3. Applied rewrites 96.2%

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

   4. Taylor expanded in y around 0

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

      1. sub-div N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 85.0

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

   6. Applied rewrites 85.0%

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

Alternative 8: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{z}{\left(t + 1\right) - z}, x\right)\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{-87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-7}:\\ \;\;\;\;x - a \cdot \frac{y}{1 + t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ z (- (+ t 1.0) z)) x)))
   (if (<= z -1.25e-87)
     t_1
     (if (<= z 2.7e-7) (- x (* a (/ y (+ 1.0 t)))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, (z / ((t + 1.0) - z)), x);
	double tmp;
	if (z <= -1.25e-87) {
		tmp = t_1;
	} else if (z <= 2.7e-7) {
		tmp = x - (a * (y / (1.0 + t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(z / Float64(Float64(t + 1.0) - z)), x)
	tmp = 0.0
	if (z <= -1.25e-87)
		tmp = t_1;
	elseif (z <= 2.7e-7)
		tmp = Float64(x - Float64(a * Float64(y / Float64(1.0 + t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(z / N[(N[(t + 1.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.25e-87], t$95$1, If[LessEqual[z, 2.7e-7], N[(x - N[(a * N[(y / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25000000000000011e-87 or 2.70000000000000009e-7 < z

   1. Initial program 95.6%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. associate--l+ N/A

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

      17. lower-/.f64 N/A

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

      18. lower--.f64 N/A

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

      19. lower-+.f64 95.6

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

   3. Applied rewrites 95.6%

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

   4. Taylor expanded in y around 0

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

      1. sub-div N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 83.4

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

   6. Applied rewrites 83.4%

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

   if -1.25000000000000011e-87 < z < 2.70000000000000009e-7

   1. Initial program 99.2%

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

   2. Taylor expanded in z around 0

      \[\leadsto x - \color{blue}{\frac{a \cdot y}{1 + t}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x - a \cdot \color{blue}{\frac{y}{1 + t}} \]

      2. lower-*.f64 N/A

         \[\leadsto x - a \cdot \color{blue}{\frac{y}{1 + t}} \]

      3. lower-/.f64 N/A

         \[\leadsto x - a \cdot \frac{y}{\color{blue}{1 + t}} \]

      4. lower-+.f64 94.0

         \[\leadsto x - a \cdot \frac{y}{1 + \color{blue}{t}} \]

   4. Applied rewrites 94.0%

      \[\leadsto x - \color{blue}{a \cdot \frac{y}{1 + t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{z}{t - z}, x\right)\\ \mathbf{if}\;z \leq -2.15 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;x - a \cdot \frac{y}{1 + t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ z (- t z)) x)))
   (if (<= z -2.15e+17) t_1 (if (<= z 0.75) (- x (* a (/ y (+ 1.0 t)))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, (z / (t - z)), x);
	double tmp;
	if (z <= -2.15e+17) {
		tmp = t_1;
	} else if (z <= 0.75) {
		tmp = x - (a * (y / (1.0 + t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(z / Float64(t - z)), x)
	tmp = 0.0
	if (z <= -2.15e+17)
		tmp = t_1;
	elseif (z <= 0.75)
		tmp = Float64(x - Float64(a * Float64(y / Float64(1.0 + t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(z / N[(t - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -2.15e+17], t$95$1, If[LessEqual[z, 0.75], N[(x - N[(a * N[(y / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.15e17 or 0.75 < z

   1. Initial program 95.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. associate--l+ N/A

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

      17. lower-/.f64 N/A

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

      18. lower--.f64 N/A

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

      19. lower-+.f64 95.1

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

   3. Applied rewrites 95.1%

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

   4. Taylor expanded in y around 0

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

      1. sub-div N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 85.6

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

   6. Applied rewrites 85.6%

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

   7. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{t - z}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 85.5%

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{t - z}, x\right) \]

   if -2.15e17 < z < 0.75

   1. Initial program 99.1%

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

   2. Taylor expanded in z around 0

      \[\leadsto x - \color{blue}{\frac{a \cdot y}{1 + t}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto x - a \cdot \color{blue}{\frac{y}{1 + t}} \]

      2. lower-*.f64 N/A

         \[\leadsto x - a \cdot \color{blue}{\frac{y}{1 + t}} \]

      3. lower-/.f64 N/A

         \[\leadsto x - a \cdot \frac{y}{\color{blue}{1 + t}} \]

      4. lower-+.f64 92.2

         \[\leadsto x - a \cdot \frac{y}{1 + \color{blue}{t}} \]

   4. Applied rewrites 92.2%

      \[\leadsto x - \color{blue}{a \cdot \frac{y}{1 + t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 77.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ t_2 := \frac{z - y}{t - z} \cdot a\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+305}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+95}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t - z}, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{1 - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y z) (/ (+ (- t z) 1.0) a)))
        (t_2 (* (/ (- z y) (- t z)) a)))
   (if (<= t_1 -1e+305)
     (- x (* a y))
     (if (<= t_1 -5e+95)
       t_2
       (if (<= t_1 1e-95)
         (fma a (/ z (- t z)) x)
         (if (<= t_1 1e+139) (fma a (/ z (- 1.0 z)) x) t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) / (((t - z) + 1.0) / a);
	double t_2 = ((z - y) / (t - z)) * a;
	double tmp;
	if (t_1 <= -1e+305) {
		tmp = x - (a * y);
	} else if (t_1 <= -5e+95) {
		tmp = t_2;
	} else if (t_1 <= 1e-95) {
		tmp = fma(a, (z / (t - z)), x);
	} else if (t_1 <= 1e+139) {
		tmp = fma(a, (z / (1.0 - z)), x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a))
	t_2 = Float64(Float64(Float64(z - y) / Float64(t - z)) * a)
	tmp = 0.0
	if (t_1 <= -1e+305)
		tmp = Float64(x - Float64(a * y));
	elseif (t_1 <= -5e+95)
		tmp = t_2;
	elseif (t_1 <= 1e-95)
		tmp = fma(a, Float64(z / Float64(t - z)), x);
	elseif (t_1 <= 1e+139)
		tmp = fma(a, Float64(z / Float64(1.0 - z)), x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z - y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+305], N[(x - N[(a * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e+95], t$95$2, If[LessEqual[t$95$1, 1e-95], N[(a * N[(z / N[(t - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+139], N[(a * N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -9.9999999999999994e304

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower--.f64 88.7

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 77.5

         \[\leadsto x - a \cdot y \]

   7. Applied rewrites 77.5%

      \[\leadsto x - a \cdot \color{blue}{y} \]

   if -9.9999999999999994e304 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -5.00000000000000025e95 or 1.00000000000000003e139 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 99.8%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 84.0

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

   4. Applied rewrites 84.0%

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

   5. Taylor expanded in t around inf

      \[\leadsto \frac{z - y}{t - z} \cdot a \]
   6. Step-by-step derivation

   7. Applied rewrites 62.0%

      \[\leadsto \frac{z - y}{t - z} \cdot a \]

   if -5.00000000000000025e95 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < 9.99999999999999989e-96

   1. Initial program 95.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. associate--l+ N/A

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

      17. lower-/.f64 N/A

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

      18. lower--.f64 N/A

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

      19. lower-+.f64 95.1

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

   3. Applied rewrites 95.1%

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

   4. Taylor expanded in y around 0

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

      1. sub-div N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 88.0

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

   6. Applied rewrites 88.0%

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

   7. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{t - z}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 85.4%

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{t - z}, x\right) \]

   if 9.99999999999999989e-96 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < 1.00000000000000003e139

   1. Initial program 99.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. associate--l+ N/A

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

      17. lower-/.f64 N/A

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

      18. lower--.f64 N/A

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

      19. lower-+.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

      1. sub-div N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 75.2

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

   6. Applied rewrites 75.2%

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

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 - z}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 65.2%

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 - z}, x\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 75.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \left(y - z\right) \cdot a\\ t_2 := \mathsf{fma}\left(a, \frac{z}{t - z}, x\right)\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{-39}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-234}:\\ \;\;\;\;x - \frac{y}{\frac{t}{a}}\\ \mathbf{elif}\;z \leq 0.085:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* (- y z) a))) (t_2 (fma a (/ z (- t z)) x)))
   (if (<= z -1.8e-39)
     t_2
     (if (<= z -1.7e-145)
       t_1
       (if (<= z 4e-234) (- x (/ y (/ t a))) (if (<= z 0.085) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((y - z) * a);
	double t_2 = fma(a, (z / (t - z)), x);
	double tmp;
	if (z <= -1.8e-39) {
		tmp = t_2;
	} else if (z <= -1.7e-145) {
		tmp = t_1;
	} else if (z <= 4e-234) {
		tmp = x - (y / (t / a));
	} else if (z <= 0.085) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(y - z) * a))
	t_2 = fma(a, Float64(z / Float64(t - z)), x)
	tmp = 0.0
	if (z <= -1.8e-39)
		tmp = t_2;
	elseif (z <= -1.7e-145)
		tmp = t_1;
	elseif (z <= 4e-234)
		tmp = Float64(x - Float64(y / Float64(t / a)));
	elseif (z <= 0.085)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(y - z), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(z / N[(t - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.8e-39], t$95$2, If[LessEqual[z, -1.7e-145], t$95$1, If[LessEqual[z, 4e-234], N[(x - N[(y / N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.085], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8e-39 or 0.0850000000000000061 < z

   1. Initial program 95.4%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. associate--l+ N/A

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

      17. lower-/.f64 N/A

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

      18. lower--.f64 N/A

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

      19. lower-+.f64 95.4

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

   3. Applied rewrites 95.4%

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

   4. Taylor expanded in y around 0

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

      1. sub-div N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 84.4

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

   6. Applied rewrites 84.4%

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

   7. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{t - z}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 83.4%

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{t - z}, x\right) \]

   if -1.8e-39 < z < -1.6999999999999999e-145 or 3.9999999999999998e-234 < z < 0.0850000000000000061

   1. Initial program 99.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower--.f64 72.8

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

   4. Applied rewrites 72.8%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 72.8

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

   6. Applied rewrites 72.8%

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

   7. Taylor expanded in z around 0

      \[\leadsto x - \left(y - z\right) \cdot a \]
   8. Step-by-step derivation

   9. Applied rewrites 72.4%

      \[\leadsto x - \left(y - z\right) \cdot a \]

   if -1.6999999999999999e-145 < z < 3.9999999999999998e-234

   1. Initial program 99.3%

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

   2. Taylor expanded in t around inf

      \[\leadsto x - \frac{y - z}{\frac{\color{blue}{t}}{a}} \]
   3. Step-by-step derivation

   4. Applied rewrites 67.4%

      \[\leadsto x - \frac{y - z}{\frac{\color{blue}{t}}{a}} \]

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 66.9%

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

Alternative 12: 75.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{\frac{t}{a}}\\ \mathbf{if}\;t \leq -1.08 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-217}:\\ \;\;\;\;x - \left(y - z\right) \cdot a\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{1 - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ y (/ t a)))))
   (if (<= t -1.08e+27)
     t_1
     (if (<= t -8e-217)
       (- x (* (- y z) a))
       (if (<= t 2.3e+43) (fma a (/ z (- 1.0 z)) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y / (t / a));
	double tmp;
	if (t <= -1.08e+27) {
		tmp = t_1;
	} else if (t <= -8e-217) {
		tmp = x - ((y - z) * a);
	} else if (t <= 2.3e+43) {
		tmp = fma(a, (z / (1.0 - z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(y / Float64(t / a)))
	tmp = 0.0
	if (t <= -1.08e+27)
		tmp = t_1;
	elseif (t <= -8e-217)
		tmp = Float64(x - Float64(Float64(y - z) * a));
	elseif (t <= 2.3e+43)
		tmp = fma(a, Float64(z / Float64(1.0 - z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(y / N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.08e+27], t$95$1, If[LessEqual[t, -8e-217], N[(x - N[(N[(y - z), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.3e+43], N[(a * N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.08e27 or 2.3000000000000002e43 < t

   1. Initial program 96.8%

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

   2. Taylor expanded in t around inf

      \[\leadsto x - \frac{y - z}{\frac{\color{blue}{t}}{a}} \]
   3. Step-by-step derivation

   4. Applied rewrites 84.7%

      \[\leadsto x - \frac{y - z}{\frac{\color{blue}{t}}{a}} \]

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 78.5%

      \[\leadsto x - \frac{\color{blue}{y}}{\frac{t}{a}} \]

   if -1.08e27 < t < -8.00000000000000066e-217

   1. Initial program 97.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower--.f64 82.8

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

   4. Applied rewrites 82.8%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 93.4

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

   6. Applied rewrites 93.4%

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

   7. Taylor expanded in z around 0

      \[\leadsto x - \left(y - z\right) \cdot a \]
   8. Step-by-step derivation

   9. Applied rewrites 60.6%

      \[\leadsto x - \left(y - z\right) \cdot a \]

   if -8.00000000000000066e-217 < t < 2.3000000000000002e43

   1. Initial program 97.7%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. associate--l+ N/A

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

      17. lower-/.f64 N/A

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

      18. lower--.f64 N/A

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

      19. lower-+.f64 97.7

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

   3. Applied rewrites 97.7%

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

   4. Taylor expanded in y around 0

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

      1. sub-div N/A

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

      2. associate--l+ N/A

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

      3. +-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 70.5

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

   6. Applied rewrites 70.5%

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

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 - z}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 69.8%

      \[\leadsto \mathsf{fma}\left(a, \frac{z}{1 - z}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{\frac{t}{a}}\\ \mathbf{if}\;t \leq -1.08 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-86}:\\ \;\;\;\;x - \left(y - z\right) \cdot a\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+43}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ y (/ t a)))))
   (if (<= t -1.08e+27)
     t_1
     (if (<= t 6.2e-86)
       (- x (* (- y z) a))
       (if (<= t 1.02e+43) (- x a) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y / (t / a));
	double tmp;
	if (t <= -1.08e+27) {
		tmp = t_1;
	} else if (t <= 6.2e-86) {
		tmp = x - ((y - z) * a);
	} else if (t <= 1.02e+43) {
		tmp = x - a;
	} else {
		tmp = t_1;
	}
	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, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x - (y / (t / a))
    if (t <= (-1.08d+27)) then
        tmp = t_1
    else if (t <= 6.2d-86) then
        tmp = x - ((y - z) * a)
    else if (t <= 1.02d+43) then
        tmp = x - a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y / (t / a));
	double tmp;
	if (t <= -1.08e+27) {
		tmp = t_1;
	} else if (t <= 6.2e-86) {
		tmp = x - ((y - z) * a);
	} else if (t <= 1.02e+43) {
		tmp = x - a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x - (y / (t / a))
	tmp = 0
	if t <= -1.08e+27:
		tmp = t_1
	elif t <= 6.2e-86:
		tmp = x - ((y - z) * a)
	elif t <= 1.02e+43:
		tmp = x - a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(y / Float64(t / a)))
	tmp = 0.0
	if (t <= -1.08e+27)
		tmp = t_1;
	elseif (t <= 6.2e-86)
		tmp = Float64(x - Float64(Float64(y - z) * a));
	elseif (t <= 1.02e+43)
		tmp = Float64(x - a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x - (y / (t / a));
	tmp = 0.0;
	if (t <= -1.08e+27)
		tmp = t_1;
	elseif (t <= 6.2e-86)
		tmp = x - ((y - z) * a);
	elseif (t <= 1.02e+43)
		tmp = x - a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(y / N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.08e+27], t$95$1, If[LessEqual[t, 6.2e-86], N[(x - N[(N[(y - z), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.02e+43], N[(x - a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.08e27 or 1.02e43 < t

   1. Initial program 96.8%

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

   2. Taylor expanded in t around inf

      \[\leadsto x - \frac{y - z}{\frac{\color{blue}{t}}{a}} \]
   3. Step-by-step derivation

   4. Applied rewrites 84.7%

      \[\leadsto x - \frac{y - z}{\frac{\color{blue}{t}}{a}} \]

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 78.5%

      \[\leadsto x - \frac{\color{blue}{y}}{\frac{t}{a}} \]

   if -1.08e27 < t < 6.19999999999999977e-86

   1. Initial program 97.4%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower--.f64 85.9

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

   4. Applied rewrites 85.9%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 95.9

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

   6. Applied rewrites 95.9%

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

   7. Taylor expanded in z around 0

      \[\leadsto x - \left(y - z\right) \cdot a \]
   8. Step-by-step derivation

   9. Applied rewrites 62.6%

      \[\leadsto x - \left(y - z\right) \cdot a \]

   if 6.19999999999999977e-86 < t < 1.02e43

   1. Initial program 98.1%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 61.8%

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

Alternative 14: 71.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -75:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 0.106:\\ \;\;\;\;x - \left(y - z\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -75.0) (- x a) (if (<= z 0.106) (- x (* (- y z) a)) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -75.0) {
		tmp = x - a;
	} else if (z <= 0.106) {
		tmp = x - ((y - z) * a);
	} else {
		tmp = x - 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, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-75.0d0)) then
        tmp = x - a
    else if (z <= 0.106d0) then
        tmp = x - ((y - z) * a)
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -75.0) {
		tmp = x - a;
	} else if (z <= 0.106) {
		tmp = x - ((y - z) * a);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -75.0:
		tmp = x - a
	elif z <= 0.106:
		tmp = x - ((y - z) * a)
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -75.0)
		tmp = Float64(x - a);
	elseif (z <= 0.106)
		tmp = Float64(x - Float64(Float64(y - z) * a));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -75.0)
		tmp = x - a;
	elseif (z <= 0.106)
		tmp = x - ((y - z) * a);
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -75.0], N[(x - a), $MachinePrecision], If[LessEqual[z, 0.106], N[(x - N[(N[(y - z), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -75 or 0.105999999999999997 < z

   1. Initial program 95.2%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 76.2%

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

   if -75 < z < 0.105999999999999997

   1. Initial program 99.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower--.f64 74.5

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

   4. Applied rewrites 74.5%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 74.5

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

   6. Applied rewrites 74.5%

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

   7. Taylor expanded in z around 0

      \[\leadsto x - \left(y - z\right) \cdot a \]
   8. Step-by-step derivation

   9. Applied rewrites 74.1%

      \[\leadsto x - \left(y - z\right) \cdot a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{-19}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 0.225:\\ \;\;\;\;x - a \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.95e-19) (- x a) (if (<= z 0.225) (- x (* a y)) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.95e-19) {
		tmp = x - a;
	} else if (z <= 0.225) {
		tmp = x - (a * y);
	} else {
		tmp = x - 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, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.95d-19)) then
        tmp = x - a
    else if (z <= 0.225d0) then
        tmp = x - (a * y)
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.95e-19) {
		tmp = x - a;
	} else if (z <= 0.225) {
		tmp = x - (a * y);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.95e-19:
		tmp = x - a
	elif z <= 0.225:
		tmp = x - (a * y)
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.95e-19)
		tmp = Float64(x - a);
	elseif (z <= 0.225)
		tmp = Float64(x - Float64(a * y));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.95e-19)
		tmp = x - a;
	elseif (z <= 0.225)
		tmp = x - (a * y);
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.95e-19], N[(x - a), $MachinePrecision], If[LessEqual[z, 0.225], N[(x - N[(a * y), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.94999999999999998e-19 or 0.225000000000000006 < z

   1. Initial program 95.3%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 75.4%

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

   if -1.94999999999999998e-19 < z < 0.225000000000000006

   1. Initial program 99.2%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower--.f64 74.5

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

   4. Applied rewrites 74.5%

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

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 71.9

         \[\leadsto x - a \cdot y \]

   7. Applied rewrites 71.9%

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

Alternative 16: 66.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.65 \cdot 10^{-14}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-11}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.65e-14) (- x a) (if (<= z 8e-11) x (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.65e-14) {
		tmp = x - a;
	} else if (z <= 8e-11) {
		tmp = x;
	} else {
		tmp = x - 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, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.65d-14)) then
        tmp = x - a
    else if (z <= 8d-11) then
        tmp = x
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.65e-14) {
		tmp = x - a;
	} else if (z <= 8e-11) {
		tmp = x;
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.65e-14:
		tmp = x - a
	elif z <= 8e-11:
		tmp = x
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.65e-14)
		tmp = Float64(x - a);
	elseif (z <= 8e-11)
		tmp = x;
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.65e-14)
		tmp = x - a;
	elseif (z <= 8e-11)
		tmp = x;
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.65e-14], N[(x - a), $MachinePrecision], If[LessEqual[z, 8e-11], x, N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.6500000000000001e-14 or 7.99999999999999952e-11 < z

   1. Initial program 95.3%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 75.2%

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

   if -2.6500000000000001e-14 < z < 7.99999999999999952e-11

   1. Initial program 99.2%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 57.2%

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

Alternative 17: 57.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ \mathbf{if}\;t\_1 \leq -6.2 \cdot 10^{+95}:\\ \;\;\;\;-1 \cdot a\\ \mathbf{elif}\;t\_1 \leq 10^{+95}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y z) (/ (+ (- t z) 1.0) a))))
   (if (<= t_1 -6.2e+95) (* -1.0 a) (if (<= t_1 1e+95) x (* -1.0 a)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_1 <= -6.2e+95) {
		tmp = -1.0 * a;
	} else if (t_1 <= 1e+95) {
		tmp = x;
	} else {
		tmp = -1.0 * 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, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y - z) / (((t - z) + 1.0d0) / a)
    if (t_1 <= (-6.2d+95)) then
        tmp = (-1.0d0) * a
    else if (t_1 <= 1d+95) then
        tmp = x
    else
        tmp = (-1.0d0) * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_1 <= -6.2e+95) {
		tmp = -1.0 * a;
	} else if (t_1 <= 1e+95) {
		tmp = x;
	} else {
		tmp = -1.0 * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - z) / (((t - z) + 1.0) / a)
	tmp = 0
	if t_1 <= -6.2e+95:
		tmp = -1.0 * a
	elif t_1 <= 1e+95:
		tmp = x
	else:
		tmp = -1.0 * a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a))
	tmp = 0.0
	if (t_1 <= -6.2e+95)
		tmp = Float64(-1.0 * a);
	elseif (t_1 <= 1e+95)
		tmp = x;
	else
		tmp = Float64(-1.0 * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - z) / (((t - z) + 1.0) / a);
	tmp = 0.0;
	if (t_1 <= -6.2e+95)
		tmp = -1.0 * a;
	elseif (t_1 <= 1e+95)
		tmp = x;
	else
		tmp = -1.0 * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -6.2e+95], N[(-1.0 * a), $MachinePrecision], If[LessEqual[t$95$1, 1e+95], x, N[(-1.0 * a), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -6.2000000000000006e95 or 1.00000000000000002e95 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 99.8%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 83.6

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

   4. Applied rewrites 83.6%

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

   5. Taylor expanded in z around inf

      \[\leadsto -1 \cdot a \]
   6. Step-by-step derivation

   7. Applied rewrites 27.2%

      \[\leadsto -1 \cdot a \]

   if -6.2000000000000006e95 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < 1.00000000000000002e95

   1. Initial program 96.0%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 71.3%

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

Alternative 18: 54.0% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.2%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 54.0%

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

Reproduce

herbie shell --seed 2025101 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64
  (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))