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

Percentage Accurate: 85.8% → 94.2%

Time: 11.4s

Alternatives: 9

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: 85.8% 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: 94.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - y \cdot z\\ t_2 := \frac{t\_1}{t - z \cdot a}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-290}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{t\_1}{z} \cdot \frac{1}{\frac{t}{z} - a}\\ \mathbf{elif}\;t\_2 \leq 10^{+269}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* y z))) (t_2 (/ t_1 (- t (* z a)))))
   (if (<= t_2 -5e-290)
     t_2
     (if (<= t_2 0.0)
       (* (/ t_1 z) (/ 1.0 (- (/ t z) a)))
       (if (<= t_2 1e+269) t_2 (/ y (- a (/ t z))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * z);
	double t_2 = t_1 / (t - (z * a));
	double tmp;
	if (t_2 <= -5e-290) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = (t_1 / z) * (1.0 / ((t / z) - a));
	} else if (t_2 <= 1e+269) {
		tmp = t_2;
	} else {
		tmp = y / (a - (t / z));
	}
	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 = x - (y * z)
    t_2 = t_1 / (t - (z * a))
    if (t_2 <= (-5d-290)) then
        tmp = t_2
    else if (t_2 <= 0.0d0) then
        tmp = (t_1 / z) * (1.0d0 / ((t / z) - a))
    else if (t_2 <= 1d+269) then
        tmp = t_2
    else
        tmp = y / (a - (t / z))
    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 - (y * z);
	double t_2 = t_1 / (t - (z * a));
	double tmp;
	if (t_2 <= -5e-290) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = (t_1 / z) * (1.0 / ((t / z) - a));
	} else if (t_2 <= 1e+269) {
		tmp = t_2;
	} else {
		tmp = y / (a - (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x - (y * z)
	t_2 = t_1 / (t - (z * a))
	tmp = 0
	if t_2 <= -5e-290:
		tmp = t_2
	elif t_2 <= 0.0:
		tmp = (t_1 / z) * (1.0 / ((t / z) - a))
	elif t_2 <= 1e+269:
		tmp = t_2
	else:
		tmp = y / (a - (t / z))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(y * z))
	t_2 = Float64(t_1 / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (t_2 <= -5e-290)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(t_1 / z) * Float64(1.0 / Float64(Float64(t / z) - a)));
	elseif (t_2 <= 1e+269)
		tmp = t_2;
	else
		tmp = Float64(y / Float64(a - Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x - (y * z);
	t_2 = t_1 / (t - (z * a));
	tmp = 0.0;
	if (t_2 <= -5e-290)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = (t_1 / z) * (1.0 / ((t / z) - a));
	elseif (t_2 <= 1e+269)
		tmp = t_2;
	else
		tmp = y / (a - (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-290], t$95$2, If[LessEqual[t$95$2, 0.0], N[(N[(t$95$1 / z), $MachinePrecision] * N[(1.0 / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+269], t$95$2, N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -5.0000000000000001e-290 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1e269

   1. Initial program 97.8%

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

   if -5.0000000000000001e-290 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 54.3%

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

      1. *-commutative 54.3%

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

   3. Simplified 54.3%

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

   5. Taylor expanded in z around inf 54.3%

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

      1. associate-/r* 99.9%

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

      2. div-inv 99.8%

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

   7. Applied egg-rr 99.8%

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

   if 1e269 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 36.4%

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

      1. *-commutative 36.4%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in z around inf 36.4%

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

   6. Taylor expanded in x around 0 83.1%

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

      1. mul-1-neg 83.1%

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

      2. distribute-neg-frac2 83.1%

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

      3. sub-neg 83.1%

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

      4. +-commutative 83.1%

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

      5. distribute-neg-in 83.1%

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

      6. remove-double-neg 83.1%

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

      7. unsub-neg 83.1%

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

   8. Simplified 83.1%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -5 \cdot 10^{-290}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{x - y \cdot z}{z} \cdot \frac{1}{\frac{t}{z} - a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 10^{+269}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a - \frac{t}{z}}\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{+116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-11}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ y (- a (/ t z)))))
   (if (<= z -1.45e+116)
     t_1
     (if (<= z -4.2e-14)
       (/ x (- t (* z a)))
       (if (<= z 2.45e-11) (/ (- x (* y z)) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y / (a - (t / z));
	double tmp;
	if (z <= -1.45e+116) {
		tmp = t_1;
	} else if (z <= -4.2e-14) {
		tmp = x / (t - (z * a));
	} else if (z <= 2.45e-11) {
		tmp = (x - (y * z)) / 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 / (a - (t / z))
    if (z <= (-1.45d+116)) then
        tmp = t_1
    else if (z <= (-4.2d-14)) then
        tmp = x / (t - (z * a))
    else if (z <= 2.45d-11) 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 a) {
	double t_1 = y / (a - (t / z));
	double tmp;
	if (z <= -1.45e+116) {
		tmp = t_1;
	} else if (z <= -4.2e-14) {
		tmp = x / (t - (z * a));
	} else if (z <= 2.45e-11) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y / (a - (t / z))
	tmp = 0
	if z <= -1.45e+116:
		tmp = t_1
	elif z <= -4.2e-14:
		tmp = x / (t - (z * a))
	elif z <= 2.45e-11:
		tmp = (x - (y * z)) / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y / Float64(a - Float64(t / z)))
	tmp = 0.0
	if (z <= -1.45e+116)
		tmp = t_1;
	elseif (z <= -4.2e-14)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 2.45e-11)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y / (a - (t / z));
	tmp = 0.0;
	if (z <= -1.45e+116)
		tmp = t_1;
	elseif (z <= -4.2e-14)
		tmp = x / (t - (z * a));
	elseif (z <= 2.45e-11)
		tmp = (x - (y * z)) / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.45e+116], t$95$1, If[LessEqual[z, -4.2e-14], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.45e-11], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.4500000000000001e116 or 2.4499999999999999e-11 < z

   1. Initial program 65.9%

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

      1. *-commutative 65.9%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in z around inf 65.9%

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

   6. Taylor expanded in x around 0 77.3%

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

      1. mul-1-neg 77.3%

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

      2. distribute-neg-frac2 77.3%

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

      3. sub-neg 77.3%

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

      4. +-commutative 77.3%

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

      5. distribute-neg-in 77.3%

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

      6. remove-double-neg 77.3%

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

      7. unsub-neg 77.3%

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

   8. Simplified 77.3%

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

   if -1.4500000000000001e116 < z < -4.1999999999999998e-14

   1. Initial program 95.9%

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

      1. *-commutative 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in x around inf 67.5%

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

      1. *-commutative 67.5%

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

   7. Simplified 67.5%

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

   if -4.1999999999999998e-14 < z < 2.4499999999999999e-11

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 83.3%

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

Alternative 3: 91.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+122} \lor \neg \left(z \leq 9.2 \cdot 10^{+165}\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.2e+122) (not (<= z 9.2e+165)))
   (/ y (- a (/ t z)))
   (/ (- x (* y z)) (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.2e+122) || !(z <= 9.2e+165)) {
		tmp = y / (a - (t / z));
	} else {
		tmp = (x - (y * z)) / (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 <= (-1.2d+122)) .or. (.not. (z <= 9.2d+165))) then
        tmp = y / (a - (t / z))
    else
        tmp = (x - (y * z)) / (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 <= -1.2e+122) || !(z <= 9.2e+165)) {
		tmp = y / (a - (t / z));
	} else {
		tmp = (x - (y * z)) / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.2e+122) or not (z <= 9.2e+165):
		tmp = y / (a - (t / z))
	else:
		tmp = (x - (y * z)) / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.2e+122) || !(z <= 9.2e+165))
		tmp = Float64(y / Float64(a - Float64(t / z)));
	else
		tmp = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(z * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -1.2e+122) || ~((z <= 9.2e+165)))
		tmp = y / (a - (t / z));
	else
		tmp = (x - (y * z)) / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.2e+122], N[Not[LessEqual[z, 9.2e+165]], $MachinePrecision]], N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2000000000000001e122 or 9.20000000000000063e165 < z

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

   3. Simplified 51.7%

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

   5. Taylor expanded in z around inf 51.8%

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

   6. Taylor expanded in x around 0 87.7%

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

      1. mul-1-neg 87.7%

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

      2. distribute-neg-frac2 87.7%

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

      3. sub-neg 87.7%

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

      4. +-commutative 87.7%

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

      5. distribute-neg-in 87.7%

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

      6. remove-double-neg 87.7%

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

      7. unsub-neg 87.7%

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

   8. Simplified 87.7%

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

   if -1.2000000000000001e122 < z < 9.20000000000000063e165

   1. Initial program 96.8%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+122} \lor \neg \left(z \leq 9.2 \cdot 10^{+165}\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+109}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -8.4 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{x}{a}}{-z}\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.4e+109)
   (/ y a)
   (if (<= z -8.4e-16)
     (/ (/ x a) (- z))
     (if (<= z 2.55e-11) (/ x t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.4e+109) {
		tmp = y / a;
	} else if (z <= -8.4e-16) {
		tmp = (x / a) / -z;
	} else if (z <= 2.55e-11) {
		tmp = x / t;
	} 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 <= (-3.4d+109)) then
        tmp = y / a
    else if (z <= (-8.4d-16)) then
        tmp = (x / a) / -z
    else if (z <= 2.55d-11) then
        tmp = x / t
    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 <= -3.4e+109) {
		tmp = y / a;
	} else if (z <= -8.4e-16) {
		tmp = (x / a) / -z;
	} else if (z <= 2.55e-11) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.4e+109:
		tmp = y / a
	elif z <= -8.4e-16:
		tmp = (x / a) / -z
	elif z <= 2.55e-11:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.4e+109)
		tmp = Float64(y / a);
	elseif (z <= -8.4e-16)
		tmp = Float64(Float64(x / a) / Float64(-z));
	elseif (z <= 2.55e-11)
		tmp = Float64(x / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.4e+109)
		tmp = y / a;
	elseif (z <= -8.4e-16)
		tmp = (x / a) / -z;
	elseif (z <= 2.55e-11)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.4e+109], N[(y / a), $MachinePrecision], If[LessEqual[z, -8.4e-16], N[(N[(x / a), $MachinePrecision] / (-z)), $MachinePrecision], If[LessEqual[z, 2.55e-11], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.40000000000000006e109 or 2.54999999999999992e-11 < 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 z around inf 54.9%

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

   if -3.40000000000000006e109 < z < -8.4000000000000004e-16

   1. Initial program 95.8%

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

      1. *-commutative 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in y around inf 87.6%

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

   7. Simplified 87.6%

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

   8. Taylor expanded in a around inf 47.8%

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

      1. associate-/l* 47.7%

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

      2. associate-*r/ 47.7%

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

      3. neg-mul-1 47.7%

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

      4. *-commutative 47.7%

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

   10. Simplified 47.7%

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

   11. Taylor expanded in y around 0 49.5%

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

       1. mul-1-neg 49.5%

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

       2. associate-/r* 53.4%

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

       3. distribute-neg-frac2 53.4%

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

   13. Simplified 53.4%

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

   if -8.4000000000000004e-16 < z < 2.54999999999999992e-11

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 62.2%

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

Alternative 5: 53.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+108}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{z \cdot \left(-a\right)}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.1e+108)
   (/ y a)
   (if (<= z -2.7e-12) (/ x (* z (- a))) (if (<= z 2.6e-11) (/ x t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.1e+108) {
		tmp = y / a;
	} else if (z <= -2.7e-12) {
		tmp = x / (z * -a);
	} else if (z <= 2.6e-11) {
		tmp = x / t;
	} 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 <= (-2.1d+108)) then
        tmp = y / a
    else if (z <= (-2.7d-12)) then
        tmp = x / (z * -a)
    else if (z <= 2.6d-11) then
        tmp = x / t
    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 <= -2.1e+108) {
		tmp = y / a;
	} else if (z <= -2.7e-12) {
		tmp = x / (z * -a);
	} else if (z <= 2.6e-11) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.1e+108:
		tmp = y / a
	elif z <= -2.7e-12:
		tmp = x / (z * -a)
	elif z <= 2.6e-11:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.1e+108)
		tmp = Float64(y / a);
	elseif (z <= -2.7e-12)
		tmp = Float64(x / Float64(z * Float64(-a)));
	elseif (z <= 2.6e-11)
		tmp = Float64(x / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.1e+108)
		tmp = y / a;
	elseif (z <= -2.7e-12)
		tmp = x / (z * -a);
	elseif (z <= 2.6e-11)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.1e+108], N[(y / a), $MachinePrecision], If[LessEqual[z, -2.7e-12], N[(x / N[(z * (-a)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.6e-11], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1000000000000001e108 or 2.6000000000000001e-11 < 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 z around inf 54.9%

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

   if -2.1000000000000001e108 < z < -2.6999999999999998e-12

   1. Initial program 95.8%

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

      1. *-commutative 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in t around 0 59.9%

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

      1. associate-*r/ 59.9%

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

      2. neg-mul-1 59.9%

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

      3. neg-sub0 59.9%

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

      4. sub-neg 59.9%

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

      5. distribute-rgt-neg-out 59.9%

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

      6. +-commutative 59.9%

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

      7. associate--r+ 59.9%

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

      8. neg-sub0 59.9%

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

      9. distribute-rgt-neg-out 59.9%

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

      10. remove-double-neg 59.9%

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

      11. *-commutative 59.9%

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

   7. Simplified 59.9%

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

   8. Taylor expanded in y around 0 49.5%

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

      1. neg-mul-1 49.5%

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

   10. Simplified 49.5%

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

   if -2.6999999999999998e-12 < z < 2.6000000000000001e-11

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 62.2%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+108}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{z \cdot \left(-a\right)}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+117} \lor \neg \left(z \leq 1.02 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -7.2e+117) (not (<= z 1.02e-21)))
   (/ y (- a (/ t z)))
   (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -7.2e+117) || !(z <= 1.02e-21)) {
		tmp = y / (a - (t / z));
	} 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 <= (-7.2d+117)) .or. (.not. (z <= 1.02d-21))) then
        tmp = y / (a - (t / z))
    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 <= -7.2e+117) || !(z <= 1.02e-21)) {
		tmp = y / (a - (t / z));
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -7.2e+117) or not (z <= 1.02e-21):
		tmp = y / (a - (t / z))
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -7.2e+117) || !(z <= 1.02e-21))
		tmp = Float64(y / Float64(a - Float64(t / z)));
	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 <= -7.2e+117) || ~((z <= 1.02e-21)))
		tmp = y / (a - (t / z));
	else
		tmp = x / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -7.2e+117], N[Not[LessEqual[z, 1.02e-21]], $MachinePrecision]], N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.20000000000000025e117 or 1.02000000000000004e-21 < z

   1. Initial program 67.1%

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

      1. *-commutative 67.1%

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

   3. Simplified 67.1%

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

   5. Taylor expanded in z around inf 67.1%

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

   6. Taylor expanded in x around 0 76.4%

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

      1. mul-1-neg 76.4%

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

      2. distribute-neg-frac2 76.4%

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

      3. sub-neg 76.4%

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

      4. +-commutative 76.4%

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

      5. distribute-neg-in 76.4%

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

      6. remove-double-neg 76.4%

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

      7. unsub-neg 76.4%

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

   8. Simplified 76.4%

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

   if -7.20000000000000025e117 < z < 1.02000000000000004e-21

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around inf 73.4%

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

      1. *-commutative 73.4%

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

   7. Simplified 73.4%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+117} \lor \neg \left(z \leq 1.02 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+118} \lor \neg \left(z \leq 1.8 \cdot 10^{+163}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -5.2e+118) (not (<= z 1.8e+163))) (/ y a) (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -5.2e+118) || !(z <= 1.8e+163)) {
		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 <= (-5.2d+118)) .or. (.not. (z <= 1.8d+163))) 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 <= -5.2e+118) || !(z <= 1.8e+163)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -5.2e+118) or not (z <= 1.8e+163):
		tmp = y / a
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -5.2e+118) || !(z <= 1.8e+163))
		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 <= -5.2e+118) || ~((z <= 1.8e+163)))
		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, -5.2e+118], N[Not[LessEqual[z, 1.8e+163]], $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 < -5.20000000000000032e118 or 1.79999999999999989e163 < z

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

   3. Simplified 51.7%

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

   5. Taylor expanded in z around inf 68.3%

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

   if -5.20000000000000032e118 < z < 1.79999999999999989e163

   1. Initial program 96.8%

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

      1. *-commutative 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in x around inf 66.5%

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

      1. *-commutative 66.5%

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

   7. Simplified 66.5%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+118} \lor \neg \left(z \leq 1.8 \cdot 10^{+163}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+50} \lor \neg \left(z \leq 5.5 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.7e+50) (not (<= z 5.5e-16))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.7e+50) || !(z <= 5.5e-16)) {
		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 <= (-1.7d+50)) .or. (.not. (z <= 5.5d-16))) 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 <= -1.7e+50) || !(z <= 5.5e-16)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.7e+50) or not (z <= 5.5e-16):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.7e+50) || !(z <= 5.5e-16))
		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 <= -1.7e+50) || ~((z <= 5.5e-16)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.7e+50], N[Not[LessEqual[z, 5.5e-16]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.6999999999999999e50 or 5.49999999999999964e-16 < z

   1. Initial program 67.9%

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

      1. *-commutative 67.9%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in z around inf 52.8%

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

   if -1.6999999999999999e50 < z < 5.49999999999999964e-16

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 58.6%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+50} \lor \neg \left(z \leq 5.5 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.4% 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 84.6%

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

   1. *-commutative 84.6%

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

3. Simplified 84.6%

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

5. Taylor expanded in z around 0 36.5%

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

Developer Target 1: 97.4% 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 2024139 
(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))))