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

Percentage Accurate: 84.9% → 91.6%

Time: 10.6s

Alternatives: 8

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: 91.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -5 \cdot 10^{+125}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -5e+125)
     t_1
     (if (<= z 1.1e+154) (/ (- x (* z y)) (- t (* z a))) 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 <= -5e+125) {
		tmp = t_1;
	} else if (z <= 1.1e+154) {
		tmp = (x - (z * y)) / (t - (z * a));
	} 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 <= (-5d+125)) then
        tmp = t_1
    else if (z <= 1.1d+154) then
        tmp = (x - (z * y)) / (t - (z * a))
    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 <= -5e+125) {
		tmp = t_1;
	} else if (z <= 1.1e+154) {
		tmp = (x - (z * y)) / (t - (z * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - (x / z)) / a
	tmp = 0
	if z <= -5e+125:
		tmp = t_1
	elif z <= 1.1e+154:
		tmp = (x - (z * y)) / (t - (z * a))
	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 <= -5e+125)
		tmp = t_1;
	elseif (z <= 1.1e+154)
		tmp = Float64(Float64(x - Float64(z * y)) / Float64(t - Float64(z * a)));
	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 <= -5e+125)
		tmp = t_1;
	elseif (z <= 1.1e+154)
		tmp = (x - (z * y)) / (t - (z * a));
	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, -5e+125], t$95$1, If[LessEqual[z, 1.1e+154], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.99999999999999962e125 or 1.1000000000000001e154 < z

   1. Initial program 51.3%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 62.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 88.3%

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

   if -4.99999999999999962e125 < z < 1.1000000000000001e154

   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 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+125}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+124}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-46}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot a - t}\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 68:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -1.05e+124)
     t_1
     (if (<= z -2.9e-46)
       (* y (/ z (- (* z a) t)))
       (if (<= z -3.2e-114)
         (/ x (- t (* z a)))
         (if (<= z 68.0) (/ (- 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 <= -1.05e+124) {
		tmp = t_1;
	} else if (z <= -2.9e-46) {
		tmp = y * (z / ((z * a) - t));
	} else if (z <= -3.2e-114) {
		tmp = x / (t - (z * a));
	} else if (z <= 68.0) {
		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 <= (-1.05d+124)) then
        tmp = t_1
    else if (z <= (-2.9d-46)) then
        tmp = y * (z / ((z * a) - t))
    else if (z <= (-3.2d-114)) then
        tmp = x / (t - (z * a))
    else if (z <= 68.0d0) 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 <= -1.05e+124) {
		tmp = t_1;
	} else if (z <= -2.9e-46) {
		tmp = y * (z / ((z * a) - t));
	} else if (z <= -3.2e-114) {
		tmp = x / (t - (z * a));
	} else if (z <= 68.0) {
		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 <= -1.05e+124:
		tmp = t_1
	elif z <= -2.9e-46:
		tmp = y * (z / ((z * a) - t))
	elif z <= -3.2e-114:
		tmp = x / (t - (z * a))
	elif z <= 68.0:
		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 <= -1.05e+124)
		tmp = t_1;
	elseif (z <= -2.9e-46)
		tmp = Float64(y * Float64(z / Float64(Float64(z * a) - t)));
	elseif (z <= -3.2e-114)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 68.0)
		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 <= -1.05e+124)
		tmp = t_1;
	elseif (z <= -2.9e-46)
		tmp = y * (z / ((z * a) - t));
	elseif (z <= -3.2e-114)
		tmp = x / (t - (z * a));
	elseif (z <= 68.0)
		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, -1.05e+124], t$95$1, If[LessEqual[z, -2.9e-46], N[(y * N[(z / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -3.2e-114], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 68.0], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.05000000000000006e124 or 68 < z

   1. Initial program 64.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 81.5%

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

   if -1.05000000000000006e124 < z < -2.90000000000000005e-46

   1. Initial program 91.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft-neg-out N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 73.4

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 79.2%

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

   if -2.90000000000000005e-46 < z < -3.2000000000000002e-114

   1. Initial program 99.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. *-commutative N/A

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

      4. lower-*.f64 90.6

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

   5. Applied rewrites 90.6%

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

   if -3.2000000000000002e-114 < z < 68

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+124}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-46}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot a - t}\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 68:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+130}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-46}:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+126}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.4e+130)
   (/ y a)
   (if (<= z -2.9e-46)
     (* z (/ y (- (* z a) t)))
     (if (<= z 4e+126) (/ x (- t (* z a))) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.4e+130) {
		tmp = y / a;
	} else if (z <= -2.9e-46) {
		tmp = z * (y / ((z * a) - t));
	} else if (z <= 4e+126) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-5.4d+130)) then
        tmp = y / a
    else if (z <= (-2.9d-46)) then
        tmp = z * (y / ((z * a) - t))
    else if (z <= 4d+126) then
        tmp = x / (t - (z * a))
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.4e+130) {
		tmp = y / a;
	} else if (z <= -2.9e-46) {
		tmp = z * (y / ((z * a) - t));
	} else if (z <= 4e+126) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.4e+130:
		tmp = y / a
	elif z <= -2.9e-46:
		tmp = z * (y / ((z * a) - t))
	elif z <= 4e+126:
		tmp = x / (t - (z * a))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.4e+130)
		tmp = Float64(y / a);
	elseif (z <= -2.9e-46)
		tmp = Float64(z * Float64(y / Float64(Float64(z * a) - t)));
	elseif (z <= 4e+126)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5.4e+130)
		tmp = y / a;
	elseif (z <= -2.9e-46)
		tmp = z * (y / ((z * a) - t));
	elseif (z <= 4e+126)
		tmp = x / (t - (z * a));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.4e+130], N[(y / a), $MachinePrecision], If[LessEqual[z, -2.9e-46], N[(z * N[(y / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e+126], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.3999999999999997e130 or 3.9999999999999997e126 < z

   1. Initial program 52.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 70.0

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

   5. Applied rewrites 70.0%

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

   if -5.3999999999999997e130 < z < -2.90000000000000005e-46

   1. Initial program 91.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft-neg-out N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 73.4

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 76.4%

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

   if -2.90000000000000005e-46 < z < 3.9999999999999997e126

   1. Initial program 99.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. *-commutative N/A

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

      4. lower-*.f64 72.7

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

   5. Applied rewrites 72.7%

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

Alternative 4: 66.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ \mathbf{if}\;x \leq -1.04 \cdot 10^{-100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{z \cdot y}{\mathsf{fma}\left(z, a, -t\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (- t (* z a)))))
   (if (<= x -1.04e-100)
     t_1
     (if (<= x 9.5e-44) (/ (* z y) (fma z a (- t))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (z * a));
	double tmp;
	if (x <= -1.04e-100) {
		tmp = t_1;
	} else if (x <= 9.5e-44) {
		tmp = (z * y) / fma(z, a, -t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (x <= -1.04e-100)
		tmp = t_1;
	elseif (x <= 9.5e-44)
		tmp = Float64(Float64(z * y) / fma(z, a, Float64(-t)));
	else
		tmp = t_1;
	end
	return 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[x, -1.04e-100], t$95$1, If[LessEqual[x, 9.5e-44], N[(N[(z * y), $MachinePrecision] / N[(z * a + (-t)), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.04e-100 or 9.49999999999999924e-44 < x

   1. Initial program 82.1%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. *-commutative N/A

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

      4. lower-*.f64 68.3

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

   5. Applied rewrites 68.3%

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

   if -1.04e-100 < x < 9.49999999999999924e-44

   1. Initial program 87.2%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft-neg-out N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 78.5

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

   5. Applied rewrites 78.5%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.04 \cdot 10^{-100}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{z \cdot y}{\mathsf{fma}\left(z, a, -t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-101}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-43}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (- t (* z a)))))
   (if (<= x -6.2e-101)
     t_1
     (if (<= x 1.4e-43) (* y (/ z (- (* z a) t))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (z * a));
	double tmp;
	if (x <= -6.2e-101) {
		tmp = t_1;
	} else if (x <= 1.4e-43) {
		tmp = y * (z / ((z * a) - 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 = x / (t - (z * a))
    if (x <= (-6.2d-101)) then
        tmp = t_1
    else if (x <= 1.4d-43) then
        tmp = y * (z / ((z * a) - 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 = x / (t - (z * a));
	double tmp;
	if (x <= -6.2e-101) {
		tmp = t_1;
	} else if (x <= 1.4e-43) {
		tmp = y * (z / ((z * a) - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x / (t - (z * a))
	tmp = 0
	if x <= -6.2e-101:
		tmp = t_1
	elif x <= 1.4e-43:
		tmp = y * (z / ((z * a) - t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (x <= -6.2e-101)
		tmp = t_1;
	elseif (x <= 1.4e-43)
		tmp = Float64(y * Float64(z / Float64(Float64(z * a) - t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x / (t - (z * a));
	tmp = 0.0;
	if (x <= -6.2e-101)
		tmp = t_1;
	elseif (x <= 1.4e-43)
		tmp = y * (z / ((z * a) - t));
	else
		tmp = t_1;
	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[x, -6.2e-101], t$95$1, If[LessEqual[x, 1.4e-43], N[(y * N[(z / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.19999999999999946e-101 or 1.3999999999999999e-43 < x

   1. Initial program 82.1%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. *-commutative N/A

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

      4. lower-*.f64 68.3

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

   5. Applied rewrites 68.3%

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

   if -6.19999999999999946e-101 < x < 1.3999999999999999e-43

   1. Initial program 87.2%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft-neg-out N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 78.5

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

   5. Applied rewrites 78.5%

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

   7. Applied rewrites 76.5%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-43}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot a - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+126}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.1e-44) (/ y a) (if (<= z 4e+126) (/ x (- t (* z a))) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.1e-44) {
		tmp = y / a;
	} else if (z <= 4e+126) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.1d-44)) then
        tmp = y / a
    else if (z <= 4d+126) then
        tmp = x / (t - (z * a))
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.1e-44) {
		tmp = y / a;
	} else if (z <= 4e+126) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.1e-44:
		tmp = y / a
	elif z <= 4e+126:
		tmp = x / (t - (z * a))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.1e-44)
		tmp = Float64(y / a);
	elseif (z <= 4e+126)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.1e-44)
		tmp = y / a;
	elseif (z <= 4e+126)
		tmp = x / (t - (z * a));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.1e-44], N[(y / a), $MachinePrecision], If[LessEqual[z, 4e+126], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.10000000000000006e-44 or 3.9999999999999997e126 < z

   1. Initial program 64.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 64.2

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

   5. Applied rewrites 64.2%

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

   if -1.10000000000000006e-44 < z < 3.9999999999999997e126

   1. Initial program 99.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. *-commutative N/A

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

      4. lower-*.f64 72.7

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

   5. Applied rewrites 72.7%

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

Alternative 7: 56.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-59}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 0.086:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9.5e-59) (/ y a) (if (<= z 0.086) (/ x t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.5e-59) {
		tmp = y / a;
	} else if (z <= 0.086) {
		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 <= (-9.5d-59)) then
        tmp = y / a
    else if (z <= 0.086d0) 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 <= -9.5e-59) {
		tmp = y / a;
	} else if (z <= 0.086) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -9.5e-59:
		tmp = y / a
	elif z <= 0.086:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9.5e-59)
		tmp = Float64(y / a);
	elseif (z <= 0.086)
		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 <= -9.5e-59)
		tmp = y / a;
	elseif (z <= 0.086)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9.5e-59], N[(y / a), $MachinePrecision], If[LessEqual[z, 0.086], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.4999999999999994e-59 or 0.085999999999999993 < z

   1. Initial program 71.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 57.5

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

   5. Applied rewrites 57.5%

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

   if -9.4999999999999994e-59 < z < 0.085999999999999993

   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 57.4

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

   5. Applied rewrites 57.4%

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

Alternative 8: 35.6% 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 84.1%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 35.3

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

5. Applied rewrites 35.3%

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

Developer Target 1: 97.1% 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 2024221 
(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))))