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

Percentage Accurate: 84.9% → 92.6%

Time: 7.1s

Alternatives: 9

Speedup: 0.7×

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: 84.9% 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: 92.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ \mathbf{if}\;\frac{x - y \cdot z}{t\_1} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z))))
   (if (<= (/ (- x (* y z)) t_1) INFINITY)
     (fma (/ z (fma a z (- t))) y (/ x t_1))
     (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double tmp;
	if (((x - (y * z)) / t_1) <= ((double) INFINITY)) {
		tmp = fma((z / fma(a, z, -t)), y, (x / t_1));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	tmp = 0.0
	if (Float64(Float64(x - Float64(y * z)) / t_1) <= Inf)
		tmp = fma(Float64(z / fma(a, z, Float64(-t))), y, Float64(x / t_1));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 88.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 91.6%

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

   if +inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{y}{a}} \]
   4. Step-by-step derivation

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{y}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 90.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+222} \lor \neg \left(z \leq 3.7 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.8e+222) (not (<= z 3.7e+173)))
   (/ (- y (/ x z)) a)
   (/ (- x (* y z)) (- t (* a z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.8e+222) || !(z <= 3.7e+173)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (y * z)) / (t - (a * 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) :: tmp
    if ((z <= (-2.8d+222)) .or. (.not. (z <= 3.7d+173))) then
        tmp = (y - (x / z)) / a
    else
        tmp = (x - (y * z)) / (t - (a * z))
    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 <= -2.8e+222) || !(z <= 3.7e+173)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (y * z)) / (t - (a * z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -2.8e+222) or not (z <= 3.7e+173):
		tmp = (y - (x / z)) / a
	else:
		tmp = (x - (y * z)) / (t - (a * z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.8e+222) || !(z <= 3.7e+173))
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	else
		tmp = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(a * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -2.8e+222) || ~((z <= 3.7e+173)))
		tmp = (y - (x / z)) / a;
	else
		tmp = (x - (y * z)) / (t - (a * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.8e+222], N[Not[LessEqual[z, 3.7e+173]], $MachinePrecision]], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8000000000000001e222 or 3.69999999999999986e173 < z

   1. Initial program 51.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. remove-double-neg N/A

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

      13. remove-double-neg N/A

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

      14. neg-mul-1 N/A

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

      15. remove-double-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-/.f64 85.5

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

   5. Applied rewrites 85.5%

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

   if -2.8000000000000001e222 < z < 3.69999999999999986e173

   1. Initial program 93.3%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+222} \lor \neg \left(z \leq 3.7 \cdot 10^{+173}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t - a \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{+90} \lor \neg \left(t \leq 2.2 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -6.6e+90) (not (<= t 2.2e+45)))
   (/ (- x (* z y)) t)
   (/ (- y (/ x z)) a)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -6.6e+90) || !(t <= 2.2e+45)) {
		tmp = (x - (z * y)) / t;
	} else {
		tmp = (y - (x / 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 ((t <= (-6.6d+90)) .or. (.not. (t <= 2.2d+45))) then
        tmp = (x - (z * y)) / t
    else
        tmp = (y - (x / 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 ((t <= -6.6e+90) || !(t <= 2.2e+45)) {
		tmp = (x - (z * y)) / t;
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -6.6e+90) or not (t <= 2.2e+45):
		tmp = (x - (z * y)) / t
	else:
		tmp = (y - (x / z)) / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -6.6e+90) || !(t <= 2.2e+45))
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	else
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -6.6e+90) || ~((t <= 2.2e+45)))
		tmp = (x - (z * y)) / t;
	else
		tmp = (y - (x / z)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -6.6e+90], N[Not[LessEqual[t, 2.2e+45]], $MachinePrecision]], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.60000000000000016e90 or 2.2e45 < t

   1. Initial program 86.1%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

   if -6.60000000000000016e90 < t < 2.2e45

   1. Initial program 83.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. remove-double-neg N/A

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

      13. remove-double-neg N/A

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

      14. neg-mul-1 N/A

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

      15. remove-double-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-/.f64 75.4

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

   5. Applied rewrites 75.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{+90} \lor \neg \left(t \leq 2.2 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-286}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{+73}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.2e+81)
   (/ y a)
   (if (<= z 5.5e-286)
     (/ (- x (* z y)) t)
     (if (<= z 2.35e+73) (/ x (- t (* a z))) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.2e+81) {
		tmp = y / a;
	} else if (z <= 5.5e-286) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 2.35e+73) {
		tmp = x / (t - (a * z));
	} 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 <= (-1.2d+81)) then
        tmp = y / a
    else if (z <= 5.5d-286) then
        tmp = (x - (z * y)) / t
    else if (z <= 2.35d+73) then
        tmp = x / (t - (a * z))
    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 <= -1.2e+81) {
		tmp = y / a;
	} else if (z <= 5.5e-286) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 2.35e+73) {
		tmp = x / (t - (a * z));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.2e+81:
		tmp = y / a
	elif z <= 5.5e-286:
		tmp = (x - (z * y)) / t
	elif z <= 2.35e+73:
		tmp = x / (t - (a * z))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.2e+81)
		tmp = Float64(y / a);
	elseif (z <= 5.5e-286)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 2.35e+73)
		tmp = Float64(x / Float64(t - Float64(a * z)));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.2e+81)
		tmp = y / a;
	elseif (z <= 5.5e-286)
		tmp = (x - (z * y)) / t;
	elseif (z <= 2.35e+73)
		tmp = x / (t - (a * z));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.2e+81], N[(y / a), $MachinePrecision], If[LessEqual[z, 5.5e-286], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 2.35e+73], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.19999999999999995e81 or 2.3500000000000001e73 < z

   1. Initial program 62.4%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{y}{a}} \]
   4. Step-by-step derivation

      1. lower-/.f64 61.8

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

   5. Applied rewrites 61.8%

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

   if -1.19999999999999995e81 < z < 5.4999999999999998e-286

   1. Initial program 98.5%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 73.7

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

   5. Applied rewrites 73.7%

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

   if 5.4999999999999998e-286 < z < 2.3500000000000001e73

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 70.9

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

   5. Applied rewrites 70.9%

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

Alternative 5: 66.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+51} \lor \neg \left(x \leq 2.7 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(a, z, -t\right)} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= x -4e+51) (not (<= x 2.7e+15)))
   (/ x (- t (* a z)))
   (* (/ z (fma a z (- t))) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x <= -4e+51) || !(x <= 2.7e+15)) {
		tmp = x / (t - (a * z));
	} else {
		tmp = (z / fma(a, z, -t)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((x <= -4e+51) || !(x <= 2.7e+15))
		tmp = Float64(x / Float64(t - Float64(a * z)));
	else
		tmp = Float64(Float64(z / fma(a, z, Float64(-t))) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[x, -4e+51], N[Not[LessEqual[x, 2.7e+15]], $MachinePrecision]], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(a * z + (-t)), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4e51 or 2.7e15 < x

   1. Initial program 84.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   if -4e51 < x < 2.7e15

   1. Initial program 84.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.4%

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

   10. Applied rewrites 72.4%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+51} \lor \neg \left(x \leq 2.7 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(a, z, -t\right)} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+51} \lor \neg \left(x \leq 2.7 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{a \cdot z - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= x -4e+51) (not (<= x 2.7e+15)))
   (/ x (- t (* a z)))
   (* y (/ z (- (* a z) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x <= -4e+51) || !(x <= 2.7e+15)) {
		tmp = x / (t - (a * z));
	} else {
		tmp = y * (z / ((a * z) - 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 ((x <= (-4d+51)) .or. (.not. (x <= 2.7d+15))) then
        tmp = x / (t - (a * z))
    else
        tmp = y * (z / ((a * z) - 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 ((x <= -4e+51) || !(x <= 2.7e+15)) {
		tmp = x / (t - (a * z));
	} else {
		tmp = y * (z / ((a * z) - t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (x <= -4e+51) or not (x <= 2.7e+15):
		tmp = x / (t - (a * z))
	else:
		tmp = y * (z / ((a * z) - t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((x <= -4e+51) || !(x <= 2.7e+15))
		tmp = Float64(x / Float64(t - Float64(a * z)));
	else
		tmp = Float64(y * Float64(z / Float64(Float64(a * z) - t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x <= -4e+51) || ~((x <= 2.7e+15)))
		tmp = x / (t - (a * z));
	else
		tmp = y * (z / ((a * z) - t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[x, -4e+51], N[Not[LessEqual[x, 2.7e+15]], $MachinePrecision]], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(z / N[(N[(a * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4e51 or 2.7e15 < x

   1. Initial program 84.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   if -4e51 < x < 2.7e15

   1. Initial program 84.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.4%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+51} \lor \neg \left(x \leq 2.7 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{a \cdot z - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+105} \lor \neg \left(z \leq 2.35 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -4.2e+105) (not (<= z 2.35e+73))) (/ y a) (/ x (- t (* a z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -4.2e+105) || !(z <= 2.35e+73)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (a * 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) :: tmp
    if ((z <= (-4.2d+105)) .or. (.not. (z <= 2.35d+73))) then
        tmp = y / a
    else
        tmp = x / (t - (a * z))
    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 <= -4.2e+105) || !(z <= 2.35e+73)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (a * z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -4.2e+105) or not (z <= 2.35e+73):
		tmp = y / a
	else:
		tmp = x / (t - (a * z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -4.2e+105) || !(z <= 2.35e+73))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / Float64(t - Float64(a * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -4.2e+105) || ~((z <= 2.35e+73)))
		tmp = y / a;
	else
		tmp = x / (t - (a * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -4.2e+105], N[Not[LessEqual[z, 2.35e+73]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.2000000000000002e105 or 2.3500000000000001e73 < z

   1. Initial program 62.2%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{y}{a}} \]
   4. Step-by-step derivation

      1. lower-/.f64 63.4

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

   5. Applied rewrites 63.4%

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

   if -4.2000000000000002e105 < z < 2.3500000000000001e73

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 68.3

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

   5. Applied rewrites 68.3%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+105} \lor \neg \left(z \leq 2.35 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-25} \lor \neg \left(z \leq 2.3 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.4e-25) (not (<= z 2.3e-21))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.4e-25) || !(z <= 2.3e-21)) {
		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 <= (-1.4d-25)) .or. (.not. (z <= 2.3d-21))) 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 <= -1.4e-25) || !(z <= 2.3e-21)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.4e-25) or not (z <= 2.3e-21):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.4e-25) || !(z <= 2.3e-21))
		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 <= -1.4e-25) || ~((z <= 2.3e-21)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.4e-25], N[Not[LessEqual[z, 2.3e-21]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.39999999999999994e-25 or 2.29999999999999999e-21 < z

   1. Initial program 73.5%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{y}{a}} \]
   4. Step-by-step derivation

      1. lower-/.f64 54.1

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

   5. Applied rewrites 54.1%

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

   if -1.39999999999999994e-25 < z < 2.29999999999999999e-21

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 60.7

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

   5. Applied rewrites 60.7%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-25} \lor \neg \left(z \leq 2.3 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.0% accurate, 2.3× 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 84.8%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 35.7

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

5. Applied rewrites 35.7%

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

Developer Target 1: 97.4% accurate, 0.5× 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 2024318 
(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))))