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

Percentage Accurate: 85.7% → 90.6%

Time: 3.9s

Alternatives: 10

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: 85.7% 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: 90.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{x}{t\_1}\\ \mathbf{if}\;a \leq -4.8 \cdot 10^{+23} \lor \neg \left(a \leq 3.2 \cdot 10^{+137}\right):\\ \;\;\;\;t\_2 - y \cdot \frac{-1}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_2 - y \cdot \frac{z}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z))) (t_2 (/ x t_1)))
   (if (or (<= a -4.8e+23) (not (<= a 3.2e+137)))
     (- t_2 (* y (/ -1.0 a)))
     (- t_2 (* y (/ z t_1))))))

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;
	double tmp;
	if ((a <= -4.8e+23) || !(a <= 3.2e+137)) {
		tmp = t_2 - (y * (-1.0 / a));
	} else {
		tmp = t_2 - (y * (z / t_1));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t - (a * z)
    t_2 = x / t_1
    if ((a <= (-4.8d+23)) .or. (.not. (a <= 3.2d+137))) then
        tmp = t_2 - (y * ((-1.0d0) / a))
    else
        tmp = t_2 - (y * (z / t_1))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = x / t_1;
	double tmp;
	if ((a <= -4.8e+23) || !(a <= 3.2e+137)) {
		tmp = t_2 - (y * (-1.0 / a));
	} else {
		tmp = t_2 - (y * (z / t_1));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (a * z)
	t_2 = x / t_1
	tmp = 0
	if (a <= -4.8e+23) or not (a <= 3.2e+137):
		tmp = t_2 - (y * (-1.0 / a))
	else:
		tmp = t_2 - (y * (z / t_1))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	t_2 = Float64(x / t_1)
	tmp = 0.0
	if ((a <= -4.8e+23) || !(a <= 3.2e+137))
		tmp = Float64(t_2 - Float64(y * Float64(-1.0 / a)));
	else
		tmp = Float64(t_2 - Float64(y * Float64(z / t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (a * z);
	t_2 = x / t_1;
	tmp = 0.0;
	if ((a <= -4.8e+23) || ~((a <= 3.2e+137)))
		tmp = t_2 - (y * (-1.0 / a));
	else
		tmp = t_2 - (y * (z / t_1));
	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[(x / t$95$1), $MachinePrecision]}, If[Or[LessEqual[a, -4.8e+23], N[Not[LessEqual[a, 3.2e+137]], $MachinePrecision]], N[(t$95$2 - N[(y * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 - N[(y * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.8e23 or 3.20000000000000019e137 < a

   1. Initial program 72.1%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 74.2

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

   4. Applied rewrites 74.2%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 90.8

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

   7. Applied rewrites 90.8%

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

   if -4.8e23 < a < 3.20000000000000019e137

   1. Initial program 90.5%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 97.3

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

   4. Applied rewrites 97.3%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+23} \lor \neg \left(a \leq 3.2 \cdot 10^{+137}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{-1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{z}{t - a \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - y \cdot z\\ t_2 := t - a \cdot z\\ t_3 := \frac{t\_1}{t\_2}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+306}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{t\_2}\\ \mathbf{elif}\;t\_3 \leq 10^{+303}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(-a, z, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} - y}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* y z))) (t_2 (- t (* a z))) (t_3 (/ t_1 t_2)))
   (if (<= t_3 -2e+306)
     (* (- y) (/ z t_2))
     (if (<= t_3 1e+303) (/ t_1 (fma (- a) z t)) (/ (- (/ x z) y) (- a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * z);
	double t_2 = t - (a * z);
	double t_3 = t_1 / t_2;
	double tmp;
	if (t_3 <= -2e+306) {
		tmp = -y * (z / t_2);
	} else if (t_3 <= 1e+303) {
		tmp = t_1 / fma(-a, z, t);
	} else {
		tmp = ((x / z) - y) / -a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(y * z))
	t_2 = Float64(t - Float64(a * z))
	t_3 = Float64(t_1 / t_2)
	tmp = 0.0
	if (t_3 <= -2e+306)
		tmp = Float64(Float64(-y) * Float64(z / t_2));
	elseif (t_3 <= 1e+303)
		tmp = Float64(t_1 / fma(Float64(-a), z, t));
	else
		tmp = Float64(Float64(Float64(x / z) - y) / Float64(-a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+306], N[((-y) * N[(z / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+303], N[(t$95$1 / N[((-a) * z + t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / (-a)), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -2.00000000000000003e306

   1. Initial program 63.3%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. sub-div N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 84.9

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

   7. Applied rewrites 84.9%

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

   if -2.00000000000000003e306 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1e303

   1. Initial program 93.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 93.2

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

   4. Applied rewrites 93.2%

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

   if 1e303 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 34.1%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

      2. sub-div N/A

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

      3. distribute-lft-out-- N/A

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

      4. associate-*r/ N/A

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

      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. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 80.8

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

   7. Applied rewrites 80.8%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - a \cdot z} \leq -2 \cdot 10^{+306}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{t - a \cdot z}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq 10^{+303}:\\ \;\;\;\;\frac{x - y \cdot z}{\mathsf{fma}\left(-a, z, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} - y}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{x - y \cdot z}{t\_1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+306}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 10^{+303}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} - y}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z))) (t_2 (/ (- x (* y z)) t_1)))
   (if (<= t_2 -2e+306)
     (* (- y) (/ z t_1))
     (if (<= t_2 1e+303) t_2 (/ (- (/ x z) y) (- a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = (x - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -2e+306) {
		tmp = -y * (z / t_1);
	} else if (t_2 <= 1e+303) {
		tmp = t_2;
	} else {
		tmp = ((x / z) - 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t - (a * z)
    t_2 = (x - (y * z)) / t_1
    if (t_2 <= (-2d+306)) then
        tmp = -y * (z / t_1)
    else if (t_2 <= 1d+303) then
        tmp = t_2
    else
        tmp = ((x / z) - y) / -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 = t - (a * z);
	double t_2 = (x - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -2e+306) {
		tmp = -y * (z / t_1);
	} else if (t_2 <= 1e+303) {
		tmp = t_2;
	} else {
		tmp = ((x / z) - y) / -a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (a * z)
	t_2 = (x - (y * z)) / t_1
	tmp = 0
	if t_2 <= -2e+306:
		tmp = -y * (z / t_1)
	elif t_2 <= 1e+303:
		tmp = t_2
	else:
		tmp = ((x / z) - y) / -a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	t_2 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_2 <= -2e+306)
		tmp = Float64(Float64(-y) * Float64(z / t_1));
	elseif (t_2 <= 1e+303)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(x / z) - y) / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (a * z);
	t_2 = (x - (y * z)) / t_1;
	tmp = 0.0;
	if (t_2 <= -2e+306)
		tmp = -y * (z / t_1);
	elseif (t_2 <= 1e+303)
		tmp = t_2;
	else
		tmp = ((x / z) - y) / -a;
	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 - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+306], N[((-y) * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+303], t$95$2, N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / (-a)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -2.00000000000000003e306

   1. Initial program 63.3%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. sub-div N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 84.9

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

   7. Applied rewrites 84.9%

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

   if -2.00000000000000003e306 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1e303

   1. Initial program 93.2%

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

   if 1e303 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 34.1%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

      2. sub-div N/A

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

      3. distribute-lft-out-- N/A

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

      4. associate-*r/ N/A

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

      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. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 80.8

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

   7. Applied rewrites 80.8%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - a \cdot z} \leq -2 \cdot 10^{+306}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{t - a \cdot z}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq 10^{+303}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} - y}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-28} \lor \neg \left(a \leq 1.1 \cdot 10^{+138}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{-1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{\mathsf{fma}\left(-a, z, t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.1e-28) (not (<= a 1.1e+138)))
   (- (/ x (- t (* a z))) (* y (/ -1.0 a)))
   (/ (- x (* y z)) (fma (- a) z t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.1e-28) || !(a <= 1.1e+138)) {
		tmp = (x / (t - (a * z))) - (y * (-1.0 / a));
	} else {
		tmp = (x - (y * z)) / fma(-a, z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.1e-28) || !(a <= 1.1e+138))
		tmp = Float64(Float64(x / Float64(t - Float64(a * z))) - Float64(y * Float64(-1.0 / a)));
	else
		tmp = Float64(Float64(x - Float64(y * z)) / fma(Float64(-a), z, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.1e-28], N[Not[LessEqual[a, 1.1e+138]], $MachinePrecision]], N[(N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[((-a) * z + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.09999999999999998e-28 or 1.1e138 < a

   1. Initial program 71.0%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 76.2

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

   4. Applied rewrites 76.2%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 90.6

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

   7. Applied rewrites 90.6%

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

   if -1.09999999999999998e-28 < a < 1.1e138

   1. Initial program 92.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 92.5

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

   4. Applied rewrites 92.5%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-28} \lor \neg \left(a \leq 1.1 \cdot 10^{+138}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z} - y \cdot \frac{-1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{\mathsf{fma}\left(-a, z, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+50} \lor \neg \left(z \leq 2.2 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{\frac{x}{z} - y}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-a, z, t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.9e+50) (not (<= z 2.2e-71)))
   (/ (- (/ x z) y) (- a))
   (/ x (fma (- a) z t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.9e+50) || !(z <= 2.2e-71)) {
		tmp = ((x / z) - y) / -a;
	} else {
		tmp = x / fma(-a, z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.9e+50) || !(z <= 2.2e-71))
		tmp = Float64(Float64(Float64(x / z) - y) / Float64(-a));
	else
		tmp = Float64(x / fma(Float64(-a), z, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.9e+50], N[Not[LessEqual[z, 2.2e-71]], $MachinePrecision]], N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / (-a)), $MachinePrecision], N[(x / N[((-a) * z + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.9000000000000002e50 or 2.19999999999999997e-71 < z

   1. Initial program 69.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 82.5

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

   4. Applied rewrites 82.5%

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

   5. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

      2. sub-div N/A

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

      3. distribute-lft-out-- N/A

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

      4. associate-*r/ N/A

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

      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. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 72.5

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

   7. Applied rewrites 72.5%

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

   if -4.9000000000000002e50 < z < 2.19999999999999997e-71

   1. Initial program 98.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 79.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+50} \lor \neg \left(z \leq 2.2 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{\frac{x}{z} - y}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-a, z, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+77}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-a, z, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{t - a \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.7e+77)
   (/ y a)
   (if (<= z 2.2e-71) (/ x (fma (- a) z t)) (* (- y) (/ z (- t (* a z)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.7e+77) {
		tmp = y / a;
	} else if (z <= 2.2e-71) {
		tmp = x / fma(-a, z, t);
	} else {
		tmp = -y * (z / (t - (a * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.7e+77)
		tmp = Float64(y / a);
	elseif (z <= 2.2e-71)
		tmp = Float64(x / fma(Float64(-a), z, t));
	else
		tmp = Float64(Float64(-y) * Float64(z / Float64(t - Float64(a * z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.7e+77], N[(y / a), $MachinePrecision], If[LessEqual[z, 2.2e-71], N[(x / N[((-a) * z + t), $MachinePrecision]), $MachinePrecision], N[((-y) * N[(z / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.69999999999999998e77

   1. Initial program 66.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 60.0

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

   5. Applied rewrites 60.0%

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

   if -1.69999999999999998e77 < z < 2.19999999999999997e-71

   1. Initial program 96.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 96.8

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 77.4%

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

   if 2.19999999999999997e-71 < z

   1. Initial program 70.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 84.0

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

   4. Applied rewrites 84.0%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. sub-div N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 67.0

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

   7. Applied rewrites 67.0%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+77}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-a, z, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{t - a \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+77} \lor \neg \left(z \leq 1.95 \cdot 10^{+51}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-a, z, t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.7e+77) (not (<= z 1.95e+51))) (/ y a) (/ x (fma (- a) z t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.7e+77) || !(z <= 1.95e+51)) {
		tmp = y / a;
	} else {
		tmp = x / fma(-a, z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.7e+77) || !(z <= 1.95e+51))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / fma(Float64(-a), z, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.7e+77], N[Not[LessEqual[z, 1.95e+51]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / N[((-a) * z + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.69999999999999998e77 or 1.94999999999999992e51 < z

   1. Initial program 65.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 61.4

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

   5. Applied rewrites 61.4%

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

   if -1.69999999999999998e77 < z < 1.94999999999999992e51

   1. Initial program 97.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 97.0

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 75.2%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+77} \lor \neg \left(z \leq 1.95 \cdot 10^{+51}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-a, z, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+77} \lor \neg \left(z \leq 1.95 \cdot 10^{+51}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.7e+77) (not (<= z 1.95e+51))) (/ y a) (/ x (- t (* a z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.7e+77) || !(z <= 1.95e+51)) {
		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 <= (-1.7d+77)) .or. (.not. (z <= 1.95d+51))) 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 <= -1.7e+77) || !(z <= 1.95e+51)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (a * z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.7e+77) or not (z <= 1.95e+51):
		tmp = y / a
	else:
		tmp = x / (t - (a * z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.7e+77) || !(z <= 1.95e+51))
		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 <= -1.7e+77) || ~((z <= 1.95e+51)))
		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, -1.7e+77], N[Not[LessEqual[z, 1.95e+51]], $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 < -1.69999999999999998e77 or 1.94999999999999992e51 < z

   1. Initial program 65.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 61.4

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

   5. Applied rewrites 61.4%

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

   if -1.69999999999999998e77 < z < 1.94999999999999992e51

   1. Initial program 97.0%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 75.2%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+77} \lor \neg \left(z \leq 1.95 \cdot 10^{+51}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+50} \lor \neg \left(z \leq 2.2 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3.8e+50) (not (<= z 2.2e-71))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.8e+50) || !(z <= 2.2e-71)) {
		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 <= (-3.8d+50)) .or. (.not. (z <= 2.2d-71))) 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 <= -3.8e+50) || !(z <= 2.2e-71)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -3.8e+50) or not (z <= 2.2e-71):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3.8e+50) || !(z <= 2.2e-71))
		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 <= -3.8e+50) || ~((z <= 2.2e-71)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3.8e+50], N[Not[LessEqual[z, 2.2e-71]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.79999999999999987e50 or 2.19999999999999997e-71 < z

   1. Initial program 69.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 56.0

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

   5. Applied rewrites 56.0%

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

   if -3.79999999999999987e50 < z < 2.19999999999999997e-71

   1. Initial program 98.1%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 55.0

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

   5. Applied rewrites 55.0%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+50} \lor \neg \left(z \leq 2.2 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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 83.0%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 32.6

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

5. Applied rewrites 32.6%

   \[\leadsto \color{blue}{\frac{x}{t}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025084 
(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))))