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

Percentage Accurate: 95.7% → 99.7%

Time: 9.0s

Alternatives: 7

Speedup: 0.3×

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

real(8) function code(x, y, z, t)
    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: 95.7% 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

real(8) function code(x, y, z, t)
    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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{-1}{\frac{z}{\frac{x}{t}}}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+244}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (/ -1.0 (/ z (/ x t)))
   (if (<= (* z t) 5e+244) (/ x (- y (* z t))) (/ (/ x t) (- z)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = -1.0 / (z / (x / t));
	} else if ((z * t) <= 5e+244) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = -1.0 / (z / (x / t));
	} else if ((z * t) <= 5e+244) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = -1.0 / (z / (x / t))
	elif (z * t) <= 5e+244:
		tmp = x / (y - (z * t))
	else:
		tmp = (x / t) / -z
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(-1.0 / Float64(z / Float64(x / t)));
	elseif (Float64(z * t) <= 5e+244)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(x / t) / Float64(-z));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = -1.0 / (z / (x / t));
	elseif ((z * t) <= 5e+244)
		tmp = x / (y - (z * t));
	else
		tmp = (x / t) / -z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[(-1.0 / N[(z / N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+244], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -inf.0

   1. Initial program 72.4%

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

   3. Taylor expanded in z around -inf 93.9%

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

      1. clear-num 93.9%

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

      2. un-div-inv 93.9%

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

      3. +-commutative 93.9%

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

      4. associate-/l* 99.8%

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

      5. *-commutative 99.8%

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

      6. fma-undefine 99.8%

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

      7. associate-/r* 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 99.8%

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

   if -inf.0 < (*.f64 z t) < 5.00000000000000022e244

   1. Initial program 99.9%

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

   if 5.00000000000000022e244 < (*.f64 z t)

   1. Initial program 73.7%

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

      1. clear-num 71.1%

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

      2. associate-/r/ 73.7%

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

   4. Applied egg-rr 73.7%

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

   5. Taylor expanded in y around 0 73.7%

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

      1. mul-1-neg 73.7%

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

      2. associate-/r* 99.9%

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

      3. distribute-neg-frac2 99.9%

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

   7. Simplified 99.9%

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

Alternative 2: 72.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+107} \lor \neg \left(z \leq 2.25 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.5e+107) (not (<= z 2.25e-54)))
   (* (/ -1.0 t) (/ x z))
   (/ x y)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.5e+107) || !(z <= 2.25e-54)) {
		tmp = (-1.0 / t) * (x / z);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.5d+107)) .or. (.not. (z <= 2.25d-54))) then
        tmp = ((-1.0d0) / t) * (x / z)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.5e+107) || !(z <= 2.25e-54)) {
		tmp = (-1.0 / t) * (x / z);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (z <= -1.5e+107) or not (z <= 2.25e-54):
		tmp = (-1.0 / t) * (x / z)
	else:
		tmp = x / y
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.5e+107) || !(z <= 2.25e-54))
		tmp = Float64(Float64(-1.0 / t) * Float64(x / z));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.5e+107) || ~((z <= 2.25e-54)))
		tmp = (-1.0 / t) * (x / z);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.5e+107], N[Not[LessEqual[z, 2.25e-54]], $MachinePrecision]], N[(N[(-1.0 / t), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.50000000000000012e107 or 2.2499999999999999e-54 < z

   1. Initial program 92.2%

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

   3. Taylor expanded in y around 0 62.3%

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

      1. associate-*r/ 62.3%

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

      2. neg-mul-1 62.3%

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

   5. Simplified 62.3%

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

      1. neg-mul-1 62.3%

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

      2. times-frac 66.3%

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

   7. Applied egg-rr 66.3%

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

   if -1.50000000000000012e107 < z < 2.2499999999999999e-54

   1. Initial program 99.1%

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

   3. Taylor expanded in y around inf 76.8%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+107} \lor \neg \left(z \leq 2.25 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+107} \lor \neg \left(z \leq 5.6 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.5e+107) (not (<= z 5.6e-56))) (/ (/ x t) (- z)) (/ x y)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.5e+107) || !(z <= 5.6e-56)) {
		tmp = (x / t) / -z;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.5d+107)) .or. (.not. (z <= 5.6d-56))) then
        tmp = (x / t) / -z
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.5e+107) || !(z <= 5.6e-56)) {
		tmp = (x / t) / -z;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (z <= -1.5e+107) or not (z <= 5.6e-56):
		tmp = (x / t) / -z
	else:
		tmp = x / y
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.5e+107) || !(z <= 5.6e-56))
		tmp = Float64(Float64(x / t) / Float64(-z));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.5e+107) || ~((z <= 5.6e-56)))
		tmp = (x / t) / -z;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.5e+107], N[Not[LessEqual[z, 5.6e-56]], $MachinePrecision]], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.50000000000000012e107 or 5.59999999999999986e-56 < z

   1. Initial program 92.2%

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

      1. clear-num 91.5%

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

      2. associate-/r/ 92.0%

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

   4. Applied egg-rr 92.0%

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

   5. Taylor expanded in y around 0 62.3%

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

      1. mul-1-neg 62.3%

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

      2. associate-/r* 67.9%

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

      3. distribute-neg-frac2 67.9%

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

   7. Simplified 67.9%

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

   if -1.50000000000000012e107 < z < 5.59999999999999986e-56

   1. Initial program 99.1%

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

   3. Taylor expanded in y around inf 76.8%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+107} \lor \neg \left(z \leq 5.6 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+107} \lor \neg \left(z \leq 3.8 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.6e+107) (not (<= z 3.8e-55))) (/ x (* t (- z))) (/ x y)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.6e+107) || !(z <= 3.8e-55)) {
		tmp = x / (t * -z);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-2.6d+107)) .or. (.not. (z <= 3.8d-55))) then
        tmp = x / (t * -z)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.6e+107) || !(z <= 3.8e-55)) {
		tmp = x / (t * -z);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (z <= -2.6e+107) or not (z <= 3.8e-55):
		tmp = x / (t * -z)
	else:
		tmp = x / y
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.6e+107) || !(z <= 3.8e-55))
		tmp = Float64(x / Float64(t * Float64(-z)));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2.6e+107) || ~((z <= 3.8e-55)))
		tmp = x / (t * -z);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.6e+107], N[Not[LessEqual[z, 3.8e-55]], $MachinePrecision]], N[(x / N[(t * (-z)), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.6000000000000001e107 or 3.7999999999999997e-55 < z

   1. Initial program 92.2%

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

   3. Taylor expanded in y around 0 62.3%

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

      1. associate-*r/ 62.3%

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

      2. neg-mul-1 62.3%

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

   5. Simplified 62.3%

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

   if -2.6000000000000001e107 < z < 3.7999999999999997e-55

   1. Initial program 99.1%

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

   3. Taylor expanded in y around inf 76.8%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+107} \lor \neg \left(z \leq 3.8 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+163} \lor \neg \left(z \leq 4.7 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.8e+163) (not (<= z 4.7e-35))) (/ x (* z t)) (/ x y)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.8e+163) || !(z <= 4.7e-35)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.8d+163)) .or. (.not. (z <= 4.7d-35))) then
        tmp = x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.8e+163) || !(z <= 4.7e-35)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (z <= -1.8e+163) or not (z <= 4.7e-35):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.8e+163) || !(z <= 4.7e-35))
		tmp = Float64(x / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.8e+163) || ~((z <= 4.7e-35)))
		tmp = x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.8e+163], N[Not[LessEqual[z, 4.7e-35]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.79999999999999989e163 or 4.7e-35 < z

   1. Initial program 90.8%

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

   3. Taylor expanded in y around 0 63.3%

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

      1. associate-*r/ 63.3%

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

      2. neg-mul-1 63.3%

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

   5. Simplified 63.3%

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

      1. neg-sub0 63.3%

         \[\leadsto \frac{\color{blue}{0 - x}}{t \cdot z} \]

      2. sub-neg 63.3%

         \[\leadsto \frac{\color{blue}{0 + \left(-x\right)}}{t \cdot z} \]

      3. add-sqr-sqrt 31.2%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{t \cdot z} \]

      4. sqrt-unprod 42.5%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{t \cdot z} \]

      5. sqr-neg 42.5%

         \[\leadsto \frac{0 + \sqrt{\color{blue}{x \cdot x}}}{t \cdot z} \]

      6. sqrt-unprod 17.1%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{t \cdot z} \]

      7. add-sqr-sqrt 30.6%

         \[\leadsto \frac{0 + \color{blue}{x}}{t \cdot z} \]

   7. Applied egg-rr 30.6%

      \[\leadsto \frac{\color{blue}{0 + x}}{t \cdot z} \]
   8. Step-by-step derivation

      1. +-lft-identity 30.6%

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

   9. Simplified 30.6%

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

   if -1.79999999999999989e163 < z < 4.7e-35

   1. Initial program 99.3%

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

   3. Taylor expanded in y around inf 72.6%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+163} \lor \neg \left(z \leq 4.7 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{x}{z}}{t}\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z -7.5e+120) (/ (/ x z) t) (if (<= z 4.7e-35) (/ x y) (/ x (* z t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7.5e+120) {
		tmp = (x / z) / t;
	} else if (z <= 4.7e-35) {
		tmp = x / y;
	} else {
		tmp = x / (z * t);
	}
	return tmp;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-7.5d+120)) then
        tmp = (x / z) / t
    else if (z <= 4.7d-35) then
        tmp = x / y
    else
        tmp = x / (z * t)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7.5e+120) {
		tmp = (x / z) / t;
	} else if (z <= 4.7e-35) {
		tmp = x / y;
	} else {
		tmp = x / (z * t);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= -7.5e+120:
		tmp = (x / z) / t
	elif z <= 4.7e-35:
		tmp = x / y
	else:
		tmp = x / (z * t)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= -7.5e+120)
		tmp = Float64(Float64(x / z) / t);
	elseif (z <= 4.7e-35)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(z * t));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -7.5e+120)
		tmp = (x / z) / t;
	elseif (z <= 4.7e-35)
		tmp = x / y;
	else
		tmp = x / (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[z, -7.5e+120], N[(N[(x / z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 4.7e-35], N[(x / y), $MachinePrecision], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.5000000000000006e120

   1. Initial program 91.1%

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

      1. clear-num 91.0%

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

      2. associate-/r/ 91.0%

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

   4. Applied egg-rr 91.0%

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

   5. Taylor expanded in y around 0 67.6%

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

      1. associate-*l/ 67.6%

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

      2. neg-mul-1 67.6%

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

      3. add-sqr-sqrt 32.2%

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

      4. sqrt-unprod 52.2%

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

      5. sqr-neg 52.2%

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

      6. sqrt-unprod 23.2%

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

      7. *-commutative 23.2%

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

      8. add-sqr-sqrt 43.0%

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

      9. associate-/r* 48.8%

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

   7. Applied egg-rr 48.8%

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

   if -7.5000000000000006e120 < z < 4.7e-35

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf 74.1%

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

   if 4.7e-35 < z

   1. Initial program 91.4%

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

   3. Taylor expanded in y around 0 61.4%

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

      1. associate-*r/ 61.4%

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

      2. neg-mul-1 61.4%

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

   5. Simplified 61.4%

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

      1. neg-sub0 61.4%

         \[\leadsto \frac{\color{blue}{0 - x}}{t \cdot z} \]

      2. sub-neg 61.4%

         \[\leadsto \frac{\color{blue}{0 + \left(-x\right)}}{t \cdot z} \]

      3. add-sqr-sqrt 30.6%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{t \cdot z} \]

      4. sqrt-unprod 39.3%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{t \cdot z} \]

      5. sqr-neg 39.3%

         \[\leadsto \frac{0 + \sqrt{\color{blue}{x \cdot x}}}{t \cdot z} \]

      6. sqrt-unprod 15.9%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{t \cdot z} \]

      7. add-sqr-sqrt 27.3%

         \[\leadsto \frac{0 + \color{blue}{x}}{t \cdot z} \]

   7. Applied egg-rr 27.3%

      \[\leadsto \frac{\color{blue}{0 + x}}{t \cdot z} \]
   8. Step-by-step derivation

      1. +-lft-identity 27.3%

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

   9. Simplified 27.3%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{x}{z}}{t}\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (/ x y))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return x / y;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x / y
end function

Java

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

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return x / y

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(x / y)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = x / y;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 95.5%

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

3. Taylor expanded in y around inf 54.7%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
4. Add Preprocessing

Developer Target 1: 96.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{if}\;x < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x < 2.1378306434876444 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (- (/ y x) (* (/ z x) t)))))
   (if (< x -1.618195973607049e+50)
     t_1
     (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    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) :: tmp
    t_1 = 1.0d0 / ((y / x) - ((z / x) * t))
    if (x < (-1.618195973607049d+50)) then
        tmp = t_1
    else if (x < 2.1378306434876444d+131) then
        tmp = x / (y - (z * t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 / ((y / x) - ((z / x) * t))
	tmp = 0
	if x < -1.618195973607049e+50:
		tmp = t_1
	elif x < 2.1378306434876444e+131:
		tmp = x / (y - (z * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(Float64(y / x) - Float64(Float64(z / x) * t)))
	tmp = 0.0
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 / ((y / x) - ((z / x) * t));
	tmp = 0.0;
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = x / (y - (z * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 / N[(N[(y / x), $MachinePrecision] - N[(N[(z / x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[x, -1.618195973607049e+50], t$95$1, If[Less[x, 2.1378306434876444e+131], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024184 
(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))))