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

Percentage Accurate: 95.6% → 95.6%

Time: 4.7s

Alternatives: 4

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.6% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

Alternative 2: 51.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y - z \cdot t} \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} + \frac{y}{{\left(t \cdot z\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x (- y (* z t))) -5e-213)
   (* (- x) (+ (pow (* t z) -1.0) (/ y (pow (* t z) 2.0))))
   (/ (/ x t) (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / (y - (z * t))) <= -5e-213) {
		tmp = -x * (pow((t * z), -1.0) + (y / pow((t * z), 2.0)));
	} else {
		tmp = (x / t) / -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)
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) :: tmp
    if ((x / (y - (z * t))) <= (-5d-213)) then
        tmp = -x * (((t * z) ** (-1.0d0)) + (y / ((t * z) ** 2.0d0)))
    else
        tmp = (x / t) / -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / (y - (z * t))) <= -5e-213) {
		tmp = -x * (Math.pow((t * z), -1.0) + (y / Math.pow((t * z), 2.0)));
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / (y - (z * t))) <= -5e-213:
		tmp = -x * (math.pow((t * z), -1.0) + (y / math.pow((t * z), 2.0)))
	else:
		tmp = (x / t) / -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / Float64(y - Float64(z * t))) <= -5e-213)
		tmp = Float64(Float64(-x) * Float64((Float64(t * z) ^ -1.0) + Float64(y / (Float64(t * z) ^ 2.0))));
	else
		tmp = Float64(Float64(x / t) / Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x / (y - (z * t))) <= -5e-213)
		tmp = -x * (((t * z) ^ -1.0) + (y / ((t * z) ^ 2.0)));
	else
		tmp = (x / t) / -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-213], N[((-x) * N[(N[Power[N[(t * z), $MachinePrecision], -1.0], $MachinePrecision] + N[(y / N[Power[N[(t * z), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (-.f64 y (*.f64 z t))) < -4.99999999999999977e-213

   1. Initial program 99.8%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{x}{z} + -1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}}}{t}} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

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

      2. associate-*r/ N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{\frac{x}{z} + \frac{x \cdot y}{t \cdot {z}^{2}}}{t}} \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{x}{z} + \frac{x \cdot y}{t \cdot {z}^{2}}}{t}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto -\frac{\frac{x}{z} + \frac{x \cdot y}{t \cdot {z}^{2}}}{t} \]

      5. lower-/.f64 N/A

         \[\leadsto -\frac{\frac{x}{z} + \frac{x \cdot y}{t \cdot {z}^{2}}}{t} \]

      6. +-commutative N/A

         \[\leadsto -\frac{\frac{x \cdot y}{t \cdot {z}^{2}} + \frac{x}{z}}{t} \]

      7. *-commutative N/A

         \[\leadsto -\frac{\frac{x \cdot y}{{z}^{2} \cdot t} + \frac{x}{z}}{t} \]

      8. times-frac N/A

         \[\leadsto -\frac{\frac{x}{{z}^{2}} \cdot \frac{y}{t} + \frac{x}{z}}{t} \]

      9. lower-fma.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{x}{{z}^{2}}, \frac{y}{t}, \frac{x}{z}\right)}{t} \]

      10. lower-/.f64 N/A

          \[\leadsto -\frac{\mathsf{fma}\left(\frac{x}{{z}^{2}}, \frac{y}{t}, \frac{x}{z}\right)}{t} \]

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 26.9

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

   5. Applied rewrites 26.9%

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

   6. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{t \cdot z} - \color{blue}{-1} \cdot \frac{y}{{t}^{2} \cdot {z}^{2}}\right) \]

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

         \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - -1 \cdot \frac{y}{{t}^{2} \cdot {z}^{2}}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - \left(\mathsf{neg}\left(\frac{y}{{t}^{2} \cdot {z}^{2}}\right)\right)\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - \left(-\frac{y}{{t}^{2} \cdot {z}^{2}}\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - \left(-\frac{y}{{t}^{2} \cdot {z}^{2}}\right)\right) \]

      12. pow-prod-down N/A

          \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - \left(-\frac{y}{{\left(t \cdot z\right)}^{2}}\right)\right) \]

      13. lower-pow.f64 N/A

          \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - \left(-\frac{y}{{\left(t \cdot z\right)}^{2}}\right)\right) \]

      14. lower-*.f64 35.3

          \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - \left(-\frac{y}{{\left(t \cdot z\right)}^{2}}\right)\right) \]

   8. Applied rewrites 35.3%

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

   if -4.99999999999999977e-213 < (/.f64 x (-.f64 y (*.f64 z t)))

   1. Initial program 97.0%

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

   3. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{x \cdot {y}^{2}}{{t}^{3} \cdot z} - \frac{x \cdot y}{{t}^{2}}}{z} + \frac{x}{t}}{z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{-1 \cdot \frac{-1 \cdot \frac{x \cdot {y}^{2}}{{t}^{3} \cdot z} - \frac{x \cdot y}{{t}^{2}}}{z} + \frac{x}{t}}{z} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{-1 \cdot \frac{-1 \cdot \frac{x \cdot {y}^{2}}{{t}^{3} \cdot z} - \frac{x \cdot y}{{t}^{2}}}{z} + \frac{x}{t}}{z} \]

   5. Applied rewrites 37.7%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\frac{\left(-\frac{y \cdot y}{z} \cdot \frac{x}{{t}^{3}}\right) - \frac{y}{t} \cdot \frac{x}{t}}{z}, -1, \frac{x}{t}\right)}{z}} \]

   6. Taylor expanded in y around 0

      \[\leadsto -\frac{\frac{x}{t}}{z} \]
   7. Step-by-step derivation

      1. lift-/.f64 57.8

         \[\leadsto -\frac{\frac{x}{t}}{z} \]

   8. Applied rewrites 57.8%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y - z \cdot t} \leq -5 \cdot 10^{-213}:\\ \;\;\;\;\left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} + \frac{y}{{\left(t \cdot z\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y - z \cdot t}\\ t_2 := {\left(t \cdot z\right)}^{2}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-321} \lor \neg \left(t\_1 \leq 0\right):\\ \;\;\;\;\left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} + \frac{y}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{x}{\left(z \cdot y\right) \cdot t} + \frac{x}{t\_2}\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (- y (* z t)))) (t_2 (pow (* t z) 2.0)))
   (if (or (<= t_1 -1e-321) (not (<= t_1 0.0)))
     (* (- x) (+ (pow (* t z) -1.0) (/ y t_2)))
     (- (* (+ (/ x (* (* z y) t)) (/ x t_2)) y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / (y - (z * t));
	double t_2 = pow((t * z), 2.0);
	double tmp;
	if ((t_1 <= -1e-321) || !(t_1 <= 0.0)) {
		tmp = -x * (pow((t * z), -1.0) + (y / t_2));
	} else {
		tmp = -(((x / ((z * y) * t)) + (x / t_2)) * y);
	}
	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)
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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / (y - (z * t))
    t_2 = (t * z) ** 2.0d0
    if ((t_1 <= (-1d-321)) .or. (.not. (t_1 <= 0.0d0))) then
        tmp = -x * (((t * z) ** (-1.0d0)) + (y / t_2))
    else
        tmp = -(((x / ((z * y) * t)) + (x / t_2)) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / (y - (z * t));
	double t_2 = Math.pow((t * z), 2.0);
	double tmp;
	if ((t_1 <= -1e-321) || !(t_1 <= 0.0)) {
		tmp = -x * (Math.pow((t * z), -1.0) + (y / t_2));
	} else {
		tmp = -(((x / ((z * y) * t)) + (x / t_2)) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / (y - (z * t))
	t_2 = math.pow((t * z), 2.0)
	tmp = 0
	if (t_1 <= -1e-321) or not (t_1 <= 0.0):
		tmp = -x * (math.pow((t * z), -1.0) + (y / t_2))
	else:
		tmp = -(((x / ((z * y) * t)) + (x / t_2)) * y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(y - Float64(z * t)))
	t_2 = Float64(t * z) ^ 2.0
	tmp = 0.0
	if ((t_1 <= -1e-321) || !(t_1 <= 0.0))
		tmp = Float64(Float64(-x) * Float64((Float64(t * z) ^ -1.0) + Float64(y / t_2)));
	else
		tmp = Float64(-Float64(Float64(Float64(x / Float64(Float64(z * y) * t)) + Float64(x / t_2)) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / (y - (z * t));
	t_2 = (t * z) ^ 2.0;
	tmp = 0.0;
	if ((t_1 <= -1e-321) || ~((t_1 <= 0.0)))
		tmp = -x * (((t * z) ^ -1.0) + (y / t_2));
	else
		tmp = -(((x / ((z * y) * t)) + (x / t_2)) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(t * z), $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e-321], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[((-x) * N[(N[Power[N[(t * z), $MachinePrecision], -1.0], $MachinePrecision] + N[(y / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[(N[(x / N[(N[(z * y), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] + N[(x / t$95$2), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision])]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (-.f64 y (*.f64 z t))) < -9.98013e-322 or -0.0 < (/.f64 x (-.f64 y (*.f64 z t)))

   1. Initial program 99.8%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{x}{z} + -1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}}}{t}} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

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

      2. associate-*r/ N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{\frac{x}{z} + \frac{x \cdot y}{t \cdot {z}^{2}}}{t}} \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{x}{z} + \frac{x \cdot y}{t \cdot {z}^{2}}}{t}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto -\frac{\frac{x}{z} + \frac{x \cdot y}{t \cdot {z}^{2}}}{t} \]

      5. lower-/.f64 N/A

         \[\leadsto -\frac{\frac{x}{z} + \frac{x \cdot y}{t \cdot {z}^{2}}}{t} \]

      6. +-commutative N/A

         \[\leadsto -\frac{\frac{x \cdot y}{t \cdot {z}^{2}} + \frac{x}{z}}{t} \]

      7. *-commutative N/A

         \[\leadsto -\frac{\frac{x \cdot y}{{z}^{2} \cdot t} + \frac{x}{z}}{t} \]

      8. times-frac N/A

         \[\leadsto -\frac{\frac{x}{{z}^{2}} \cdot \frac{y}{t} + \frac{x}{z}}{t} \]

      9. lower-fma.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{x}{{z}^{2}}, \frac{y}{t}, \frac{x}{z}\right)}{t} \]

      10. lower-/.f64 N/A

          \[\leadsto -\frac{\mathsf{fma}\left(\frac{x}{{z}^{2}}, \frac{y}{t}, \frac{x}{z}\right)}{t} \]

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 32.0

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

   5. Applied rewrites 32.0%

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

   6. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{t \cdot z} - \color{blue}{-1} \cdot \frac{y}{{t}^{2} \cdot {z}^{2}}\right) \]

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 N/A

         \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - -1 \cdot \frac{y}{{t}^{2} \cdot {z}^{2}}\right) \]

      9. mul-1-neg N/A

         \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - \left(\mathsf{neg}\left(\frac{y}{{t}^{2} \cdot {z}^{2}}\right)\right)\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - \left(-\frac{y}{{t}^{2} \cdot {z}^{2}}\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - \left(-\frac{y}{{t}^{2} \cdot {z}^{2}}\right)\right) \]

      12. pow-prod-down N/A

          \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - \left(-\frac{y}{{\left(t \cdot z\right)}^{2}}\right)\right) \]

      13. lower-pow.f64 N/A

          \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - \left(-\frac{y}{{\left(t \cdot z\right)}^{2}}\right)\right) \]

      14. lower-*.f64 41.5

          \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - \left(-\frac{y}{{\left(t \cdot z\right)}^{2}}\right)\right) \]

   8. Applied rewrites 41.5%

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

   if -9.98013e-322 < (/.f64 x (-.f64 y (*.f64 z t))) < -0.0

   1. Initial program 92.8%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{x}{z} + -1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}}}{t}} \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

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

      2. associate-*r/ N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{\frac{x}{z} + \frac{x \cdot y}{t \cdot {z}^{2}}}{t}} \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{x}{z} + \frac{x \cdot y}{t \cdot {z}^{2}}}{t}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto -\frac{\frac{x}{z} + \frac{x \cdot y}{t \cdot {z}^{2}}}{t} \]

      5. lower-/.f64 N/A

         \[\leadsto -\frac{\frac{x}{z} + \frac{x \cdot y}{t \cdot {z}^{2}}}{t} \]

      6. +-commutative N/A

         \[\leadsto -\frac{\frac{x \cdot y}{t \cdot {z}^{2}} + \frac{x}{z}}{t} \]

      7. *-commutative N/A

         \[\leadsto -\frac{\frac{x \cdot y}{{z}^{2} \cdot t} + \frac{x}{z}}{t} \]

      8. times-frac N/A

         \[\leadsto -\frac{\frac{x}{{z}^{2}} \cdot \frac{y}{t} + \frac{x}{z}}{t} \]

      9. lower-fma.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{x}{{z}^{2}}, \frac{y}{t}, \frac{x}{z}\right)}{t} \]

      10. lower-/.f64 N/A

          \[\leadsto -\frac{\mathsf{fma}\left(\frac{x}{{z}^{2}}, \frac{y}{t}, \frac{x}{z}\right)}{t} \]

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 71.6

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto -y \cdot \left(\frac{x}{t \cdot \left(y \cdot z\right)} + \frac{x}{{t}^{2} \cdot {z}^{2}}\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -\left(\frac{x}{t \cdot \left(y \cdot z\right)} + \frac{x}{{t}^{2} \cdot {z}^{2}}\right) \cdot y \]

      2. lower-*.f64 N/A

         \[\leadsto -\left(\frac{x}{t \cdot \left(y \cdot z\right)} + \frac{x}{{t}^{2} \cdot {z}^{2}}\right) \cdot y \]

      3. lower-+.f64 N/A

         \[\leadsto -\left(\frac{x}{t \cdot \left(y \cdot z\right)} + \frac{x}{{t}^{2} \cdot {z}^{2}}\right) \cdot y \]

      4. lower-/.f64 N/A

         \[\leadsto -\left(\frac{x}{t \cdot \left(y \cdot z\right)} + \frac{x}{{t}^{2} \cdot {z}^{2}}\right) \cdot y \]

      5. *-commutative N/A

         \[\leadsto -\left(\frac{x}{\left(y \cdot z\right) \cdot t} + \frac{x}{{t}^{2} \cdot {z}^{2}}\right) \cdot y \]

      6. lower-*.f64 N/A

         \[\leadsto -\left(\frac{x}{\left(y \cdot z\right) \cdot t} + \frac{x}{{t}^{2} \cdot {z}^{2}}\right) \cdot y \]

      7. *-commutative N/A

         \[\leadsto -\left(\frac{x}{\left(z \cdot y\right) \cdot t} + \frac{x}{{t}^{2} \cdot {z}^{2}}\right) \cdot y \]

      8. lower-*.f64 N/A

         \[\leadsto -\left(\frac{x}{\left(z \cdot y\right) \cdot t} + \frac{x}{{t}^{2} \cdot {z}^{2}}\right) \cdot y \]

      9. lower-/.f64 N/A

         \[\leadsto -\left(\frac{x}{\left(z \cdot y\right) \cdot t} + \frac{x}{{t}^{2} \cdot {z}^{2}}\right) \cdot y \]

      10. pow-prod-down N/A

          \[\leadsto -\left(\frac{x}{\left(z \cdot y\right) \cdot t} + \frac{x}{{\left(t \cdot z\right)}^{2}}\right) \cdot y \]

      11. lower-pow.f64 N/A

          \[\leadsto -\left(\frac{x}{\left(z \cdot y\right) \cdot t} + \frac{x}{{\left(t \cdot z\right)}^{2}}\right) \cdot y \]

      12. lower-*.f64 73.5

          \[\leadsto -\left(\frac{x}{\left(z \cdot y\right) \cdot t} + \frac{x}{{\left(t \cdot z\right)}^{2}}\right) \cdot y \]

   8. Applied rewrites 73.5%

      \[\leadsto -\left(\frac{x}{\left(z \cdot y\right) \cdot t} + \frac{x}{{\left(t \cdot z\right)}^{2}}\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y - z \cdot t} \leq -1 \cdot 10^{-321} \lor \neg \left(\frac{x}{y - z \cdot t} \leq 0\right):\\ \;\;\;\;\left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} + \frac{y}{{\left(t \cdot z\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{x}{\left(z \cdot y\right) \cdot t} + \frac{x}{{\left(t \cdot z\right)}^{2}}\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} + \frac{y}{{\left(t \cdot z\right)}^{2}}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (- x) (+ (pow (* t z) -1.0) (/ y (pow (* t z) 2.0)))))

C

double code(double x, double y, double z, double t) {
	return -x * (pow((t * z), -1.0) + (y / pow((t * z), 2.0)));
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = -x * (((t * z) ** (-1.0d0)) + (y / ((t * z) ** 2.0d0)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return -x * (Math.pow((t * z), -1.0) + (y / Math.pow((t * z), 2.0)));
}

Python

def code(x, y, z, t):
	return -x * (math.pow((t * z), -1.0) + (y / math.pow((t * z), 2.0)))

Julia

function code(x, y, z, t)
	return Float64(Float64(-x) * Float64((Float64(t * z) ^ -1.0) + Float64(y / (Float64(t * z) ^ 2.0))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = -x * (((t * z) ^ -1.0) + (y / ((t * z) ^ 2.0)));
end

Wolfram

code[x_, y_, z_, t_] := N[((-x) * N[(N[Power[N[(t * z), $MachinePrecision], -1.0], $MachinePrecision] + N[(y / N[Power[N[(t * z), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\frac{-1 \cdot \frac{x}{z} + -1 \cdot \frac{x \cdot y}{t \cdot {z}^{2}}}{t}} \]
4. Step-by-step derivation

   1. distribute-lft-out N/A

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

   2. associate-*r/ N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{\frac{x}{z} + \frac{x \cdot y}{t \cdot {z}^{2}}}{t}} \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\frac{x}{z} + \frac{x \cdot y}{t \cdot {z}^{2}}}{t}\right) \]

   4. lower-neg.f64 N/A

      \[\leadsto -\frac{\frac{x}{z} + \frac{x \cdot y}{t \cdot {z}^{2}}}{t} \]

   5. lower-/.f64 N/A

      \[\leadsto -\frac{\frac{x}{z} + \frac{x \cdot y}{t \cdot {z}^{2}}}{t} \]

   6. +-commutative N/A

      \[\leadsto -\frac{\frac{x \cdot y}{t \cdot {z}^{2}} + \frac{x}{z}}{t} \]

   7. *-commutative N/A

      \[\leadsto -\frac{\frac{x \cdot y}{{z}^{2} \cdot t} + \frac{x}{z}}{t} \]

   8. times-frac N/A

      \[\leadsto -\frac{\frac{x}{{z}^{2}} \cdot \frac{y}{t} + \frac{x}{z}}{t} \]

   9. lower-fma.f64 N/A

      \[\leadsto -\frac{\mathsf{fma}\left(\frac{x}{{z}^{2}}, \frac{y}{t}, \frac{x}{z}\right)}{t} \]

   10. lower-/.f64 N/A

       \[\leadsto -\frac{\mathsf{fma}\left(\frac{x}{{z}^{2}}, \frac{y}{t}, \frac{x}{z}\right)}{t} \]

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 43.1

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

5. Applied rewrites 43.1%

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

6. Taylor expanded in x around -inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \left(\frac{1}{t \cdot z} - \color{blue}{-1} \cdot \frac{y}{{t}^{2} \cdot {z}^{2}}\right) \]

   4. lower-neg.f64 N/A

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

   5. lower--.f64 N/A

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

   6. inv-pow N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-*.f64 N/A

      \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - -1 \cdot \frac{y}{{t}^{2} \cdot {z}^{2}}\right) \]

   9. mul-1-neg N/A

      \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - \left(\mathsf{neg}\left(\frac{y}{{t}^{2} \cdot {z}^{2}}\right)\right)\right) \]

   10. lower-neg.f64 N/A

       \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - \left(-\frac{y}{{t}^{2} \cdot {z}^{2}}\right)\right) \]

   11. lower-/.f64 N/A

       \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - \left(-\frac{y}{{t}^{2} \cdot {z}^{2}}\right)\right) \]

   12. pow-prod-down N/A

       \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - \left(-\frac{y}{{\left(t \cdot z\right)}^{2}}\right)\right) \]

   13. lower-pow.f64 N/A

       \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - \left(-\frac{y}{{\left(t \cdot z\right)}^{2}}\right)\right) \]

   14. lower-*.f64 47.2

       \[\leadsto \left(-x\right) \cdot \left({\left(t \cdot z\right)}^{-1} - \left(-\frac{y}{{\left(t \cdot z\right)}^{2}}\right)\right) \]

8. Applied rewrites 47.2%

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

9. Final simplification 47.2%

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

Reproduce

herbie shell --seed 2025057 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -161819597360704900000000000000000000000000000000000) (/ 1 (- (/ y x) (* (/ z x) t))) (if (< x 213783064348764440000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ x (- y (* z t))) (/ 1 (- (/ y x) (* (/ z x) t))))))

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