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

Percentage Accurate: 85.1% → 90.8%

Time: 11.1s

Alternatives: 8

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+111}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -3.6e+84)
     t_1
     (if (<= z 3.3e+111) (/ (- 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 <= -3.6e+84) {
		tmp = t_1;
	} else if (z <= 3.3e+111) {
		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 <= (-3.6d+84)) then
        tmp = t_1
    else if (z <= 3.3d+111) 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 <= -3.6e+84) {
		tmp = t_1;
	} else if (z <= 3.3e+111) {
		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 <= -3.6e+84:
		tmp = t_1
	elif z <= 3.3e+111:
		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 <= -3.6e+84)
		tmp = t_1;
	elseif (z <= 3.3e+111)
		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 <= -3.6e+84)
		tmp = t_1;
	elseif (z <= 3.3e+111)
		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, -3.6e+84], t$95$1, If[LessEqual[z, 3.3e+111], 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 < -3.5999999999999999e84 or 3.3000000000000001e111 < z

   1. Initial program 48.7%

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

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x - y \cdot z}{z}\right), \color{blue}{a}\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(\frac{x}{z} - \frac{y \cdot z}{z}\right)\right), a\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(\frac{x}{z} + \left(\mathsf{neg}\left(\frac{y \cdot z}{z}\right)\right)\right)\right), a\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(\frac{y \cdot z}{z}\right)\right)\right), a\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(y \cdot \frac{z}{z}\right)\right)\right), a\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(y \cdot 1\right)\right)\right), a\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(y\right)\right)\right), a\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right)\right), a\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(y\right)\right)\right), a\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), a\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + y\right), a\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y + -1 \cdot \frac{x}{z}\right), a\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y + \left(\mathsf{neg}\left(\frac{x}{z}\right)\right)\right), a\right) \]

      18. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - \frac{x}{z}\right), a\right) \]

      19. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{x}{z}\right)\right), a\right) \]

      20. /-lowering-/.f64 84.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(x, z\right)\right), a\right) \]

   5. Simplified 84.9%

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

   if -3.5999999999999999e84 < z < 3.3000000000000001e111

   1. Initial program 97.6%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+84}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+111}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-62}:\\ \;\;\;\;\frac{y}{\frac{z \cdot a - t}{z}}\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+20}:\\ \;\;\;\;\frac{z \cdot \left(\frac{x}{z} - y\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -3.4e+18)
     t_1
     (if (<= z -6e-62)
       (/ y (/ (- (* z a) t) z))
       (if (<= z 8.8e-51)
         (/ x (- t (* z a)))
         (if (<= z 1.25e+20) (/ (* z (- (/ x z) y)) t) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -3.4e+18) {
		tmp = t_1;
	} else if (z <= -6e-62) {
		tmp = y / (((z * a) - t) / z);
	} else if (z <= 8.8e-51) {
		tmp = x / (t - (z * a));
	} else if (z <= 1.25e+20) {
		tmp = (z * ((x / z) - y)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y - (x / z)) / a
    if (z <= (-3.4d+18)) then
        tmp = t_1
    else if (z <= (-6d-62)) then
        tmp = y / (((z * a) - t) / z)
    else if (z <= 8.8d-51) then
        tmp = x / (t - (z * a))
    else if (z <= 1.25d+20) then
        tmp = (z * ((x / z) - y)) / t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -3.4e+18) {
		tmp = t_1;
	} else if (z <= -6e-62) {
		tmp = y / (((z * a) - t) / z);
	} else if (z <= 8.8e-51) {
		tmp = x / (t - (z * a));
	} else if (z <= 1.25e+20) {
		tmp = (z * ((x / z) - y)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - (x / z)) / a
	tmp = 0
	if z <= -3.4e+18:
		tmp = t_1
	elif z <= -6e-62:
		tmp = y / (((z * a) - t) / z)
	elif z <= 8.8e-51:
		tmp = x / (t - (z * a))
	elif z <= 1.25e+20:
		tmp = (z * ((x / z) - y)) / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - Float64(x / z)) / a)
	tmp = 0.0
	if (z <= -3.4e+18)
		tmp = t_1;
	elseif (z <= -6e-62)
		tmp = Float64(y / Float64(Float64(Float64(z * a) - t) / z));
	elseif (z <= 8.8e-51)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 1.25e+20)
		tmp = Float64(Float64(z * Float64(Float64(x / z) - y)) / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - (x / z)) / a;
	tmp = 0.0;
	if (z <= -3.4e+18)
		tmp = t_1;
	elseif (z <= -6e-62)
		tmp = y / (((z * a) - t) / z);
	elseif (z <= 8.8e-51)
		tmp = x / (t - (z * a));
	elseif (z <= 1.25e+20)
		tmp = (z * ((x / z) - y)) / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -3.4e+18], t$95$1, If[LessEqual[z, -6e-62], N[(y / N[(N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.8e-51], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.25e+20], N[(N[(z * N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.4e18 or 1.25e20 < z

   1. Initial program 58.6%

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

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x - y \cdot z}{z}\right), \color{blue}{a}\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(\frac{x}{z} - \frac{y \cdot z}{z}\right)\right), a\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(\frac{x}{z} + \left(\mathsf{neg}\left(\frac{y \cdot z}{z}\right)\right)\right)\right), a\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(\frac{y \cdot z}{z}\right)\right)\right), a\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(y \cdot \frac{z}{z}\right)\right)\right), a\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(y \cdot 1\right)\right)\right), a\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(y\right)\right)\right), a\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right)\right), a\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(y\right)\right)\right), a\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), a\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + y\right), a\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y + -1 \cdot \frac{x}{z}\right), a\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y + \left(\mathsf{neg}\left(\frac{x}{z}\right)\right)\right), a\right) \]

      18. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - \frac{x}{z}\right), a\right) \]

      19. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{x}{z}\right)\right), a\right) \]

      20. /-lowering-/.f64 82.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(x, z\right)\right), a\right) \]

   5. Simplified 82.1%

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

   if -3.4e18 < z < -6.0000000000000002e-62

   1. Initial program 99.3%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{t - a \cdot z}{x - y \cdot z}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(t - a \cdot z\right), \color{blue}{\left(x - y \cdot z\right)}\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(t, \left(a \cdot z\right)\right), \left(\color{blue}{x} - y \cdot z\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(t, \left(z \cdot a\right)\right), \left(x - y \cdot z\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, a\right)\right), \left(x - y \cdot z\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, a\right)\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(y \cdot z\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 99.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, a\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right)\right)\right) \]

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{y \cdot z}{t - a \cdot z}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{y \cdot z}{t - a \cdot z}\right)}\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(y \cdot \color{blue}{\frac{z}{t - a \cdot z}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{z}{t - a \cdot z}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(z, \color{blue}{\left(t - a \cdot z\right)}\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(z, \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(z, \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 67.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(z, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right)\right)\right) \]

   7. Simplified 67.7%

      \[\leadsto \color{blue}{0 - y \cdot \frac{z}{t - z \cdot a}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{t - z \cdot a}{z}\right)\right)}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(y, \left(\frac{\mathsf{neg}\left(\left(t - z \cdot a\right)\right)}{\color{blue}{z}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(t - z \cdot a\right)\right)\right), \color{blue}{z}\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(0 - \left(t - z \cdot a\right)\right), z\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(0 - \left(t + \left(\mathsf{neg}\left(z \cdot a\right)\right)\right)\right), z\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(0 - \left(\left(\mathsf{neg}\left(z \cdot a\right)\right) + t\right)\right), z\right)\right) \]

      11. associate--r+ N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(\left(0 - \left(\mathsf{neg}\left(z \cdot a\right)\right)\right) - t\right), z\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot a\right)\right)\right)\right) - t\right), z\right)\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot a - t\right), z\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot a\right), t\right), z\right)\right) \]

      15. *-lowering-*.f64 67.9%

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, a\right), t\right), z\right)\right) \]

   9. Applied egg-rr 67.9%

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

   if -6.0000000000000002e-62 < z < 8.8000000000000001e-51

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      4. *-lowering-*.f64 82.1%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   5. Simplified 82.1%

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

   if 8.8000000000000001e-51 < z < 1.25e20

   1. Initial program 99.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{t}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), t\right) \]

      3. *-lowering-*.f64 75.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), t\right) \]

   5. Simplified 75.4%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(z \cdot \left(\frac{x}{z} - y\right)\right)}, t\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(\frac{x}{z} - y\right)\right), t\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(\left(\frac{x}{z}\right), y\right)\right), t\right) \]

      3. /-lowering-/.f64 75.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x, z\right), y\right)\right), t\right) \]

   8. Simplified 75.4%

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

Alternative 3: 71.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -33000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{y}{\frac{z \cdot a - t}{z}}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -33000000.0)
     t_1
     (if (<= z -7.2e-62)
       (/ y (/ (- (* z a) t) z))
       (if (<= z 9e-53)
         (/ x (- t (* z a)))
         (if (<= z 9.8e+16) (/ (- 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 <= -33000000.0) {
		tmp = t_1;
	} else if (z <= -7.2e-62) {
		tmp = y / (((z * a) - t) / z);
	} else if (z <= 9e-53) {
		tmp = x / (t - (z * a));
	} else if (z <= 9.8e+16) {
		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 <= (-33000000.0d0)) then
        tmp = t_1
    else if (z <= (-7.2d-62)) then
        tmp = y / (((z * a) - t) / z)
    else if (z <= 9d-53) then
        tmp = x / (t - (z * a))
    else if (z <= 9.8d+16) 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 <= -33000000.0) {
		tmp = t_1;
	} else if (z <= -7.2e-62) {
		tmp = y / (((z * a) - t) / z);
	} else if (z <= 9e-53) {
		tmp = x / (t - (z * a));
	} else if (z <= 9.8e+16) {
		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 <= -33000000.0:
		tmp = t_1
	elif z <= -7.2e-62:
		tmp = y / (((z * a) - t) / z)
	elif z <= 9e-53:
		tmp = x / (t - (z * a))
	elif z <= 9.8e+16:
		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 <= -33000000.0)
		tmp = t_1;
	elseif (z <= -7.2e-62)
		tmp = Float64(y / Float64(Float64(Float64(z * a) - t) / z));
	elseif (z <= 9e-53)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 9.8e+16)
		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 <= -33000000.0)
		tmp = t_1;
	elseif (z <= -7.2e-62)
		tmp = y / (((z * a) - t) / z);
	elseif (z <= 9e-53)
		tmp = x / (t - (z * a));
	elseif (z <= 9.8e+16)
		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, -33000000.0], t$95$1, If[LessEqual[z, -7.2e-62], N[(y / N[(N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9e-53], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.8e+16], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.3e7 or 9.8e16 < z

   1. Initial program 58.6%

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

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x - y \cdot z}{z}\right), \color{blue}{a}\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(\frac{x}{z} - \frac{y \cdot z}{z}\right)\right), a\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(\frac{x}{z} + \left(\mathsf{neg}\left(\frac{y \cdot z}{z}\right)\right)\right)\right), a\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(\frac{y \cdot z}{z}\right)\right)\right), a\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(y \cdot \frac{z}{z}\right)\right)\right), a\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(y \cdot 1\right)\right)\right), a\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(y\right)\right)\right), a\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right)\right), a\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(y\right)\right)\right), a\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), a\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + y\right), a\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y + -1 \cdot \frac{x}{z}\right), a\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y + \left(\mathsf{neg}\left(\frac{x}{z}\right)\right)\right), a\right) \]

      18. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - \frac{x}{z}\right), a\right) \]

      19. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{x}{z}\right)\right), a\right) \]

      20. /-lowering-/.f64 82.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(x, z\right)\right), a\right) \]

   5. Simplified 82.1%

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

   if -3.3e7 < z < -7.1999999999999999e-62

   1. Initial program 99.3%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{t - a \cdot z}{x - y \cdot z}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(t - a \cdot z\right), \color{blue}{\left(x - y \cdot z\right)}\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(t, \left(a \cdot z\right)\right), \left(\color{blue}{x} - y \cdot z\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(t, \left(z \cdot a\right)\right), \left(x - y \cdot z\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, a\right)\right), \left(x - y \cdot z\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, a\right)\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(y \cdot z\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 99.2%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, a\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right)\right)\right) \]

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{y \cdot z}{t - a \cdot z}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{y \cdot z}{t - a \cdot z}\right)}\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(y \cdot \color{blue}{\frac{z}{t - a \cdot z}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{z}{t - a \cdot z}\right)}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(z, \color{blue}{\left(t - a \cdot z\right)}\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(z, \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(z, \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 67.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(z, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right)\right)\right) \]

   7. Simplified 67.7%

      \[\leadsto \color{blue}{0 - y \cdot \frac{z}{t - z \cdot a}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. distribute-neg-frac2 N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{t - z \cdot a}{z}\right)\right)}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(y, \left(\frac{\mathsf{neg}\left(\left(t - z \cdot a\right)\right)}{\color{blue}{z}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(t - z \cdot a\right)\right)\right), \color{blue}{z}\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(0 - \left(t - z \cdot a\right)\right), z\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(0 - \left(t + \left(\mathsf{neg}\left(z \cdot a\right)\right)\right)\right), z\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(0 - \left(\left(\mathsf{neg}\left(z \cdot a\right)\right) + t\right)\right), z\right)\right) \]

      11. associate--r+ N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(\left(0 - \left(\mathsf{neg}\left(z \cdot a\right)\right)\right) - t\right), z\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot a\right)\right)\right)\right) - t\right), z\right)\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(z \cdot a - t\right), z\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot a\right), t\right), z\right)\right) \]

      15. *-lowering-*.f64 67.9%

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, a\right), t\right), z\right)\right) \]

   9. Applied egg-rr 67.9%

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

   if -7.1999999999999999e-62 < z < 8.9999999999999997e-53

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      4. *-lowering-*.f64 82.1%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   5. Simplified 82.1%

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

   if 8.9999999999999997e-53 < z < 9.8e16

   1. Initial program 99.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{t}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), t\right) \]

      3. *-lowering-*.f64 75.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), t\right) \]

   5. Simplified 75.4%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -33000000:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq -7.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{y}{\frac{z \cdot a - t}{z}}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{-62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+17}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -6.8e-62)
     t_1
     (if (<= z 8e-52)
       (/ x (- t (* z a)))
       (if (<= z 5.6e+17) (/ (- 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 <= -6.8e-62) {
		tmp = t_1;
	} else if (z <= 8e-52) {
		tmp = x / (t - (z * a));
	} else if (z <= 5.6e+17) {
		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 <= (-6.8d-62)) then
        tmp = t_1
    else if (z <= 8d-52) then
        tmp = x / (t - (z * a))
    else if (z <= 5.6d+17) 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 <= -6.8e-62) {
		tmp = t_1;
	} else if (z <= 8e-52) {
		tmp = x / (t - (z * a));
	} else if (z <= 5.6e+17) {
		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 <= -6.8e-62:
		tmp = t_1
	elif z <= 8e-52:
		tmp = x / (t - (z * a))
	elif z <= 5.6e+17:
		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 <= -6.8e-62)
		tmp = t_1;
	elseif (z <= 8e-52)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 5.6e+17)
		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 <= -6.8e-62)
		tmp = t_1;
	elseif (z <= 8e-52)
		tmp = x / (t - (z * a));
	elseif (z <= 5.6e+17)
		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, -6.8e-62], t$95$1, If[LessEqual[z, 8e-52], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.6e+17], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.79999999999999975e-62 or 5.6e17 < z

   1. Initial program 64.2%

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

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x - y \cdot z}{z}\right), \color{blue}{a}\right) \]

      6. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(\frac{x}{z} - \frac{y \cdot z}{z}\right)\right), a\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \left(\frac{x}{z} + \left(\mathsf{neg}\left(\frac{y \cdot z}{z}\right)\right)\right)\right), a\right) \]

      8. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(\frac{y \cdot z}{z}\right)\right)\right), a\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(y \cdot \frac{z}{z}\right)\right)\right), a\right) \]

      10. *-inverses N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(y \cdot 1\right)\right)\right), a\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(y\right)\right)\right), a\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right)\right), a\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + -1 \cdot \left(\mathsf{neg}\left(y\right)\right)\right), a\right) \]

      14. neg-mul-1 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), a\right) \]

      15. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{x}{z} + y\right), a\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y + -1 \cdot \frac{x}{z}\right), a\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y + \left(\mathsf{neg}\left(\frac{x}{z}\right)\right)\right), a\right) \]

      18. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(y - \frac{x}{z}\right), a\right) \]

      19. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\frac{x}{z}\right)\right), a\right) \]

      20. /-lowering-/.f64 76.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{/.f64}\left(x, z\right)\right), a\right) \]

   5. Simplified 76.7%

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

   if -6.79999999999999975e-62 < z < 8.0000000000000001e-52

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      4. *-lowering-*.f64 82.1%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   5. Simplified 82.1%

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

   if 8.0000000000000001e-52 < z < 5.6e17

   1. Initial program 99.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x - y \cdot z\right), \color{blue}{t}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \left(y \cdot z\right)\right), t\right) \]

      3. *-lowering-*.f64 75.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), t\right) \]

   5. Simplified 75.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+17}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{+60}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.1e+60) (/ y a) (if (<= z 8.5e+33) (/ x (- t (* z a))) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.1e+60) {
		tmp = y / a;
	} else if (z <= 8.5e+33) {
		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 <= (-5.1d+60)) then
        tmp = y / a
    else if (z <= 8.5d+33) 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 <= -5.1e+60) {
		tmp = y / a;
	} else if (z <= 8.5e+33) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.1e+60:
		tmp = y / a
	elif z <= 8.5e+33:
		tmp = x / (t - (z * a))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.1e+60)
		tmp = Float64(y / a);
	elseif (z <= 8.5e+33)
		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 <= -5.1e+60)
		tmp = y / a;
	elseif (z <= 8.5e+33)
		tmp = x / (t - (z * a));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.1e+60], N[(y / a), $MachinePrecision], If[LessEqual[z, 8.5e+33], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.09999999999999996e60 or 8.4999999999999998e33 < z

   1. Initial program 55.4%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 65.8%

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{a}\right) \]

   5. Simplified 65.8%

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

   if -5.09999999999999996e60 < z < 8.4999999999999998e33

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(t - a \cdot z\right)}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \color{blue}{\left(a \cdot z\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \left(z \cdot \color{blue}{a}\right)\right)\right) \]

      4. *-lowering-*.f64 72.9%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(z, \color{blue}{a}\right)\right)\right) \]

   5. Simplified 72.9%

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

Alternative 6: 54.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-62}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 10^{+20}:\\ \;\;\;\;\frac{1}{\frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7e-62) (/ y a) (if (<= z 1e+20) (/ 1.0 (/ t x)) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7e-62) {
		tmp = y / a;
	} else if (z <= 1e+20) {
		tmp = 1.0 / (t / x);
	} 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 <= (-7d-62)) then
        tmp = y / a
    else if (z <= 1d+20) then
        tmp = 1.0d0 / (t / x)
    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 <= -7e-62) {
		tmp = y / a;
	} else if (z <= 1e+20) {
		tmp = 1.0 / (t / x);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -7e-62:
		tmp = y / a
	elif z <= 1e+20:
		tmp = 1.0 / (t / x)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7e-62)
		tmp = Float64(y / a);
	elseif (z <= 1e+20)
		tmp = Float64(1.0 / Float64(t / x));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7e-62)
		tmp = y / a;
	elseif (z <= 1e+20)
		tmp = 1.0 / (t / x);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7e-62], N[(y / a), $MachinePrecision], If[LessEqual[z, 1e+20], N[(1.0 / N[(t / x), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.0000000000000003e-62 or 1e20 < z

   1. Initial program 64.2%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 59.4%

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{a}\right) \]

   5. Simplified 59.4%

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

   if -7.0000000000000003e-62 < z < 1e20

   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. /-lowering-/.f64 56.2%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{t}\right) \]

   5. Simplified 56.2%

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

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{t}{x}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{t}{x}\right)}\right) \]

      3. /-lowering-/.f64 56.4%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \color{blue}{x}\right)\right) \]

   7. Applied egg-rr 56.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{t}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 54.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{-62}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.4e-62) (/ y a) (if (<= z 2e+21) (/ x t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.4e-62) {
		tmp = y / a;
	} else if (z <= 2e+21) {
		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 <= (-6.4d-62)) then
        tmp = y / a
    else if (z <= 2d+21) 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 <= -6.4e-62) {
		tmp = y / a;
	} else if (z <= 2e+21) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -6.4e-62:
		tmp = y / a
	elif z <= 2e+21:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.4e-62)
		tmp = Float64(y / a);
	elseif (z <= 2e+21)
		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 <= -6.4e-62)
		tmp = y / a;
	elseif (z <= 2e+21)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.4e-62], N[(y / a), $MachinePrecision], If[LessEqual[z, 2e+21], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.40000000000000043e-62 or 2e21 < z

   1. Initial program 64.2%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 59.4%

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{a}\right) \]

   5. Simplified 59.4%

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

   if -6.40000000000000043e-62 < z < 2e21

   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. /-lowering-/.f64 56.2%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{t}\right) \]

   5. Simplified 56.2%

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

Alternative 8: 35.7% 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 81.7%

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

3. Taylor expanded in z around 0

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

   1. /-lowering-/.f64 34.5%

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{t}\right) \]

5. Simplified 34.5%

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

Developer Target 1: 97.5% 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 2024191 
(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))))