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

Percentage Accurate: 85.1% → 91.0%

Time: 12.3s

Alternatives: 7

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{+135} \lor \neg \left(z \leq 8.2 \cdot 10^{+139}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.75e+135) (not (<= z 8.2e+139)))
   (/ (- y (/ x 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 <= -2.75e+135) || !(z <= 8.2e+139)) {
		tmp = (y - (x / 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 <= (-2.75d+135)) .or. (.not. (z <= 8.2d+139))) then
        tmp = (y - (x / 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 <= -2.75e+135) || !(z <= 8.2e+139)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (z * y)) / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -2.75e+135) or not (z <= 8.2e+139):
		tmp = (y - (x / 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 <= -2.75e+135) || !(z <= 8.2e+139))
		tmp = Float64(Float64(y - Float64(x / 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 <= -2.75e+135) || ~((z <= 8.2e+139)))
		tmp = (y - (x / 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, -2.75e+135], N[Not[LessEqual[z, 8.2e+139]], $MachinePrecision]], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $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 < -2.7499999999999999e135 or 8.2000000000000004e139 < z

   1. Initial program 55.5%

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

      1. *-commutative 55.5%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in x around 0 55.5%

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

      1. *-commutative 55.5%

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

      2. *-commutative 55.5%

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

      3. *-un-lft-identity 55.5%

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

      4. times-frac 63.5%

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

   7. Applied egg-rr 63.5%

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

      1. /-rgt-identity 63.5%

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

   9. Simplified 63.5%

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

   10. Taylor expanded in a around inf 88.9%

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

       1. mul-1-neg 88.9%

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

       2. unsub-neg 88.9%

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

   12. Simplified 88.9%

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

   if -2.7499999999999999e135 < z < 8.2000000000000004e139

   1. Initial program 97.7%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{+135} \lor \neg \left(z \leq 8.2 \cdot 10^{+139}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 64.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ \mathbf{if}\;z \leq -5 \cdot 10^{+138}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-163}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.00285:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+122}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (- t (* z a)))))
   (if (<= z -5e+138)
     (/ y a)
     (if (<= z 7.8e-163)
       t_1
       (if (<= z 0.00285)
         (/ (- x (* z y)) t)
         (if (<= z 5e+122) 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 <= -5e+138) {
		tmp = y / a;
	} else if (z <= 7.8e-163) {
		tmp = t_1;
	} else if (z <= 0.00285) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 5e+122) {
		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 <= (-5d+138)) then
        tmp = y / a
    else if (z <= 7.8d-163) then
        tmp = t_1
    else if (z <= 0.00285d0) then
        tmp = (x - (z * y)) / t
    else if (z <= 5d+122) 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 <= -5e+138) {
		tmp = y / a;
	} else if (z <= 7.8e-163) {
		tmp = t_1;
	} else if (z <= 0.00285) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 5e+122) {
		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 <= -5e+138:
		tmp = y / a
	elif z <= 7.8e-163:
		tmp = t_1
	elif z <= 0.00285:
		tmp = (x - (z * y)) / t
	elif z <= 5e+122:
		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 <= -5e+138)
		tmp = Float64(y / a);
	elseif (z <= 7.8e-163)
		tmp = t_1;
	elseif (z <= 0.00285)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 5e+122)
		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 <= -5e+138)
		tmp = y / a;
	elseif (z <= 7.8e-163)
		tmp = t_1;
	elseif (z <= 0.00285)
		tmp = (x - (z * y)) / t;
	elseif (z <= 5e+122)
		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, -5e+138], N[(y / a), $MachinePrecision], If[LessEqual[z, 7.8e-163], t$95$1, If[LessEqual[z, 0.00285], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 5e+122], t$95$1, N[(y / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.00000000000000016e138 or 4.99999999999999989e122 < z

   1. Initial program 57.2%

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

      1. *-commutative 57.2%

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

   3. Simplified 57.2%

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

   5. Taylor expanded in z around inf 67.7%

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

   if -5.00000000000000016e138 < z < 7.8000000000000004e-163 or 0.0028500000000000001 < z < 4.99999999999999989e122

   1. Initial program 97.0%

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

      1. *-commutative 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in x around inf 72.5%

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

      1. *-commutative 72.5%

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

   7. Simplified 72.5%

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

   if 7.8000000000000004e-163 < z < 0.0028500000000000001

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in t around inf 76.4%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+138}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-163}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 0.00285:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+122}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -4.3 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-163}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{t} - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -4.3e+52)
     t_1
     (if (<= z 7.2e-163)
       (/ x (- t (* z a)))
       (if (<= z 4.5e+29) (- (/ x t) (/ (* z y) t)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -4.3e+52) {
		tmp = t_1;
	} else if (z <= 7.2e-163) {
		tmp = x / (t - (z * a));
	} else if (z <= 4.5e+29) {
		tmp = (x / t) - ((z * y) / t);
	} else {
		tmp = t_1;
	}
	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 = (y - (x / z)) / a
    if (z <= (-4.3d+52)) then
        tmp = t_1
    else if (z <= 7.2d-163) then
        tmp = x / (t - (z * a))
    else if (z <= 4.5d+29) then
        tmp = (x / t) - ((z * y) / 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 a) {
	double t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -4.3e+52) {
		tmp = t_1;
	} else if (z <= 7.2e-163) {
		tmp = x / (t - (z * a));
	} else if (z <= 4.5e+29) {
		tmp = (x / t) - ((z * y) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - (x / z)) / a
	tmp = 0
	if z <= -4.3e+52:
		tmp = t_1
	elif z <= 7.2e-163:
		tmp = x / (t - (z * a))
	elif z <= 4.5e+29:
		tmp = (x / t) - ((z * y) / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - Float64(x / z)) / a)
	tmp = 0.0
	if (z <= -4.3e+52)
		tmp = t_1;
	elseif (z <= 7.2e-163)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 4.5e+29)
		tmp = Float64(Float64(x / t) - Float64(Float64(z * y) / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - (x / z)) / a;
	tmp = 0.0;
	if (z <= -4.3e+52)
		tmp = t_1;
	elseif (z <= 7.2e-163)
		tmp = x / (t - (z * a));
	elseif (z <= 4.5e+29)
		tmp = (x / t) - ((z * y) / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -4.3e+52], t$95$1, If[LessEqual[z, 7.2e-163], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.5e+29], N[(N[(x / t), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.3e52 or 4.5000000000000002e29 < z

   1. Initial program 66.2%

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

      1. *-commutative 66.2%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in x around 0 66.3%

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

      1. *-commutative 66.3%

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

      2. *-commutative 66.3%

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

      3. *-un-lft-identity 66.3%

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

      4. times-frac 71.9%

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

   7. Applied egg-rr 71.9%

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

      1. /-rgt-identity 71.9%

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

   9. Simplified 71.9%

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

   10. Taylor expanded in a around inf 79.9%

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

       1. mul-1-neg 79.9%

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

       2. unsub-neg 79.9%

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

   12. Simplified 79.9%

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

   if -4.3e52 < z < 7.1999999999999996e-163

   1. Initial program 99.0%

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

      1. *-commutative 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around inf 74.8%

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

      1. *-commutative 74.8%

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

   7. Simplified 74.8%

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

   if 7.1999999999999996e-163 < z < 4.5000000000000002e29

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. times-frac 91.1%

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

   7. Applied egg-rr 91.1%

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

      1. /-rgt-identity 91.1%

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

   9. Simplified 91.1%

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

       1. +-commutative 91.1%

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

       2. mul-1-neg 91.1%

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

       3. unsub-neg 91.1%

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

       4. *-commutative 91.1%

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

       5. clear-num 91.0%

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

       6. un-div-inv 91.6%

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

   11. Applied egg-rr 91.6%

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

   12. Taylor expanded in a around 0 75.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{+52}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-163}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{t} - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -3.9 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{-162}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+27}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -3.9e+52)
     t_1
     (if (<= z 1.08e-162)
       (/ x (- t (* z a)))
       (if (<= z 3.9e+27) (/ (- x (* z y)) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -3.9e+52) {
		tmp = t_1;
	} else if (z <= 1.08e-162) {
		tmp = x / (t - (z * a));
	} else if (z <= 3.9e+27) {
		tmp = (x - (z * y)) / t;
	} else {
		tmp = t_1;
	}
	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 = (y - (x / z)) / a
    if (z <= (-3.9d+52)) then
        tmp = t_1
    else if (z <= 1.08d-162) then
        tmp = x / (t - (z * a))
    else if (z <= 3.9d+27) then
        tmp = (x - (z * y)) / 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 a) {
	double t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -3.9e+52) {
		tmp = t_1;
	} else if (z <= 1.08e-162) {
		tmp = x / (t - (z * a));
	} else if (z <= 3.9e+27) {
		tmp = (x - (z * y)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - (x / z)) / a
	tmp = 0
	if z <= -3.9e+52:
		tmp = t_1
	elif z <= 1.08e-162:
		tmp = x / (t - (z * a))
	elif z <= 3.9e+27:
		tmp = (x - (z * y)) / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - Float64(x / z)) / a)
	tmp = 0.0
	if (z <= -3.9e+52)
		tmp = t_1;
	elseif (z <= 1.08e-162)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 3.9e+27)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - (x / z)) / a;
	tmp = 0.0;
	if (z <= -3.9e+52)
		tmp = t_1;
	elseif (z <= 1.08e-162)
		tmp = x / (t - (z * a));
	elseif (z <= 3.9e+27)
		tmp = (x - (z * y)) / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -3.9e+52], t$95$1, If[LessEqual[z, 1.08e-162], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.9e+27], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.9e52 or 3.8999999999999999e27 < z

   1. Initial program 66.2%

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

      1. *-commutative 66.2%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in x around 0 66.3%

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

      1. *-commutative 66.3%

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

      2. *-commutative 66.3%

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

      3. *-un-lft-identity 66.3%

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

      4. times-frac 71.9%

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

   7. Applied egg-rr 71.9%

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

      1. /-rgt-identity 71.9%

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

   9. Simplified 71.9%

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

   10. Taylor expanded in a around inf 79.9%

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

       1. mul-1-neg 79.9%

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

       2. unsub-neg 79.9%

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

   12. Simplified 79.9%

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

   if -3.9e52 < z < 1.08000000000000006e-162

   1. Initial program 99.0%

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

      1. *-commutative 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around inf 74.8%

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

      1. *-commutative 74.8%

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

   7. Simplified 74.8%

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

   if 1.08000000000000006e-162 < z < 3.8999999999999999e27

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in t around inf 75.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+52}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{-162}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+27}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+126} \lor \neg \left(z \leq 4.8 \cdot 10^{+122}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -9.5e+126) (not (<= z 4.8e+122))) (/ y a) (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -9.5e+126) || !(z <= 4.8e+122)) {
		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 <= (-9.5d+126)) .or. (.not. (z <= 4.8d+122))) 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 <= -9.5e+126) || !(z <= 4.8e+122)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -9.5e+126) or not (z <= 4.8e+122):
		tmp = y / a
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -9.5e+126) || !(z <= 4.8e+122))
		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 <= -9.5e+126) || ~((z <= 4.8e+122)))
		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, -9.5e+126], N[Not[LessEqual[z, 4.8e+122]], $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 < -9.49999999999999951e126 or 4.8000000000000004e122 < z

   1. Initial program 57.2%

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

      1. *-commutative 57.2%

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

   3. Simplified 57.2%

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

   5. Taylor expanded in z around inf 67.7%

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

   if -9.49999999999999951e126 < z < 4.8000000000000004e122

   1. Initial program 97.6%

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

      1. *-commutative 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in x around inf 68.3%

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

      1. *-commutative 68.3%

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

   7. Simplified 68.3%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+126} \lor \neg \left(z \leq 4.8 \cdot 10^{+122}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.75 \cdot 10^{-28} \lor \neg \left(z \leq 4.2 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3.75e-28) (not (<= z 4.2e-23))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.75e-28) || !(z <= 4.2e-23)) {
		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 <= (-3.75d-28)) .or. (.not. (z <= 4.2d-23))) 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 <= -3.75e-28) || !(z <= 4.2e-23)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -3.75e-28) or not (z <= 4.2e-23):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3.75e-28) || !(z <= 4.2e-23))
		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 <= -3.75e-28) || ~((z <= 4.2e-23)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3.75e-28], N[Not[LessEqual[z, 4.2e-23]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.7500000000000001e-28 or 4.2000000000000002e-23 < z

   1. Initial program 71.6%

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

      1. *-commutative 71.6%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in z around inf 53.3%

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

   if -3.7500000000000001e-28 < z < 4.2000000000000002e-23

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 51.7%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.75 \cdot 10^{-28} \lor \neg \left(z \leq 4.2 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.5% 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 85.1%

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

   1. *-commutative 85.1%

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

3. Simplified 85.1%

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

5. Taylor expanded in z around 0 34.4%

   \[\leadsto \color{blue}{\frac{x}{t}} \]
6. Add Preprocessing

Developer Target 1: 97.2% 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 2024131 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -32113435955957344) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 4392440296622287/125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))))))

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