Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A

Percentage Accurate: 84.8% → 89.2%

Time: 4.1s

Alternatives: 7

Speedup: 0.5×

Specification

Math

\[\begin{array}{l} \\ \frac{x - y \cdot z}{t - a \cdot z} \end{array} \]

FPCore

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

C

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

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 - (a * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y \cdot z}{t - a \cdot z} \end{array} \]

FPCore

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

C

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

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 - (a * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ \mathbf{if}\;\frac{x - y \cdot z}{t\_1} \leq 10^{+270}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-y, z, x\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z))))
   (if (<= (/ (- x (* y z)) t_1) 1e+270)
     (/ (fma (- y) z x) t_1)
     (/ (- y (/ x z)) a))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double tmp;
	if (((x - (y * z)) / t_1) <= 1e+270) {
		tmp = fma(-y, z, x) / t_1;
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	tmp = 0.0
	if (Float64(Float64(x - Float64(y * z)) / t_1) <= 1e+270)
		tmp = Float64(fma(Float64(-y), z, x) / t_1);
	else
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], 1e+270], N[(N[((-y) * z + x), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1e270

   1. Initial program 93.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 93.8

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

   4. Applied rewrites 93.8%

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

   if 1e270 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 34.0%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 86.0

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

   7. Applied rewrites 86.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 86.0

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

   9. Applied rewrites 86.0%

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

   10. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lift-/.f64 86.0

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

   12. Applied rewrites 86.0%

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

Alternative 2: 89.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{if}\;t\_1 \leq 10^{+270}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* y z)) (- t (* a z)))))
   (if (<= t_1 1e+270) t_1 (/ (- y (/ x z)) a))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (y * z)) / (t - (a * z));
	double tmp;
	if (t_1 <= 1e+270) {
		tmp = t_1;
	} else {
		tmp = (y - (x / z)) / 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 = (x - (y * z)) / (t - (a * z))
    if (t_1 <= 1d+270) then
        tmp = t_1
    else
        tmp = (y - (x / z)) / 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 = (x - (y * z)) / (t - (a * z));
	double tmp;
	if (t_1 <= 1e+270) {
		tmp = t_1;
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - (y * z)) / (t - (a * z))
	tmp = 0
	if t_1 <= 1e+270:
		tmp = t_1
	else:
		tmp = (y - (x / z)) / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(a * z)))
	tmp = 0.0
	if (t_1 <= 1e+270)
		tmp = t_1;
	else
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x - (y * z)) / (t - (a * z));
	tmp = 0.0;
	if (t_1 <= 1e+270)
		tmp = t_1;
	else
		tmp = (y - (x / z)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+270], t$95$1, N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1e270

   1. Initial program 93.8%

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

   if 1e270 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 34.0%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 61.9

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 86.0

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

   7. Applied rewrites 86.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 86.0

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

   9. Applied rewrites 86.0%

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

   10. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lift-/.f64 86.0

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

   12. Applied rewrites 86.0%

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

Alternative 3: 69.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{+54} \lor \neg \left(a \leq 5.4 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -8.5e+54) (not (<= a 5.4e-58)))
   (/ (- y (/ x z)) a)
   (/ (- x (* y z)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -8.5e+54) || !(a <= 5.4e-58)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (y * z)) / t;
	}
	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 ((a <= (-8.5d+54)) .or. (.not. (a <= 5.4d-58))) then
        tmp = (y - (x / z)) / a
    else
        tmp = (x - (y * z)) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -8.5e+54) || !(a <= 5.4e-58)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (y * z)) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -8.5e+54) or not (a <= 5.4e-58):
		tmp = (y - (x / z)) / a
	else:
		tmp = (x - (y * z)) / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -8.5e+54) || !(a <= 5.4e-58))
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	else
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -8.5e+54) || ~((a <= 5.4e-58)))
		tmp = (y - (x / z)) / a;
	else
		tmp = (x - (y * z)) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -8.5e+54], N[Not[LessEqual[a, 5.4e-58]], $MachinePrecision]], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.4999999999999995e54 or 5.3999999999999998e-58 < a

   1. Initial program 80.5%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift--.f64 82.8

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

   4. Applied rewrites 82.8%

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

   5. Taylor expanded in a around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 77.0

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

   7. Applied rewrites 77.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-/.f64 77.0

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

   9. Applied rewrites 77.0%

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

   10. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lift-/.f64 77.0

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

   12. Applied rewrites 77.0%

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

   if -8.4999999999999995e54 < a < 5.3999999999999998e-58

   1. Initial program 94.5%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 78.1%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{+54} \lor \neg \left(a \leq 5.4 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+67}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-116}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+112}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.25e+67)
   (/ y a)
   (if (<= z 1.6e-116)
     (/ (- x (* y z)) t)
     (if (<= z 6e+112) (/ x (- t (* a z))) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.25e+67) {
		tmp = y / a;
	} else if (z <= 1.6e-116) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 6e+112) {
		tmp = x / (t - (a * z));
	} else {
		tmp = y / 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 <= (-1.25d+67)) then
        tmp = y / a
    else if (z <= 1.6d-116) then
        tmp = (x - (y * z)) / t
    else if (z <= 6d+112) then
        tmp = x / (t - (a * z))
    else
        tmp = y / 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 <= -1.25e+67) {
		tmp = y / a;
	} else if (z <= 1.6e-116) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 6e+112) {
		tmp = x / (t - (a * z));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.25e+67:
		tmp = y / a
	elif z <= 1.6e-116:
		tmp = (x - (y * z)) / t
	elif z <= 6e+112:
		tmp = x / (t - (a * z))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.25e+67)
		tmp = Float64(y / a);
	elseif (z <= 1.6e-116)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	elseif (z <= 6e+112)
		tmp = Float64(x / Float64(t - Float64(a * z)));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.25e+67)
		tmp = y / a;
	elseif (z <= 1.6e-116)
		tmp = (x - (y * z)) / t;
	elseif (z <= 6e+112)
		tmp = x / (t - (a * z));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.25e+67], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.6e-116], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 6e+112], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.24999999999999994e67 or 5.99999999999999958e112 < z

   1. Initial program 68.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 66.2

         \[\leadsto \frac{y}{\color{blue}{a}} \]

   5. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\frac{y}{a}} \]

   if -1.24999999999999994e67 < z < 1.60000000000000005e-116

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 76.8%

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

   if 1.60000000000000005e-116 < z < 5.99999999999999958e112

   1. Initial program 97.7%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 67.4%

      \[\leadsto \frac{\color{blue}{x}}{t - a \cdot z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+67}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-116}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+112}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+163} \lor \neg \left(z \leq 6 \cdot 10^{+112}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -5.4e+163) (not (<= z 6e+112))) (/ y a) (/ x (- t (* a z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -5.4e+163) || !(z <= 6e+112)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (a * z));
	}
	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 <= (-5.4d+163)) .or. (.not. (z <= 6d+112))) then
        tmp = y / a
    else
        tmp = x / (t - (a * z))
    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 <= -5.4e+163) || !(z <= 6e+112)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (a * z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -5.4e+163) or not (z <= 6e+112):
		tmp = y / a
	else:
		tmp = x / (t - (a * z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -5.4e+163) || !(z <= 6e+112))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / Float64(t - Float64(a * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -5.4e+163) || ~((z <= 6e+112)))
		tmp = y / a;
	else
		tmp = x / (t - (a * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -5.4e+163], N[Not[LessEqual[z, 6e+112]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.39999999999999998e163 or 5.99999999999999958e112 < z

   1. Initial program 63.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 71.9

         \[\leadsto \frac{y}{\color{blue}{a}} \]

   5. Applied rewrites 71.9%

      \[\leadsto \color{blue}{\frac{y}{a}} \]

   if -5.39999999999999998e163 < z < 5.99999999999999958e112

   1. Initial program 98.1%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 65.0%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+163} \lor \neg \left(z \leq 6 \cdot 10^{+112}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.13 \cdot 10^{-25} \lor \neg \left(z \leq 30000000000000\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.13e-25) (not (<= z 30000000000000.0))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.13e-25) || !(z <= 30000000000000.0)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	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 <= (-4.13d-25)) .or. (.not. (z <= 30000000000000.0d0))) then
        tmp = y / a
    else
        tmp = x / t
    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 <= -4.13e-25) || !(z <= 30000000000000.0)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -4.13e-25) or not (z <= 30000000000000.0):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.13e-25) || !(z <= 30000000000000.0))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -4.13e-25) || ~((z <= 30000000000000.0)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.13e-25], N[Not[LessEqual[z, 30000000000000.0]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.12999999999999979e-25 or 3e13 < z

   1. Initial program 77.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 57.5

         \[\leadsto \frac{y}{\color{blue}{a}} \]

   5. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\frac{y}{a}} \]

   if -4.12999999999999979e-25 < z < 3e13

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 54.1

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

   5. Applied rewrites 54.1%

      \[\leadsto \color{blue}{\frac{x}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.13 \cdot 10^{-25} \lor \neg \left(z \leq 30000000000000\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

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 / t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x / t), $MachinePrecision]

Derivation

1. Initial program 87.3%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 30.8

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

5. Applied rewrites 30.8%

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

6. Final simplification 30.8%

   \[\leadsto \frac{x}{t} \]
7. Add Preprocessing

Developer Target 1: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{x}{t\_1} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z))) (t_2 (- (/ x t_1) (/ y (- (/ t z) a)))))
   (if (< z -32113435955957344.0)
     t_2
     (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 t_1)) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = (x / t_1) - (y / ((t / z) - a));
	double tmp;
	if (z < -32113435955957344.0) {
		tmp = t_2;
	} else if (z < 3.5139522372978296e-86) {
		tmp = (x - (y * z)) * (1.0 / t_1);
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = t - (a * z)
    t_2 = (x / t_1) - (y / ((t / z) - a))
    if (z < (-32113435955957344.0d0)) then
        tmp = t_2
    else if (z < 3.5139522372978296d-86) then
        tmp = (x - (y * z)) * (1.0d0 / t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = (x / t_1) - (y / ((t / z) - a));
	double tmp;
	if (z < -32113435955957344.0) {
		tmp = t_2;
	} else if (z < 3.5139522372978296e-86) {
		tmp = (x - (y * z)) * (1.0 / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (a * z)
	t_2 = (x / t_1) - (y / ((t / z) - a))
	tmp = 0
	if z < -32113435955957344.0:
		tmp = t_2
	elif z < 3.5139522372978296e-86:
		tmp = (x - (y * z)) * (1.0 / t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	t_2 = Float64(Float64(x / t_1) - Float64(y / Float64(Float64(t / z) - a)))
	tmp = 0.0
	if (z < -32113435955957344.0)
		tmp = t_2;
	elseif (z < 3.5139522372978296e-86)
		tmp = Float64(Float64(x - Float64(y * z)) * Float64(1.0 / t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (a * z);
	t_2 = (x / t_1) - (y / ((t / z) - a));
	tmp = 0.0;
	if (z < -32113435955957344.0)
		tmp = t_2;
	elseif (z < 3.5139522372978296e-86)
		tmp = (x - (y * z)) * (1.0 / t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / t$95$1), $MachinePrecision] - N[(y / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -32113435955957344.0], t$95$2, If[Less[z, 3.5139522372978296e-86], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2025037 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -32113435955957344) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 4392440296622287/125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))))))

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