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

Percentage Accurate: 85.1% → 91.0%

Time: 14.1s

Alternatives: 8

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = (x - (y * z)) / (t - (a * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = (x - (y * z)) / (t - (a * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.12 \cdot 10^{+182} \lor \neg \left(z \leq 4.5 \cdot 10^{+182}\right):\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{\mathsf{fma}\left(-z, a, t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.12e+182) (not (<= z 4.5e+182)))
   (/ (- y) (- (/ t z) a))
   (/ (- x (* z y)) (fma (- z) a t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.12e+182) || !(z <= 4.5e+182)) {
		tmp = -y / ((t / z) - a);
	} else {
		tmp = (x - (z * y)) / fma(-z, a, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.12e+182) || !(z <= 4.5e+182))
		tmp = Float64(Float64(-y) / Float64(Float64(t / z) - a));
	else
		tmp = Float64(Float64(x - Float64(z * y)) / fma(Float64(-z), a, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.12e+182], N[Not[LessEqual[z, 4.5e+182]], $MachinePrecision]], N[((-y) / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.11999999999999994e182 or 4.50000000000000029e182 < z

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

   3. Simplified 49.5%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 49.5%

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

      2. inv-pow 49.5%

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

      3. sub-neg 49.5%

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

      4. +-commutative 49.5%

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

      5. *-commutative 49.5%

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

      6. distribute-rgt-neg-in 49.5%

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

      7. fma-def 49.5%

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

   6. Applied egg-rr 49.5%

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

   7. Taylor expanded in a around 0 43.4%

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

      1. +-commutative 43.4%

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

      2. mul-1-neg 43.4%

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

      3. unsub-neg 43.4%

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

      4. associate-/l* 55.2%

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

   9. Simplified 55.2%

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

   10. Taylor expanded in y around -inf 91.6%

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

       1. associate-*r/ 91.6%

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

       2. mul-1-neg 91.6%

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

   12. Simplified 91.6%

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

   if -1.11999999999999994e182 < z < 4.50000000000000029e182

   1. Initial program 95.5%

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

      1. *-commutative 95.5%

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

   3. Simplified 95.5%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sub-neg 95.5%

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

      2. +-commutative 95.5%

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

      3. distribute-lft-neg-in 95.5%

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

      4. fma-def 95.6%

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

   6. Applied egg-rr 95.6%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.12 \cdot 10^{+182} \lor \neg \left(z \leq 4.5 \cdot 10^{+182}\right):\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{\mathsf{fma}\left(-z, a, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.65 \cdot 10^{+115} \lor \neg \left(y \leq -6.2 \cdot 10^{+71}\right) \land \left(y \leq -1.95 \cdot 10^{+31} \lor \neg \left(y \leq 9.5 \cdot 10^{-9}\right)\right):\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= y -3.65e+115)
         (and (not (<= y -6.2e+71)) (or (<= y -1.95e+31) (not (<= y 9.5e-9)))))
   (/ (- y) (- (/ t z) a))
   (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -3.65e+115) || (!(y <= -6.2e+71) && ((y <= -1.95e+31) || !(y <= 9.5e-9)))) {
		tmp = -y / ((t / z) - a);
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y <= (-3.65d+115)) .or. (.not. (y <= (-6.2d+71))) .and. (y <= (-1.95d+31)) .or. (.not. (y <= 9.5d-9))) then
        tmp = -y / ((t / z) - a)
    else
        tmp = x / (t - (z * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -3.65e+115) || (!(y <= -6.2e+71) && ((y <= -1.95e+31) || !(y <= 9.5e-9)))) {
		tmp = -y / ((t / z) - a);
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (y <= -3.65e+115) or (not (y <= -6.2e+71) and ((y <= -1.95e+31) or not (y <= 9.5e-9))):
		tmp = -y / ((t / z) - a)
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((y <= -3.65e+115) || (!(y <= -6.2e+71) && ((y <= -1.95e+31) || !(y <= 9.5e-9))))
		tmp = Float64(Float64(-y) / Float64(Float64(t / z) - a));
	else
		tmp = Float64(x / Float64(t - Float64(z * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((y <= -3.65e+115) || (~((y <= -6.2e+71)) && ((y <= -1.95e+31) || ~((y <= 9.5e-9)))))
		tmp = -y / ((t / z) - a);
	else
		tmp = x / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[y, -3.65e+115], And[N[Not[LessEqual[y, -6.2e+71]], $MachinePrecision], Or[LessEqual[y, -1.95e+31], N[Not[LessEqual[y, 9.5e-9]], $MachinePrecision]]]], N[((-y) / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.64999999999999984e115 or -6.20000000000000036e71 < y < -1.95e31 or 9.5000000000000007e-9 < y

   1. Initial program 74.6%

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

      1. *-commutative 74.6%

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

   3. Simplified 74.6%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 74.6%

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

      2. inv-pow 74.6%

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

      3. sub-neg 74.6%

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

      4. +-commutative 74.6%

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

      5. *-commutative 74.6%

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

      6. distribute-rgt-neg-in 74.6%

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

      7. fma-def 74.6%

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

   6. Applied egg-rr 74.6%

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

   7. Taylor expanded in a around 0 70.3%

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

      1. +-commutative 70.3%

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

      2. mul-1-neg 70.3%

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

      3. unsub-neg 70.3%

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

      4. associate-/l* 73.7%

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

   9. Simplified 73.7%

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

   10. Taylor expanded in y around -inf 72.1%

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

       1. associate-*r/ 72.1%

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

       2. mul-1-neg 72.1%

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

   12. Simplified 72.1%

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

   if -3.64999999999999984e115 < y < -6.20000000000000036e71 or -1.95e31 < y < 9.5000000000000007e-9

   1. Initial program 96.9%

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

      1. *-commutative 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in x around inf 83.3%

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

      1. *-commutative 83.3%

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

   7. Simplified 83.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.65 \cdot 10^{+115} \lor \neg \left(y \leq -6.2 \cdot 10^{+71}\right) \land \left(y \leq -1.95 \cdot 10^{+31} \lor \neg \left(y \leq 9.5 \cdot 10^{-9}\right)\right):\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ \mathbf{if}\;z \leq -1.02 \cdot 10^{+182}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-111}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (- t (* z a)))))
   (if (<= z -1.02e+182)
     (/ y a)
     (if (<= z -1e-73)
       t_1
       (if (<= z -1.5e-111)
         (/ (- x (* z y)) t)
         (if (<= z 1e+80) t_1 (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (z * a));
	double tmp;
	if (z <= -1.02e+182) {
		tmp = y / a;
	} else if (z <= -1e-73) {
		tmp = t_1;
	} else if (z <= -1.5e-111) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 1e+80) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (t - (z * a))
    if (z <= (-1.02d+182)) then
        tmp = y / a
    else if (z <= (-1d-73)) then
        tmp = t_1
    else if (z <= (-1.5d-111)) then
        tmp = (x - (z * y)) / t
    else if (z <= 1d+80) then
        tmp = t_1
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (z * a));
	double tmp;
	if (z <= -1.02e+182) {
		tmp = y / a;
	} else if (z <= -1e-73) {
		tmp = t_1;
	} else if (z <= -1.5e-111) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 1e+80) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x / (t - (z * a))
	tmp = 0
	if z <= -1.02e+182:
		tmp = y / a
	elif z <= -1e-73:
		tmp = t_1
	elif z <= -1.5e-111:
		tmp = (x - (z * y)) / t
	elif z <= 1e+80:
		tmp = t_1
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (z <= -1.02e+182)
		tmp = Float64(y / a);
	elseif (z <= -1e-73)
		tmp = t_1;
	elseif (z <= -1.5e-111)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 1e+80)
		tmp = t_1;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x / (t - (z * a));
	tmp = 0.0;
	if (z <= -1.02e+182)
		tmp = y / a;
	elseif (z <= -1e-73)
		tmp = t_1;
	elseif (z <= -1.5e-111)
		tmp = (x - (z * y)) / t;
	elseif (z <= 1e+80)
		tmp = t_1;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.02e+182], N[(y / a), $MachinePrecision], If[LessEqual[z, -1e-73], t$95$1, If[LessEqual[z, -1.5e-111], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 1e+80], t$95$1, N[(y / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.02e182 or 1e80 < z

   1. Initial program 60.4%

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

      1. *-commutative 60.4%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in z around inf 73.9%

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

   if -1.02e182 < z < -9.99999999999999997e-74 or -1.50000000000000004e-111 < z < 1e80

   1. Initial program 96.5%

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

      1. *-commutative 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in x around inf 72.3%

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

      1. *-commutative 72.3%

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

   7. Simplified 72.3%

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

   if -9.99999999999999997e-74 < z < -1.50000000000000004e-111

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in t around inf 87.3%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+182}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-73}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-111}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 10^{+80}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+75}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+80}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.6e+75)
   (/ y a)
   (if (<= z 1.05e-95)
     (/ x t)
     (if (<= z 1.05e+80) (/ (- x) (* z a)) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.6e+75) {
		tmp = y / a;
	} else if (z <= 1.05e-95) {
		tmp = x / t;
	} else if (z <= 1.05e+80) {
		tmp = -x / (z * a);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-6.6d+75)) then
        tmp = y / a
    else if (z <= 1.05d-95) then
        tmp = x / t
    else if (z <= 1.05d+80) then
        tmp = -x / (z * a)
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.6e+75) {
		tmp = y / a;
	} else if (z <= 1.05e-95) {
		tmp = x / t;
	} else if (z <= 1.05e+80) {
		tmp = -x / (z * a);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -6.6e+75:
		tmp = y / a
	elif z <= 1.05e-95:
		tmp = x / t
	elif z <= 1.05e+80:
		tmp = -x / (z * a)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.6e+75)
		tmp = Float64(y / a);
	elseif (z <= 1.05e-95)
		tmp = Float64(x / t);
	elseif (z <= 1.05e+80)
		tmp = Float64(Float64(-x) / Float64(z * a));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -6.6e+75)
		tmp = y / a;
	elseif (z <= 1.05e-95)
		tmp = x / t;
	elseif (z <= 1.05e+80)
		tmp = -x / (z * a);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.6e+75], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.05e-95], N[(x / t), $MachinePrecision], If[LessEqual[z, 1.05e+80], N[((-x) / N[(z * a), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.59999999999999996e75 or 1.05000000000000001e80 < z

   1. Initial program 67.5%

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

      1. *-commutative 67.5%

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

   3. Simplified 67.5%

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

   5. Taylor expanded in z around inf 65.9%

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

   if -6.59999999999999996e75 < z < 1.05e-95

   1. Initial program 98.3%

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

      1. *-commutative 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in z around 0 56.6%

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

   if 1.05e-95 < z < 1.05000000000000001e80

   1. Initial program 96.3%

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

      1. *-commutative 96.3%

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

   3. Simplified 96.3%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 96.0%

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

      2. inv-pow 96.0%

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

      3. sub-neg 96.0%

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

      4. +-commutative 96.0%

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

      5. *-commutative 96.0%

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

      6. distribute-rgt-neg-in 96.0%

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

      7. fma-def 96.1%

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

   6. Applied egg-rr 96.1%

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

   7. Taylor expanded in a around 0 92.3%

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

      1. +-commutative 92.3%

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

      2. mul-1-neg 92.3%

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

      3. unsub-neg 92.3%

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

      4. associate-/l* 88.7%

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

   9. Simplified 88.7%

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

   10. Taylor expanded in y around 0 64.2%

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

       1. *-commutative 64.2%

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

   12. Simplified 64.2%

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

   13. Taylor expanded in t around 0 45.4%

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

       1. associate-*r/ 45.4%

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

       2. neg-mul-1 45.4%

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

       3. *-commutative 45.4%

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

   15. Simplified 45.4%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+75}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+80}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+182} \lor \neg \left(z \leq 4.5 \cdot 10^{+182}\right):\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3.3e+182) (not (<= z 4.5e+182)))
   (/ (- y) (- (/ t z) a))
   (/ (- x (* z y)) (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.3e+182) || !(z <= 4.5e+182)) {
		tmp = -y / ((t / z) - a);
	} else {
		tmp = (x - (z * y)) / (t - (z * a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-3.3d+182)) .or. (.not. (z <= 4.5d+182))) then
        tmp = -y / ((t / z) - a)
    else
        tmp = (x - (z * y)) / (t - (z * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.3e+182) || !(z <= 4.5e+182)) {
		tmp = -y / ((t / z) - a);
	} else {
		tmp = (x - (z * y)) / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -3.3e+182) or not (z <= 4.5e+182):
		tmp = -y / ((t / z) - a)
	else:
		tmp = (x - (z * y)) / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3.3e+182) || !(z <= 4.5e+182))
		tmp = Float64(Float64(-y) / Float64(Float64(t / z) - a));
	else
		tmp = Float64(Float64(x - Float64(z * y)) / Float64(t - Float64(z * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -3.3e+182) || ~((z <= 4.5e+182)))
		tmp = -y / ((t / z) - a);
	else
		tmp = (x - (z * y)) / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3.3e+182], N[Not[LessEqual[z, 4.5e+182]], $MachinePrecision]], N[((-y) / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.3000000000000001e182 or 4.50000000000000029e182 < z

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

   3. Simplified 49.5%

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t - z \cdot a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 49.5%

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

      2. inv-pow 49.5%

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

      3. sub-neg 49.5%

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

      4. +-commutative 49.5%

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

      5. *-commutative 49.5%

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

      6. distribute-rgt-neg-in 49.5%

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

      7. fma-def 49.5%

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

   6. Applied egg-rr 49.5%

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

   7. Taylor expanded in a around 0 43.4%

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

      1. +-commutative 43.4%

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

      2. mul-1-neg 43.4%

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

      3. unsub-neg 43.4%

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

      4. associate-/l* 55.2%

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

   9. Simplified 55.2%

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

   10. Taylor expanded in y around -inf 91.6%

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

       1. associate-*r/ 91.6%

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

       2. mul-1-neg 91.6%

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

   12. Simplified 91.6%

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

   if -3.3000000000000001e182 < z < 4.50000000000000029e182

   1. Initial program 95.5%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+182} \lor \neg \left(z \leq 4.5 \cdot 10^{+182}\right):\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+182} \lor \neg \left(z \leq 7 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.02e+182) (not (<= z 7e+79))) (/ y a) (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.02e+182) || !(z <= 7e+79)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-1.02d+182)) .or. (.not. (z <= 7d+79))) then
        tmp = y / a
    else
        tmp = x / (t - (z * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.02e+182) || !(z <= 7e+79)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.02e+182) or not (z <= 7e+79):
		tmp = y / a
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.02e+182) || !(z <= 7e+79))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / Float64(t - Float64(z * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -1.02e+182) || ~((z <= 7e+79)))
		tmp = y / a;
	else
		tmp = x / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.02e+182], N[Not[LessEqual[z, 7e+79]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.02e182 or 6.99999999999999961e79 < z

   1. Initial program 60.4%

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

      1. *-commutative 60.4%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in z around inf 73.9%

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

   if -1.02e182 < z < 6.99999999999999961e79

   1. Initial program 96.6%

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

      1. *-commutative 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in x around inf 70.7%

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

      1. *-commutative 70.7%

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

   7. Simplified 70.7%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+182} \lor \neg \left(z \leq 7 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+75} \lor \neg \left(z \leq 8 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -6.6e+75) (not (<= z 8e+79))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -6.6e+75) || !(z <= 8e+79)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-6.6d+75)) .or. (.not. (z <= 8d+79))) then
        tmp = y / a
    else
        tmp = x / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -6.6e+75) || !(z <= 8e+79)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -6.6e+75) or not (z <= 8e+79):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -6.6e+75) || !(z <= 8e+79))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -6.6e+75) || ~((z <= 8e+79)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -6.6e+75], N[Not[LessEqual[z, 8e+79]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.59999999999999996e75 or 7.99999999999999974e79 < z

   1. Initial program 67.5%

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

      1. *-commutative 67.5%

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

   3. Simplified 67.5%

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

   5. Taylor expanded in z around inf 65.9%

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

   if -6.59999999999999996e75 < z < 7.99999999999999974e79

   1. Initial program 98.0%

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

      1. *-commutative 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in z around 0 51.9%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+75} \lor \neg \left(z \leq 8 \cdot 10^{+79}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x / t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.6%

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

   1. *-commutative 86.6%

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

3. Simplified 86.6%

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

5. Taylor expanded in z around 0 38.7%

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

6. Final simplification 38.7%

   \[\leadsto \frac{x}{t} \]
7. Add Preprocessing

Developer target: 97.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{x}{t_1} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z))) (t_2 (- (/ x t_1) (/ y (- (/ t z) a)))))
   (if (< z -32113435955957344.0)
     t_2
     (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 t_1)) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = (x / t_1) - (y / ((t / z) - a));
	double tmp;
	if (z < -32113435955957344.0) {
		tmp = t_2;
	} else if (z < 3.5139522372978296e-86) {
		tmp = (x - (y * z)) * (1.0 / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t - (a * z)
    t_2 = (x / t_1) - (y / ((t / z) - a))
    if (z < (-32113435955957344.0d0)) then
        tmp = t_2
    else if (z < 3.5139522372978296d-86) then
        tmp = (x - (y * z)) * (1.0d0 / t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = (x / t_1) - (y / ((t / z) - a));
	double tmp;
	if (z < -32113435955957344.0) {
		tmp = t_2;
	} else if (z < 3.5139522372978296e-86) {
		tmp = (x - (y * z)) * (1.0 / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (a * z)
	t_2 = (x / t_1) - (y / ((t / z) - a))
	tmp = 0
	if z < -32113435955957344.0:
		tmp = t_2
	elif z < 3.5139522372978296e-86:
		tmp = (x - (y * z)) * (1.0 / t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	t_2 = Float64(Float64(x / t_1) - Float64(y / Float64(Float64(t / z) - a)))
	tmp = 0.0
	if (z < -32113435955957344.0)
		tmp = t_2;
	elseif (z < 3.5139522372978296e-86)
		tmp = Float64(Float64(x - Float64(y * z)) * Float64(1.0 / t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (a * z);
	t_2 = (x / t_1) - (y / ((t / z) - a));
	tmp = 0.0;
	if (z < -32113435955957344.0)
		tmp = t_2;
	elseif (z < 3.5139522372978296e-86)
		tmp = (x - (y * z)) * (1.0 / t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / t$95$1), $MachinePrecision] - N[(y / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -32113435955957344.0], t$95$2, If[Less[z, 3.5139522372978296e-86], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024020 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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