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

Percentage Accurate: 95.8% → 99.7%

Time: 9.7s

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: 95.8% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+226}\right):\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) (- INFINITY)) (not (<= (* z t) 5e+226)))
   (/ (/ x (- t)) z)
   (/ x (fma z (- t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -((double) INFINITY)) || !((z * t) <= 5e+226)) {
		tmp = (x / -t) / z;
	} else {
		tmp = x / fma(z, -t, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= Float64(-Inf)) || !(Float64(z * t) <= 5e+226))
		tmp = Float64(Float64(x / Float64(-t)) / z);
	else
		tmp = Float64(x / fma(z, Float64(-t), y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[(z * t), $MachinePrecision], 5e+226]], $MachinePrecision]], N[(N[(x / (-t)), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(z * (-t) + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 5.0000000000000005e226 < (*.f64 z t)

   1. Initial program 63.6%

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

   3. Taylor expanded in z around inf 89.7%

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

      1. distribute-lft-out 89.7%

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

      2. associate-*r/ 89.7%

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

      3. mul-1-neg 89.7%

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

      4. *-commutative 89.7%

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

   5. Simplified 89.7%

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

   6. Taylor expanded in t around inf 99.9%

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

   if -inf.0 < (*.f64 z t) < 5.0000000000000005e226

   1. Initial program 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-neg-out 99.8%

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

      4. distribute-rgt-neg-out 99.8%

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

      5. fma-define 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+226}\right):\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+226}\right):\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) (- INFINITY)) (not (<= (* z t) 5e+226)))
   (/ (/ x (- t)) z)
   (/ x (- y (* z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -((double) INFINITY)) || !((z * t) <= 5e+226)) {
		tmp = (x / -t) / z;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -Double.POSITIVE_INFINITY) || !((z * t) <= 5e+226)) {
		tmp = (x / -t) / z;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -math.inf) or not ((z * t) <= 5e+226):
		tmp = (x / -t) / z
	else:
		tmp = x / (y - (z * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= Float64(-Inf)) || !(Float64(z * t) <= 5e+226))
		tmp = Float64(Float64(x / Float64(-t)) / z);
	else
		tmp = Float64(x / Float64(y - Float64(z * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -Inf) || ~(((z * t) <= 5e+226)))
		tmp = (x / -t) / z;
	else
		tmp = x / (y - (z * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[(z * t), $MachinePrecision], 5e+226]], $MachinePrecision]], N[(N[(x / (-t)), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 5.0000000000000005e226 < (*.f64 z t)

   1. Initial program 63.6%

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

   3. Taylor expanded in z around inf 89.7%

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

      1. distribute-lft-out 89.7%

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

      2. associate-*r/ 89.7%

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

      3. mul-1-neg 89.7%

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

      4. *-commutative 89.7%

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

   5. Simplified 89.7%

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

   6. Taylor expanded in t around inf 99.9%

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

   if -inf.0 < (*.f64 z t) < 5.0000000000000005e226

   1. Initial program 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+226}\right):\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+45} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{-91}\right):\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -1e+45) (not (<= (* z t) 2e-91)))
   (/ (/ x (- t)) z)
   (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -1e+45) || !((z * t) <= 2e-91)) {
		tmp = (x / -t) / z;
	} else {
		tmp = x / y;
	}
	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) :: tmp
    if (((z * t) <= (-1d+45)) .or. (.not. ((z * t) <= 2d-91))) then
        tmp = (x / -t) / z
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -1e+45) || !((z * t) <= 2e-91)) {
		tmp = (x / -t) / z;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -1e+45) or not ((z * t) <= 2e-91):
		tmp = (x / -t) / z
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -1e+45) || !(Float64(z * t) <= 2e-91))
		tmp = Float64(Float64(x / Float64(-t)) / z);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -1e+45) || ~(((z * t) <= 2e-91)))
		tmp = (x / -t) / z;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -1e+45], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e-91]], $MachinePrecision]], N[(N[(x / (-t)), $MachinePrecision] / z), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.9999999999999993e44 or 2.00000000000000004e-91 < (*.f64 z t)

   1. Initial program 88.7%

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

   3. Taylor expanded in z around inf 70.3%

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

      1. distribute-lft-out 70.3%

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

      2. associate-*r/ 70.3%

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

      3. mul-1-neg 70.3%

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

      4. *-commutative 70.3%

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

   5. Simplified 70.3%

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

   6. Taylor expanded in t around inf 77.2%

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

   if -9.9999999999999993e44 < (*.f64 z t) < 2.00000000000000004e-91

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 86.2%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+45} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{-91}\right):\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+57}:\\ \;\;\;\;\frac{-1}{t \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-91}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -1e+57)
   (/ -1.0 (* t (/ z x)))
   (if (<= (* z t) 2e-91) (/ x y) (/ (/ x (- t)) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -1e+57) {
		tmp = -1.0 / (t * (z / x));
	} else if ((z * t) <= 2e-91) {
		tmp = x / y;
	} else {
		tmp = (x / -t) / z;
	}
	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) :: tmp
    if ((z * t) <= (-1d+57)) then
        tmp = (-1.0d0) / (t * (z / x))
    else if ((z * t) <= 2d-91) then
        tmp = x / y
    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 ((z * t) <= -1e+57) {
		tmp = -1.0 / (t * (z / x));
	} else if ((z * t) <= 2e-91) {
		tmp = x / y;
	} else {
		tmp = (x / -t) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -1e+57:
		tmp = -1.0 / (t * (z / x))
	elif (z * t) <= 2e-91:
		tmp = x / y
	else:
		tmp = (x / -t) / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -1e+57)
		tmp = Float64(-1.0 / Float64(t * Float64(z / x)));
	elseif (Float64(z * t) <= 2e-91)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / Float64(-t)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -1e+57)
		tmp = -1.0 / (t * (z / x));
	elseif ((z * t) <= 2e-91)
		tmp = x / y;
	else
		tmp = (x / -t) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+57], N[(-1.0 / N[(t * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-91], N[(x / y), $MachinePrecision], N[(N[(x / (-t)), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.00000000000000005e57

   1. Initial program 85.7%

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

   3. Taylor expanded in z around inf 76.9%

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

      1. distribute-lft-out 76.9%

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

      2. associate-*r/ 76.9%

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

      3. mul-1-neg 76.9%

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

      4. *-commutative 76.9%

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

   5. Simplified 76.9%

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

   6. Taylor expanded in t around inf 83.9%

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

      1. neg-mul-1 83.9%

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

      2. clear-num 84.6%

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

      3. un-div-inv 84.6%

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

      4. clear-num 84.6%

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

      5. associate-/l/ 70.5%

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

      6. clear-num 70.4%

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

      7. *-commutative 70.4%

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

      8. associate-/l* 84.5%

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

   8. Applied egg-rr 84.5%

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

   if -1.00000000000000005e57 < (*.f64 z t) < 2.00000000000000004e-91

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 85.6%

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

   if 2.00000000000000004e-91 < (*.f64 z t)

   1. Initial program 90.7%

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

   3. Taylor expanded in z around inf 65.2%

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

      1. distribute-lft-out 65.2%

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

      2. associate-*r/ 65.2%

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

      3. mul-1-neg 65.2%

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

      4. *-commutative 65.2%

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

   5. Simplified 65.2%

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

   6. Taylor expanded in t around inf 72.2%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+57}:\\ \;\;\;\;\frac{-1}{t \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-91}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-91}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -1e+57)
   (/ (/ x z) (- t))
   (if (<= (* z t) 2e-91) (/ x y) (/ (/ x (- t)) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -1e+57) {
		tmp = (x / z) / -t;
	} else if ((z * t) <= 2e-91) {
		tmp = x / y;
	} else {
		tmp = (x / -t) / z;
	}
	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) :: tmp
    if ((z * t) <= (-1d+57)) then
        tmp = (x / z) / -t
    else if ((z * t) <= 2d-91) then
        tmp = x / y
    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 ((z * t) <= -1e+57) {
		tmp = (x / z) / -t;
	} else if ((z * t) <= 2e-91) {
		tmp = x / y;
	} else {
		tmp = (x / -t) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -1e+57:
		tmp = (x / z) / -t
	elif (z * t) <= 2e-91:
		tmp = x / y
	else:
		tmp = (x / -t) / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -1e+57)
		tmp = Float64(Float64(x / z) / Float64(-t));
	elseif (Float64(z * t) <= 2e-91)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / Float64(-t)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -1e+57)
		tmp = (x / z) / -t;
	elseif ((z * t) <= 2e-91)
		tmp = x / y;
	else
		tmp = (x / -t) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+57], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-91], N[(x / y), $MachinePrecision], N[(N[(x / (-t)), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.00000000000000005e57

   1. Initial program 85.7%

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

   3. Taylor expanded in z around inf 83.8%

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

      1. clear-num 83.1%

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

      2. inv-pow 83.1%

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

   5. Applied egg-rr 83.1%

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

      1. unpow-1 83.1%

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

      2. associate-/l* 97.2%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in x around 0 83.8%

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

      1. associate-/r* 94.4%

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

   10. Simplified 94.4%

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

   11. Taylor expanded in y around 0 85.1%

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

       1. neg-mul-1 85.1%

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

   13. Simplified 85.1%

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

   if -1.00000000000000005e57 < (*.f64 z t) < 2.00000000000000004e-91

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 85.6%

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

   if 2.00000000000000004e-91 < (*.f64 z t)

   1. Initial program 90.7%

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

   3. Taylor expanded in z around inf 65.2%

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

      1. distribute-lft-out 65.2%

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

      2. associate-*r/ 65.2%

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

      3. mul-1-neg 65.2%

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

      4. *-commutative 65.2%

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

   5. Simplified 65.2%

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

   6. Taylor expanded in t around inf 72.2%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+57}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-91}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+204} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+226}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -5e+204) (not (<= (* z t) 5e+226)))
   (/ x (* z t))
   (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e+204) || !((z * t) <= 5e+226)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	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) :: tmp
    if (((z * t) <= (-5d+204)) .or. (.not. ((z * t) <= 5d+226))) then
        tmp = x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e+204) || !((z * t) <= 5e+226)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -5e+204) or not ((z * t) <= 5e+226):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -5e+204) || !(Float64(z * t) <= 5e+226))
		tmp = Float64(x / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -5e+204) || ~(((z * t) <= 5e+226)))
		tmp = x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -5e+204], N[Not[LessEqual[N[(z * t), $MachinePrecision], 5e+226]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5.00000000000000008e204 or 5.0000000000000005e226 < (*.f64 z t)

   1. Initial program 71.0%

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

   3. Taylor expanded in t around inf 70.9%

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

   4. Taylor expanded in t around inf 68.1%

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

      1. associate-*r/ 68.1%

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

      2. neg-mul-1 68.1%

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

      3. *-commutative 68.1%

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

   6. Simplified 68.1%

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

      1. neg-sub0 68.1%

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

      2. sub-neg 68.1%

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

      3. add-sqr-sqrt 36.6%

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

      4. sqrt-unprod 57.6%

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

      5. sqr-neg 57.6%

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

      6. sqrt-unprod 25.6%

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

      7. add-sqr-sqrt 55.5%

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

   8. Applied egg-rr 55.5%

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

      1. +-lft-identity 55.5%

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

   10. Simplified 55.5%

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

   if -5.00000000000000008e204 < (*.f64 z t) < 5.0000000000000005e226

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 66.3%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+204} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+226}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

3. Taylor expanded in y around inf 56.2%

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

Developer Target 1: 96.6% 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 2024157 
(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))))