Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A

Percentage Accurate: 98.1% → 98.1%
Time: 3.6s
Alternatives: 11
Speedup: 1.0×

Specification

?
\[\begin{array}{l} \\ x + y \cdot \frac{z - t}{z - a} \end{array} \]
(FPCore (x y z t a) :precision binary64 (+ x (* y (/ (- z t) (- z a)))))
double code(double x, double y, double z, double t, double a) {
	return x + (y * ((z - t) / (z - a)));
}

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 - a)))
end function

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

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

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

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

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

\begin{array}{l}

\\
x + y \cdot \frac{z - t}{z - a}
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 11 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 98.1% accurate, 1.0× speedup?

\[\begin{array}{l} \\ x + y \cdot \frac{z - t}{z - a} \end{array} \]
(FPCore (x y z t a) :precision binary64 (+ x (* y (/ (- z t) (- z a)))))
double code(double x, double y, double z, double t, double a) {
	return x + (y * ((z - t) / (z - a)));
}

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 - a)))
end function

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

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

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

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

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

\begin{array}{l}

\\
x + y \cdot \frac{z - t}{z - a}
\end{array}

Alternative 1: 98.1% accurate, 1.0× speedup?

\[\begin{array}{l} \\ x + y \cdot \frac{z - t}{z - a} \end{array} \]
(FPCore (x y z t a) :precision binary64 (+ x (* y (/ (- z t) (- z a)))))
double code(double x, double y, double z, double t, double a) {
	return x + (y * ((z - t) / (z - a)));
}

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 - a)))
end function

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

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

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

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

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

\begin{array}{l}

\\
x + y \cdot \frac{z - t}{z - a}
\end{array}
Derivation

  1. Initial program 98.3%

    \[x + y \cdot \frac{z - t}{z - a} \]
  2. Add Preprocessing
  3. Add Preprocessing

Alternative 2: 82.9% accurate, 0.3× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-102}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+132}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot y}{\left(-z\right) + a}\\ \end{array} \end{array} \]
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma t (/ y a) x)))
   (if (<= t_1 5e-102)
     t_2
     (if (<= t_1 1e+19)
       (fma y (/ z (- z a)) x)
       (if (<= t_1 2e+132) t_2 (/ (* t y) (+ (- z) a)))))))
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma(t, (y / a), x);
	double tmp;
	if (t_1 <= 5e-102) {
		tmp = t_2;
	} else if (t_1 <= 1e+19) {
		tmp = fma(y, (z / (z - a)), x);
	} else if (t_1 <= 2e+132) {
		tmp = t_2;
	} else {
		tmp = (t * y) / (-z + a);
	}
	return tmp;
}

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(t, Float64(y / a), x)
	tmp = 0.0
	if (t_1 <= 5e-102)
		tmp = t_2;
	elseif (t_1 <= 1e+19)
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	elseif (t_1 <= 2e+132)
		tmp = t_2;
	else
		tmp = Float64(Float64(t * y) / Float64(Float64(-z) + a));
	end
	return tmp
end

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-102], t$95$2, If[LessEqual[t$95$1, 1e+19], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+132], t$95$2, N[(N[(t * y), $MachinePrecision] / N[((-z) + a), $MachinePrecision]), $MachinePrecision]]]]]]

\begin{array}{l}

\\
\begin{array}{l}
t_1 := \frac{z - t}{z - a}\\
t_2 := \mathsf{fma}\left(t, \frac{y}{a}, x\right)\\
\mathbf{if}\;t\_1 \leq 5 \cdot 10^{-102}:\\
\;\;\;\;t\_2\\

\mathbf{elif}\;t\_1 \leq 10^{+19}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\

\mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+132}:\\
\;\;\;\;t\_2\\

\mathbf{else}:\\
\;\;\;\;\frac{t \cdot y}{\left(-z\right) + a}\\


\end{array}
\end{array}
Derivation
  1. Split input into 3 regimes

  2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 5.00000000000000026e-102 or 1e19 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.99999999999999998e132

    1. Initial program 98.9%

      \[x + y \cdot \frac{z - t}{z - a} \]
    2. Add Preprocessing

    3. Taylor expanded in z around 0

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

      1. +-commutativeN/A

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

      2. associate-/l*N/A

        \[\leadsto t \cdot \frac{y}{a} + x \]

      3. lower-fma.f64N/A

        \[\leadsto \mathsf{fma}\left(t, \color{blue}{\frac{y}{a}}, x\right) \]

      4. lower-/.f6482.0

        \[\leadsto \mathsf{fma}\left(t, \frac{y}{\color{blue}{a}}, x\right) \]

    5. Applied rewrites82.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{y}{a}, x\right)} \]

    if 5.00000000000000026e-102 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e19

    1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{z - a} \]
    2. Add Preprocessing

    3. Taylor expanded in t around 0

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

      1. +-commutativeN/A

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

      2. associate-/l*N/A

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

      3. lower-fma.f64N/A

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

      4. lower-/.f64N/A

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

      5. lift--.f6495.9

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

    5. Applied rewrites95.9%

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

    if 1.99999999999999998e132 < (/.f64 (-.f64 z t) (-.f64 z a))

    1. Initial program 88.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
    2. Add Preprocessing

    3. Taylor expanded in t around inf

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

      1. associate-*r/N/A

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

      2. lower-/.f64N/A

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

      3. associate-*r*N/A

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

      4. lower-*.f64N/A

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

      5. mul-1-negN/A

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

      6. lower-neg.f64N/A

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

      7. lift--.f6496.2

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

    5. Applied rewrites96.2%

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

  4. Final simplification89.5%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 5 \cdot 10^{-102}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 2 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot y}{\left(-z\right) + a}\\ \end{array} \]
  5. Add Preprocessing

Alternative 3: 80.9% accurate, 0.3× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-102}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+165}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t\right) \cdot y}{z}\\ \end{array} \end{array} \]
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma t (/ y a) x)))
   (if (<= t_1 5e-102)
     t_2
     (if (<= t_1 1e+19)
       (fma y (/ z (- z a)) x)
       (if (<= t_1 5e+165) t_2 (/ (* (- t) y) z))))))
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma(t, (y / a), x);
	double tmp;
	if (t_1 <= 5e-102) {
		tmp = t_2;
	} else if (t_1 <= 1e+19) {
		tmp = fma(y, (z / (z - a)), x);
	} else if (t_1 <= 5e+165) {
		tmp = t_2;
	} else {
		tmp = (-t * y) / z;
	}
	return tmp;
}

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(t, Float64(y / a), x)
	tmp = 0.0
	if (t_1 <= 5e-102)
		tmp = t_2;
	elseif (t_1 <= 1e+19)
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	elseif (t_1 <= 5e+165)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(-t) * y) / z);
	end
	return tmp
end

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-102], t$95$2, If[LessEqual[t$95$1, 1e+19], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+165], t$95$2, N[(N[((-t) * y), $MachinePrecision] / z), $MachinePrecision]]]]]]

\begin{array}{l}

\\
\begin{array}{l}
t_1 := \frac{z - t}{z - a}\\
t_2 := \mathsf{fma}\left(t, \frac{y}{a}, x\right)\\
\mathbf{if}\;t\_1 \leq 5 \cdot 10^{-102}:\\
\;\;\;\;t\_2\\

\mathbf{elif}\;t\_1 \leq 10^{+19}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\

\mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+165}:\\
\;\;\;\;t\_2\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-t\right) \cdot y}{z}\\


\end{array}
\end{array}
Derivation
  1. Split input into 3 regimes

  2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 5.00000000000000026e-102 or 1e19 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.9999999999999997e165

    1. Initial program 98.9%

      \[x + y \cdot \frac{z - t}{z - a} \]
    2. Add Preprocessing

    3. Taylor expanded in z around 0

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

      1. +-commutativeN/A

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

      2. associate-/l*N/A

        \[\leadsto t \cdot \frac{y}{a} + x \]

      3. lower-fma.f64N/A

        \[\leadsto \mathsf{fma}\left(t, \color{blue}{\frac{y}{a}}, x\right) \]

      4. lower-/.f6482.3

        \[\leadsto \mathsf{fma}\left(t, \frac{y}{\color{blue}{a}}, x\right) \]

    5. Applied rewrites82.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{y}{a}, x\right)} \]

    if 5.00000000000000026e-102 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e19

    1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{z - a} \]
    2. Add Preprocessing

    3. Taylor expanded in t around 0

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

      1. +-commutativeN/A

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

      2. associate-/l*N/A

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

      3. lower-fma.f64N/A

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

      4. lower-/.f64N/A

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

      5. lift--.f6495.9

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

    5. Applied rewrites95.9%

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

    if 4.9999999999999997e165 < (/.f64 (-.f64 z t) (-.f64 z a))

    1. Initial program 87.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
    2. Add Preprocessing

    3. Taylor expanded in a around 0

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

      1. +-commutativeN/A

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

      2. associate-/l*N/A

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

      3. lower-fma.f64N/A

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

      4. lower-/.f64N/A

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

      5. lift--.f6469.9

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

    5. Applied rewrites69.9%

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

    6. Taylor expanded in z around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot y}{z}} \]
    7. Step-by-step derivation

      1. mul-1-negN/A

        \[\leadsto \mathsf{neg}\left(\frac{t \cdot y}{z}\right) \]

      2. lower-neg.f64N/A

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

      3. lower-/.f64N/A

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

      4. lift-*.f6474.3

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

    8. Applied rewrites74.3%

      \[\leadsto -\frac{t \cdot y}{z} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification87.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 5 \cdot 10^{-102}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 5 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t\right) \cdot y}{z}\\ \end{array} \]
  5. Add Preprocessing

Alternative 4: 80.4% accurate, 0.3× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq 10^{-12}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+19}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+165}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t\right) \cdot y}{z}\\ \end{array} \end{array} \]
(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma t (/ y a) x)))
   (if (<= t_1 1e-12)
     t_2
     (if (<= t_1 1e+19) (+ x y) (if (<= t_1 5e+165) t_2 (/ (* (- t) y) z))))))
double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma(t, (y / a), x);
	double tmp;
	if (t_1 <= 1e-12) {
		tmp = t_2;
	} else if (t_1 <= 1e+19) {
		tmp = x + y;
	} else if (t_1 <= 5e+165) {
		tmp = t_2;
	} else {
		tmp = (-t * y) / z;
	}
	return tmp;
}

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(t, Float64(y / a), x)
	tmp = 0.0
	if (t_1 <= 1e-12)
		tmp = t_2;
	elseif (t_1 <= 1e+19)
		tmp = Float64(x + y);
	elseif (t_1 <= 5e+165)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(-t) * y) / z);
	end
	return tmp
end

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-12], t$95$2, If[LessEqual[t$95$1, 1e+19], N[(x + y), $MachinePrecision], If[LessEqual[t$95$1, 5e+165], t$95$2, N[(N[((-t) * y), $MachinePrecision] / z), $MachinePrecision]]]]]]

\begin{array}{l}

\\
\begin{array}{l}
t_1 := \frac{z - t}{z - a}\\
t_2 := \mathsf{fma}\left(t, \frac{y}{a}, x\right)\\
\mathbf{if}\;t\_1 \leq 10^{-12}:\\
\;\;\;\;t\_2\\

\mathbf{elif}\;t\_1 \leq 10^{+19}:\\
\;\;\;\;x + y\\

\mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+165}:\\
\;\;\;\;t\_2\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-t\right) \cdot y}{z}\\


\end{array}
\end{array}
Derivation
  1. Split input into 3 regimes

  2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999998e-13 or 1e19 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.9999999999999997e165

    1. Initial program 99.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
    2. Add Preprocessing

    3. Taylor expanded in z around 0

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

      1. +-commutativeN/A

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

      2. associate-/l*N/A

        \[\leadsto t \cdot \frac{y}{a} + x \]

      3. lower-fma.f64N/A

        \[\leadsto \mathsf{fma}\left(t, \color{blue}{\frac{y}{a}}, x\right) \]

      4. lower-/.f6481.5

        \[\leadsto \mathsf{fma}\left(t, \frac{y}{\color{blue}{a}}, x\right) \]

    5. Applied rewrites81.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{y}{a}, x\right)} \]

    if 9.9999999999999998e-13 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e19

    1. Initial program 100.0%

      \[x + y \cdot \frac{z - t}{z - a} \]
    2. Add Preprocessing

    3. Taylor expanded in z around inf

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

      1. Applied rewrites95.9%

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

      if 4.9999999999999997e165 < (/.f64 (-.f64 z t) (-.f64 z a))

      1. Initial program 87.0%

        \[x + y \cdot \frac{z - t}{z - a} \]
      2. Add Preprocessing

      3. Taylor expanded in a around 0

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

        1. +-commutativeN/A

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

        2. associate-/l*N/A

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

        3. lower-fma.f64N/A

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

        4. lower-/.f64N/A

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

        5. lift--.f6469.9

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

      5. Applied rewrites69.9%

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

      6. Taylor expanded in z around 0

        \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot y}{z}} \]
      7. Step-by-step derivation

        1. mul-1-negN/A

          \[\leadsto \mathsf{neg}\left(\frac{t \cdot y}{z}\right) \]

        2. lower-neg.f64N/A

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

        3. lower-/.f64N/A

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

        4. lift-*.f6474.3

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

      8. Applied rewrites74.3%

        \[\leadsto -\frac{t \cdot y}{z} \]
    5. Recombined 3 regimes into one program.

    6. Final simplification86.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{+19}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 5 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t\right) \cdot y}{z}\\ \end{array} \]
    7. Add Preprocessing

    Alternative 5: 80.5% accurate, 0.3× speedup?

    \[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq 10^{-12}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+19}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+165}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z}\\ \end{array} \end{array} \]
    (FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma t (/ y a) x)))
   (if (<= t_1 1e-12)
     t_2
     (if (<= t_1 1e+19) (+ x y) (if (<= t_1 5e+165) t_2 (* (- t) (/ y z)))))))
    double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma(t, (y / a), x);
	double tmp;
	if (t_1 <= 1e-12) {
		tmp = t_2;
	} else if (t_1 <= 1e+19) {
		tmp = x + y;
	} else if (t_1 <= 5e+165) {
		tmp = t_2;
	} else {
		tmp = -t * (y / z);
	}
	return tmp;
}

    function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(t, Float64(y / a), x)
	tmp = 0.0
	if (t_1 <= 1e-12)
		tmp = t_2;
	elseif (t_1 <= 1e+19)
		tmp = Float64(x + y);
	elseif (t_1 <= 5e+165)
		tmp = t_2;
	else
		tmp = Float64(Float64(-t) * Float64(y / z));
	end
	return tmp
end

    code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-12], t$95$2, If[LessEqual[t$95$1, 1e+19], N[(x + y), $MachinePrecision], If[LessEqual[t$95$1, 5e+165], t$95$2, N[((-t) * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]]]

    \begin{array}{l}

\\
\begin{array}{l}
t_1 := \frac{z - t}{z - a}\\
t_2 := \mathsf{fma}\left(t, \frac{y}{a}, x\right)\\
\mathbf{if}\;t\_1 \leq 10^{-12}:\\
\;\;\;\;t\_2\\

\mathbf{elif}\;t\_1 \leq 10^{+19}:\\
\;\;\;\;x + y\\

\mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+165}:\\
\;\;\;\;t\_2\\

\mathbf{else}:\\
\;\;\;\;\left(-t\right) \cdot \frac{y}{z}\\


\end{array}
\end{array}
    Derivation
    1. Split input into 3 regimes

    2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999998e-13 or 1e19 < (/.f64 (-.f64 z t) (-.f64 z a)) < 4.9999999999999997e165

      1. Initial program 99.0%

        \[x + y \cdot \frac{z - t}{z - a} \]
      2. Add Preprocessing

      3. Taylor expanded in z around 0

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

        1. +-commutativeN/A

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

        2. associate-/l*N/A

          \[\leadsto t \cdot \frac{y}{a} + x \]

        3. lower-fma.f64N/A

          \[\leadsto \mathsf{fma}\left(t, \color{blue}{\frac{y}{a}}, x\right) \]

        4. lower-/.f6481.5

          \[\leadsto \mathsf{fma}\left(t, \frac{y}{\color{blue}{a}}, x\right) \]

      5. Applied rewrites81.5%

        \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{y}{a}, x\right)} \]

      if 9.9999999999999998e-13 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e19

      1. Initial program 100.0%

        \[x + y \cdot \frac{z - t}{z - a} \]
      2. Add Preprocessing

      3. Taylor expanded in z around inf

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

        1. Applied rewrites95.9%

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

        if 4.9999999999999997e165 < (/.f64 (-.f64 z t) (-.f64 z a))

        1. Initial program 87.0%

          \[x + y \cdot \frac{z - t}{z - a} \]
        2. Add Preprocessing

        3. Taylor expanded in a around 0

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

          1. +-commutativeN/A

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

          2. associate-/l*N/A

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

          3. lower-fma.f64N/A

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

          4. lower-/.f64N/A

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

          5. lift--.f6469.9

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

        5. Applied rewrites69.9%

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

        6. Taylor expanded in x around 0

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

          1. associate-*r/N/A

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

          2. *-commutativeN/A

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

          3. lower-*.f64N/A

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

          4. lift-/.f64N/A

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

          5. lift--.f6469.9

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

        8. Applied rewrites69.9%

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

        9. Taylor expanded in z around 0

          \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot y}{z}} \]
        10. Step-by-step derivation

          1. mul-1-negN/A

            \[\leadsto \mathsf{neg}\left(\frac{t \cdot y}{z}\right) \]

          2. lower-neg.f64N/A

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

          3. lift-/.f64N/A

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

          4. *-commutativeN/A

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

          5. lower-*.f6474.3

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

        11. Applied rewrites74.3%

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

          1. lift-*.f64N/A

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

          2. lift-/.f64N/A

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

          3. *-commutativeN/A

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

          4. associate-/l*N/A

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

          5. lower-*.f64N/A

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

          6. lower-/.f6474.3

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

        13. Applied rewrites74.3%

          \[\leadsto -t \cdot \frac{y}{z} \]
      5. Recombined 3 regimes into one program.

      6. Final simplification86.4%

        \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 10^{+19}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;\frac{z - t}{z - a} \leq 5 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \frac{y}{z}\\ \end{array} \]
      7. Add Preprocessing

      Alternative 6: 83.7% accurate, 0.4× speedup?

      \[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -100000000000 \lor \neg \left(t\_1 \leq 50000\right):\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{z - a}\\ \end{array} \end{array} \]
      (FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (or (<= t_1 -100000000000.0) (not (<= t_1 50000.0)))
     (* (- z t) (/ y (- z a)))
     (+ x (* y (/ z (- z a)))))))
      double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if ((t_1 <= -100000000000.0) || !(t_1 <= 50000.0)) {
		tmp = (z - t) * (y / (z - a));
	} else {
		tmp = x + (y * (z / (z - a)));
	}
	return tmp;
}

      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 = (z - t) / (z - a)
    if ((t_1 <= (-100000000000.0d0)) .or. (.not. (t_1 <= 50000.0d0))) then
        tmp = (z - t) * (y / (z - a))
    else
        tmp = x + (y * (z / (z - a)))
    end if
    code = tmp
end function

      public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if ((t_1 <= -100000000000.0) || !(t_1 <= 50000.0)) {
		tmp = (z - t) * (y / (z - a));
	} else {
		tmp = x + (y * (z / (z - a)));
	}
	return tmp;
}

      def code(x, y, z, t, a):
	t_1 = (z - t) / (z - a)
	tmp = 0
	if (t_1 <= -100000000000.0) or not (t_1 <= 50000.0):
		tmp = (z - t) * (y / (z - a))
	else:
		tmp = x + (y * (z / (z - a)))
	return tmp

      function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if ((t_1 <= -100000000000.0) || !(t_1 <= 50000.0))
		tmp = Float64(Float64(z - t) * Float64(y / Float64(z - a)));
	else
		tmp = Float64(x + Float64(y * Float64(z / Float64(z - a))));
	end
	return tmp
end

      function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (z - a);
	tmp = 0.0;
	if ((t_1 <= -100000000000.0) || ~((t_1 <= 50000.0)))
		tmp = (z - t) * (y / (z - a));
	else
		tmp = x + (y * (z / (z - a)));
	end
	tmp_2 = tmp;
end

      code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -100000000000.0], N[Not[LessEqual[t$95$1, 50000.0]], $MachinePrecision]], N[(N[(z - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

      \begin{array}{l}

\\
\begin{array}{l}
t_1 := \frac{z - t}{z - a}\\
\mathbf{if}\;t\_1 \leq -100000000000 \lor \neg \left(t\_1 \leq 50000\right):\\
\;\;\;\;\left(z - t\right) \cdot \frac{y}{z - a}\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \frac{z}{z - a}\\


\end{array}
\end{array}
      Derivation
      1. Split input into 2 regimes

      2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1e11 or 5e4 < (/.f64 (-.f64 z t) (-.f64 z a))

        1. Initial program 95.2%

          \[x + y \cdot \frac{z - t}{z - a} \]
        2. Add Preprocessing

        3. Taylor expanded in x around 0

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

          1. lower-/.f64N/A

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

          2. *-commutativeN/A

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

          3. lower-*.f64N/A

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

          4. lift--.f64N/A

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

          5. lift--.f6468.5

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

        5. Applied rewrites68.5%

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

          1. lift--.f64N/A

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

          2. lift-/.f64N/A

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

          3. lift--.f64N/A

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

          4. lift-*.f64N/A

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

          5. associate-/l*N/A

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

          6. lower-*.f64N/A

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

          7. lift--.f64N/A

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

          8. lift-/.f64N/A

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

          9. lift--.f6478.6

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

        7. Applied rewrites78.6%

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

        if -1e11 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e4

        1. Initial program 99.8%

          \[x + y \cdot \frac{z - t}{z - a} \]
        2. Add Preprocessing

        3. Taylor expanded in z around inf

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

          1. Applied rewrites94.0%

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

        6. Final simplification89.1%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -100000000000 \lor \neg \left(\frac{z - t}{z - a} \leq 50000\right):\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{z - a}\\ \end{array} \]
        7. Add Preprocessing

        Alternative 7: 83.7% accurate, 0.4× speedup?

        \[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -100000000000 \lor \neg \left(t\_1 \leq 50000\right):\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \end{array} \end{array} \]
        (FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (or (<= t_1 -100000000000.0) (not (<= t_1 50000.0)))
     (* (- z t) (/ y (- z a)))
     (fma y (/ z (- z a)) x))))
        double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if ((t_1 <= -100000000000.0) || !(t_1 <= 50000.0)) {
		tmp = (z - t) * (y / (z - a));
	} else {
		tmp = fma(y, (z / (z - a)), x);
	}
	return tmp;
}

        function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if ((t_1 <= -100000000000.0) || !(t_1 <= 50000.0))
		tmp = Float64(Float64(z - t) * Float64(y / Float64(z - a)));
	else
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	end
	return tmp
end

        code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -100000000000.0], N[Not[LessEqual[t$95$1, 50000.0]], $MachinePrecision]], N[(N[(z - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

        \begin{array}{l}

\\
\begin{array}{l}
t_1 := \frac{z - t}{z - a}\\
\mathbf{if}\;t\_1 \leq -100000000000 \lor \neg \left(t\_1 \leq 50000\right):\\
\;\;\;\;\left(z - t\right) \cdot \frac{y}{z - a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\


\end{array}
\end{array}
        Derivation
        1. Split input into 2 regimes

        2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -1e11 or 5e4 < (/.f64 (-.f64 z t) (-.f64 z a))

          1. Initial program 95.2%

            \[x + y \cdot \frac{z - t}{z - a} \]
          2. Add Preprocessing

          3. Taylor expanded in x around 0

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

            1. lower-/.f64N/A

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

            2. *-commutativeN/A

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

            3. lower-*.f64N/A

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

            4. lift--.f64N/A

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

            5. lift--.f6468.5

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

          5. Applied rewrites68.5%

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

            1. lift--.f64N/A

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

            2. lift-/.f64N/A

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

            3. lift--.f64N/A

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

            4. lift-*.f64N/A

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

            5. associate-/l*N/A

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

            6. lower-*.f64N/A

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

            7. lift--.f64N/A

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

            8. lift-/.f64N/A

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

            9. lift--.f6478.6

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

          7. Applied rewrites78.6%

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

          if -1e11 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e4

          1. Initial program 99.8%

            \[x + y \cdot \frac{z - t}{z - a} \]
          2. Add Preprocessing

          3. Taylor expanded in t around 0

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

            1. +-commutativeN/A

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

            2. associate-/l*N/A

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

            3. lower-fma.f64N/A

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

            4. lower-/.f64N/A

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

            5. lift--.f6494.0

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

          5. Applied rewrites94.0%

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

        4. Final simplification89.1%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq -100000000000 \lor \neg \left(\frac{z - t}{z - a} \leq 50000\right):\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{z - a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \end{array} \]
        5. Add Preprocessing

        Alternative 8: 81.1% accurate, 0.4× speedup?

        \[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 10^{-12} \lor \neg \left(t\_1 \leq 10^{+19}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]
        (FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (or (<= t_1 1e-12) (not (<= t_1 1e+19))) (fma t (/ y a) x) (+ x y))))
        double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if ((t_1 <= 1e-12) || !(t_1 <= 1e+19)) {
		tmp = fma(t, (y / a), x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

        function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if ((t_1 <= 1e-12) || !(t_1 <= 1e+19))
		tmp = fma(t, Float64(y / a), x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

        code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 1e-12], N[Not[LessEqual[t$95$1, 1e+19]], $MachinePrecision]], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]

        \begin{array}{l}

\\
\begin{array}{l}
t_1 := \frac{z - t}{z - a}\\
\mathbf{if}\;t\_1 \leq 10^{-12} \lor \neg \left(t\_1 \leq 10^{+19}\right):\\
\;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\

\mathbf{else}:\\
\;\;\;\;x + y\\


\end{array}
\end{array}
        Derivation
        1. Split input into 2 regimes

        2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999998e-13 or 1e19 < (/.f64 (-.f64 z t) (-.f64 z a))

          1. Initial program 97.4%

            \[x + y \cdot \frac{z - t}{z - a} \]
          2. Add Preprocessing

          3. Taylor expanded in z around 0

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

            1. +-commutativeN/A

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

            2. associate-/l*N/A

              \[\leadsto t \cdot \frac{y}{a} + x \]

            3. lower-fma.f64N/A

              \[\leadsto \mathsf{fma}\left(t, \color{blue}{\frac{y}{a}}, x\right) \]

            4. lower-/.f6476.3

              \[\leadsto \mathsf{fma}\left(t, \frac{y}{\color{blue}{a}}, x\right) \]

          5. Applied rewrites76.3%

            \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{y}{a}, x\right)} \]

          if 9.9999999999999998e-13 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e19

          1. Initial program 100.0%

            \[x + y \cdot \frac{z - t}{z - a} \]
          2. Add Preprocessing

          3. Taylor expanded in z around inf

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

            1. Applied rewrites95.9%

              \[\leadsto x + \color{blue}{y} \]
          5. Recombined 2 regimes into one program.

          6. Final simplification83.7%

            \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 10^{-12} \lor \neg \left(\frac{z - t}{z - a} \leq 10^{+19}\right):\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
          7. Add Preprocessing

          Alternative 9: 55.3% accurate, 0.5× speedup?

          \[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+67}:\\ \;\;\;\;y\\ \mathbf{elif}\;t\_1 \leq 10^{+100}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]
          (FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- z a)))))
   (if (<= t_1 -2e+67) y (if (<= t_1 1e+100) x y))))
          double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (z - a));
	double tmp;
	if (t_1 <= -2e+67) {
		tmp = y;
	} else if (t_1 <= 1e+100) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

          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 - a))
    if (t_1 <= (-2d+67)) then
        tmp = y
    else if (t_1 <= 1d+100) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

          public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (z - a));
	double tmp;
	if (t_1 <= -2e+67) {
		tmp = y;
	} else if (t_1 <= 1e+100) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

          def code(x, y, z, t, a):
	t_1 = y * ((z - t) / (z - a))
	tmp = 0
	if t_1 <= -2e+67:
		tmp = y
	elif t_1 <= 1e+100:
		tmp = x
	else:
		tmp = y
	return tmp

          function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(z - a)))
	tmp = 0.0
	if (t_1 <= -2e+67)
		tmp = y;
	elseif (t_1 <= 1e+100)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

          function tmp_2 = code(x, y, z, t, a)
	t_1 = y * ((z - t) / (z - a));
	tmp = 0.0;
	if (t_1 <= -2e+67)
		tmp = y;
	elseif (t_1 <= 1e+100)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

          code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+67], y, If[LessEqual[t$95$1, 1e+100], x, y]]]

          \begin{array}{l}

\\
\begin{array}{l}
t_1 := y \cdot \frac{z - t}{z - a}\\
\mathbf{if}\;t\_1 \leq -2 \cdot 10^{+67}:\\
\;\;\;\;y\\

\mathbf{elif}\;t\_1 \leq 10^{+100}:\\
\;\;\;\;x\\

\mathbf{else}:\\
\;\;\;\;y\\


\end{array}
\end{array}
          Derivation
          1. Split input into 2 regimes

          2. if (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < -1.99999999999999997e67 or 1.00000000000000002e100 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a)))

            1. Initial program 95.9%

              \[x + y \cdot \frac{z - t}{z - a} \]
            2. Add Preprocessing

            3. Taylor expanded in a around 0

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

              1. +-commutativeN/A

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

              2. associate-/l*N/A

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

              3. lower-fma.f64N/A

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

              4. lower-/.f64N/A

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

              5. lift--.f6468.1

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

            5. Applied rewrites68.1%

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

            6. Taylor expanded in x around 0

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

              1. associate-*r/N/A

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

              2. *-commutativeN/A

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

              3. lower-*.f64N/A

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

              4. lift-/.f64N/A

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

              5. lift--.f6464.0

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

            8. Applied rewrites64.0%

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

            9. Taylor expanded in z around inf

              \[\leadsto y \]
            10. Step-by-step derivation

              1. Applied rewrites33.8%

                \[\leadsto y \]

              if -1.99999999999999997e67 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < 1.00000000000000002e100

              1. Initial program 99.8%

                \[x + y \cdot \frac{z - t}{z - a} \]
              2. Add Preprocessing

              3. Taylor expanded in x around inf

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

                1. Applied rewrites73.3%

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

              Alternative 10: 67.4% accurate, 1.0× speedup?

              \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \leq 6.2 \cdot 10^{-36}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]
              (FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- z t) (- z a)) 6.2e-36) x (+ x y)))
              double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (z - a)) <= 6.2e-36) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

              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 - t) / (z - a)) <= 6.2d-36) then
        tmp = x
    else
        tmp = x + y
    end if
    code = tmp
end function

              public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z - t) / (z - a)) <= 6.2e-36) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

              def code(x, y, z, t, a):
	tmp = 0
	if ((z - t) / (z - a)) <= 6.2e-36:
		tmp = x
	else:
		tmp = x + y
	return tmp

              function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z - t) / Float64(z - a)) <= 6.2e-36)
		tmp = x;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

              function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((z - t) / (z - a)) <= 6.2e-36)
		tmp = x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

              code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision], 6.2e-36], x, N[(x + y), $MachinePrecision]]

              \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\frac{z - t}{z - a} \leq 6.2 \cdot 10^{-36}:\\
\;\;\;\;x\\

\mathbf{else}:\\
\;\;\;\;x + y\\


\end{array}
\end{array}
              Derivation
              1. Split input into 2 regimes

              2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 6.1999999999999997e-36

                1. Initial program 98.9%

                  \[x + y \cdot \frac{z - t}{z - a} \]
                2. Add Preprocessing

                3. Taylor expanded in x around inf

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

                  1. Applied rewrites62.5%

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

                  if 6.1999999999999997e-36 < (/.f64 (-.f64 z t) (-.f64 z a))

                  1. Initial program 98.0%

                    \[x + y \cdot \frac{z - t}{z - a} \]
                  2. Add Preprocessing

                  3. Taylor expanded in z around inf

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

                    1. Applied rewrites72.6%

                      \[\leadsto x + \color{blue}{y} \]
                  5. Recombined 2 regimes into one program.
                  6. Add Preprocessing

                  Alternative 11: 51.5% accurate, 26.0× speedup?

                  \[\begin{array}{l} \\ x \end{array} \]
                  (FPCore (x y z t a) :precision binary64 x)
                  double code(double x, double y, double z, double t, double a) {
	return x;
}

                  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

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

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

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

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

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

                  \begin{array}{l}

\\
x
\end{array}
                  Derivation

                  1. Initial program 98.3%

                    \[x + y \cdot \frac{z - t}{z - a} \]
                  2. Add Preprocessing

                  3. Taylor expanded in x around inf

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

                    1. Applied rewrites49.6%

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

                    Developer Target 1: 98.3% accurate, 0.8× speedup?

                    \[\begin{array}{l} \\ x + \frac{y}{\frac{z - a}{z - t}} \end{array} \]
                    (FPCore (x y z t a) :precision binary64 (+ x (/ y (/ (- z a) (- z t)))))
                    double code(double x, double y, double z, double t, double a) {
	return x + (y / ((z - a) / (z - t)));
}

                    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 - a) / (z - t)))
end function

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

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

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

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

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

                    \begin{array}{l}

\\
x + \frac{y}{\frac{z - a}{z - t}}
\end{array}

                    Reproduce

                    ?
                    herbie shell --seed 2025061 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ x (/ y (/ (- z a) (- z t)))))

  (+ x (* y (/ (- z t) (- z a)))))