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

Percentage Accurate: 95.7% → 97.8%

Time: 17.2s

Alternatives: 6

Speedup: 0.5×

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: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY)) (/ (/ x (- z)) t) (/ x (- y (* z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = (x / -z) / t;
	} 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) {
		tmp = (x / -z) / t;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = (x / -z) / t
	else:
		tmp = x / (y - (z * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(Float64(x / Float64(-z)) / t);
	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)
		tmp = (x / -z) / t;
	else
		tmp = x / (y - (z * t));
	end
	tmp_2 = tmp;
end

Wolfram

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

   1. Initial program 56.4%

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

   3. Taylor expanded in y around 0 56.4%

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

      1. associate-*r/ 56.4%

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

      2. neg-mul-1 56.4%

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

   5. Simplified 56.4%

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

      1. add-sqr-sqrt 8.8%

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

      2. sqrt-unprod 54.7%

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

      3. sqr-neg 54.7%

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

      4. sqrt-unprod 47.6%

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

      5. add-sqr-sqrt 56.4%

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

      6. *-un-lft-identity 56.4%

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

   7. Applied egg-rr 56.4%

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

      1. *-lft-identity 56.4%

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

      2. associate-/l/ 55.7%

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

   9. Simplified 55.7%

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

       1. frac-2neg 55.7%

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

       2. neg-sub0 55.7%

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

       3. div-sub 55.7%

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

       4. add-sqr-sqrt 47.2%

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

       5. sqrt-unprod 55.2%

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

       6. sqr-neg 55.2%

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

       7. sqrt-unprod 30.5%

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

       8. add-sqr-sqrt 100.0%

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

       9. frac-2neg 100.0%

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

   11. Applied egg-rr 100.0%

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

       1. div0 100.0%

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

       2. neg-sub0 100.0%

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

       3. distribute-neg-frac 100.0%

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

   13. Simplified 100.0%

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

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

   1. Initial program 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+43} \lor \neg \left(z \leq 1.3 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.7e+43) (not (<= z 1.3e-79))) (/ (/ x (- z)) t) (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.7e+43) || !(z <= 1.3e-79)) {
		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 <= (-1.7d+43)) .or. (.not. (z <= 1.3d-79))) 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 <= -1.7e+43) || !(z <= 1.3e-79)) {
		tmp = (x / -z) / t;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.7e+43) or not (z <= 1.3e-79):
		tmp = (x / -z) / t
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.7e+43) || !(z <= 1.3e-79))
		tmp = Float64(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 <= -1.7e+43) || ~((z <= 1.3e-79)))
		tmp = (x / -z) / t;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.7e+43], N[Not[LessEqual[z, 1.3e-79]], $MachinePrecision]], N[(N[(x / (-z)), $MachinePrecision] / t), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.70000000000000006e43 or 1.29999999999999997e-79 < z

   1. Initial program 94.7%

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

   3. Taylor expanded in y around 0 65.9%

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

      1. associate-*r/ 65.9%

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

      2. neg-mul-1 65.9%

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

   5. Simplified 65.9%

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

      1. add-sqr-sqrt 28.3%

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

      2. sqrt-unprod 40.2%

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

      3. sqr-neg 40.2%

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

      4. sqrt-unprod 16.5%

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

      5. add-sqr-sqrt 29.6%

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

      6. *-un-lft-identity 29.6%

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

   7. Applied egg-rr 29.6%

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

      1. *-lft-identity 29.6%

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

      2. associate-/l/ 31.2%

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

   9. Simplified 31.2%

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

       1. frac-2neg 31.2%

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

       2. neg-sub0 31.2%

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

       3. div-sub 31.2%

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

       4. add-sqr-sqrt 16.5%

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

       5. sqrt-unprod 40.1%

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

       6. sqr-neg 40.1%

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

       7. sqrt-unprod 31.2%

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

       8. add-sqr-sqrt 68.7%

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

       9. frac-2neg 68.7%

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

   11. Applied egg-rr 68.7%

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

       1. div0 68.7%

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

       2. neg-sub0 68.7%

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

       3. distribute-neg-frac 68.7%

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

   13. Simplified 68.7%

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

   if -1.70000000000000006e43 < z < 1.29999999999999997e-79

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 71.5%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+43} \lor \neg \left(z \leq 1.3 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+43} \lor \neg \left(z \leq 1.28 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.9e+43) (not (<= z 1.28e-79))) (/ x (* t (- z))) (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.9e+43) || !(z <= 1.28e-79)) {
		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 <= (-2.9d+43)) .or. (.not. (z <= 1.28d-79))) 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 <= -2.9e+43) || !(z <= 1.28e-79)) {
		tmp = x / (t * -z);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.9e+43) or not (z <= 1.28e-79):
		tmp = x / (t * -z)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.9e+43) || !(z <= 1.28e-79))
		tmp = Float64(x / Float64(t * Float64(-z)));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2.9e+43) || ~((z <= 1.28e-79)))
		tmp = x / (t * -z);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.9e+43], N[Not[LessEqual[z, 1.28e-79]], $MachinePrecision]], N[(x / N[(t * (-z)), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9000000000000002e43 or 1.28e-79 < z

   1. Initial program 94.7%

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

   3. Taylor expanded in y around 0 65.9%

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

      1. associate-*r/ 65.9%

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

      2. neg-mul-1 65.9%

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

   5. Simplified 65.9%

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

   if -2.9000000000000002e43 < z < 1.28e-79

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 71.5%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+43} \lor \neg \left(z \leq 1.28 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.76 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-80}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.76e+43)
   (/ x (* t (- z)))
   (if (<= z 4.4e-80) (/ x y) (/ (/ x t) (- z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.76e+43) {
		tmp = x / (t * -z);
	} else if (z <= 4.4e-80) {
		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 <= (-1.76d+43)) then
        tmp = x / (t * -z)
    else if (z <= 4.4d-80) 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 <= -1.76e+43) {
		tmp = x / (t * -z);
	} else if (z <= 4.4e-80) {
		tmp = x / y;
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.76e+43:
		tmp = x / (t * -z)
	elif z <= 4.4e-80:
		tmp = x / y
	else:
		tmp = (x / t) / -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.76e+43)
		tmp = Float64(x / Float64(t * Float64(-z)));
	elseif (z <= 4.4e-80)
		tmp = Float64(x / y);
	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 (z <= -1.76e+43)
		tmp = x / (t * -z);
	elseif (z <= 4.4e-80)
		tmp = x / y;
	else
		tmp = (x / t) / -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.76e+43], N[(x / N[(t * (-z)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.4e-80], N[(x / y), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.7600000000000001e43

   1. Initial program 90.1%

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

   3. Taylor expanded in y around 0 72.5%

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

      1. associate-*r/ 72.5%

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

      2. neg-mul-1 72.5%

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

   5. Simplified 72.5%

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

   if -1.7600000000000001e43 < z < 4.4000000000000002e-80

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 72.0%

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

   if 4.4000000000000002e-80 < z

   1. Initial program 97.5%

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

      1. clear-num 95.9%

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

      2. associate-/r/ 97.4%

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

   4. Applied egg-rr 97.4%

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

   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* 64.0%

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

      3. distribute-neg-frac2 64.0%

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

   7. Simplified 64.0%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.76 \cdot 10^{+43}:\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-80}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.95 \cdot 10^{+173} \lor \neg \left(z \leq 3.7 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.95e+173) (not (<= z 3.7e-22))) (/ x (* z t)) (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.95e+173) || !(z <= 3.7e-22)) {
		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 <= (-3.95d+173)) .or. (.not. (z <= 3.7d-22))) 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 <= -3.95e+173) || !(z <= 3.7e-22)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -3.95e+173) or not (z <= 3.7e-22):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -3.95e+173) || !(z <= 3.7e-22))
		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 <= -3.95e+173) || ~((z <= 3.7e-22)))
		tmp = x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -3.95e+173], N[Not[LessEqual[z, 3.7e-22]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.94999999999999993e173 or 3.7e-22 < z

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0 68.0%

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

      1. associate-*r/ 68.0%

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

      2. neg-mul-1 68.0%

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

   5. Simplified 68.0%

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

      1. neg-sub0 68.0%

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

      2. sub-neg 68.0%

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

      3. add-sqr-sqrt 29.8%

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

      4. sqrt-unprod 45.2%

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

      5. sqr-neg 45.2%

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

      6. sqrt-unprod 19.4%

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

      7. add-sqr-sqrt 34.8%

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

   7. Applied egg-rr 34.8%

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

      1. +-lft-identity 34.8%

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

   9. Simplified 34.8%

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

   if -3.94999999999999993e173 < z < 3.7e-22

   1. Initial program 97.5%

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

   3. Taylor expanded in y around inf 63.8%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.95 \cdot 10^{+173} \lor \neg \left(z \leq 3.7 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.1% 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 97.3%

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

3. Taylor expanded in y around inf 54.9%

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

Developer Target 1: 96.5% 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 2024143 
(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))))