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

Percentage Accurate: 85.1% → 91.7%

Time: 16.7s

Alternatives: 11

Speedup: 0.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x - y \cdot z}{t\_1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-305}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{y}{a} + \frac{-1}{z \cdot \frac{a}{x}}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{x}{t\_1} - \frac{y \cdot z}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a))) (t_2 (/ (- x (* y z)) t_1)))
   (if (<= t_2 -5e-305)
     t_2
     (if (<= t_2 0.0)
       (+ (/ y a) (/ -1.0 (* z (/ a x))))
       (if (<= t_2 2e+292)
         (- (/ x t_1) (/ (* y z) t_1))
         (/ (- y (/ x z)) a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = (x - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -5e-305) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = (y / a) + (-1.0 / (z * (a / x)));
	} else if (t_2 <= 2e+292) {
		tmp = (x / t_1) - ((y * z) / t_1);
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t - (z * a)
    t_2 = (x - (y * z)) / t_1
    if (t_2 <= (-5d-305)) then
        tmp = t_2
    else if (t_2 <= 0.0d0) then
        tmp = (y / a) + ((-1.0d0) / (z * (a / x)))
    else if (t_2 <= 2d+292) then
        tmp = (x / t_1) - ((y * z) / t_1)
    else
        tmp = (y - (x / z)) / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = (x - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -5e-305) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = (y / a) + (-1.0 / (z * (a / x)));
	} else if (t_2 <= 2e+292) {
		tmp = (x / t_1) - ((y * z) / t_1);
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (z * a)
	t_2 = (x - (y * z)) / t_1
	tmp = 0
	if t_2 <= -5e-305:
		tmp = t_2
	elif t_2 <= 0.0:
		tmp = (y / a) + (-1.0 / (z * (a / x)))
	elif t_2 <= 2e+292:
		tmp = (x / t_1) - ((y * z) / t_1)
	else:
		tmp = (y - (x / z)) / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(z * a))
	t_2 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_2 <= -5e-305)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(y / a) + Float64(-1.0 / Float64(z * Float64(a / x))));
	elseif (t_2 <= 2e+292)
		tmp = Float64(Float64(x / t_1) - Float64(Float64(y * z) / t_1));
	else
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (z * a);
	t_2 = (x - (y * z)) / t_1;
	tmp = 0.0;
	if (t_2 <= -5e-305)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = (y / a) + (-1.0 / (z * (a / x)));
	elseif (t_2 <= 2e+292)
		tmp = (x / t_1) - ((y * z) / t_1);
	else
		tmp = (y - (x / z)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-305], t$95$2, If[LessEqual[t$95$2, 0.0], N[(N[(y / a), $MachinePrecision] + N[(-1.0 / N[(z * N[(a / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+292], N[(N[(x / t$95$1), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -4.99999999999999985e-305

   1. Initial program 96.7%

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

   if -4.99999999999999985e-305 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 52.6%

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

      1. *-commutative 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in z around inf 65.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{x}{a \cdot z} + \frac{y}{a}\right) - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}} \]
   6. Step-by-step derivation

      1. +-commutative 65.7%

         \[\leadsto \color{blue}{\left(\frac{y}{a} + -1 \cdot \frac{x}{a \cdot z}\right)} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z} \]

      2. associate--l+ 65.7%

         \[\leadsto \color{blue}{\frac{y}{a} + \left(-1 \cdot \frac{x}{a \cdot z} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}\right)} \]

      3. associate-/r* 81.0%

         \[\leadsto \frac{y}{a} + \left(-1 \cdot \color{blue}{\frac{\frac{x}{a}}{z}} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}\right) \]

      4. associate-*r/ 81.0%

         \[\leadsto \frac{y}{a} + \left(\color{blue}{\frac{-1 \cdot \frac{x}{a}}{z}} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}\right) \]

      5. associate-/r* 81.0%

         \[\leadsto \frac{y}{a} + \left(\frac{-1 \cdot \frac{x}{a}}{z} - -1 \cdot \color{blue}{\frac{\frac{t \cdot y}{{a}^{2}}}{z}}\right) \]

      6. associate-*r/ 81.0%

         \[\leadsto \frac{y}{a} + \left(\frac{-1 \cdot \frac{x}{a}}{z} - \color{blue}{\frac{-1 \cdot \frac{t \cdot y}{{a}^{2}}}{z}}\right) \]

      7. div-sub 81.0%

         \[\leadsto \frac{y}{a} + \color{blue}{\frac{-1 \cdot \frac{x}{a} - -1 \cdot \frac{t \cdot y}{{a}^{2}}}{z}} \]

      8. distribute-lft-out-- 81.0%

         \[\leadsto \frac{y}{a} + \frac{\color{blue}{-1 \cdot \left(\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}\right)}}{z} \]

      9. associate-*r/ 81.0%

         \[\leadsto \frac{y}{a} + \color{blue}{-1 \cdot \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}} \]

      10. mul-1-neg 81.0%

          \[\leadsto \frac{y}{a} + \color{blue}{\left(-\frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}\right)} \]

      11. unsub-neg 81.0%

          \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}} \]

   7. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{t}{{a}^{2}} \cdot y}{z}} \]

   8. Taylor expanded in x around inf 65.9%

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

      1. associate-/r* 81.2%

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

   10. Simplified 81.2%

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

       1. clear-num 81.2%

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

       2. inv-pow 81.2%

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

       3. div-inv 81.3%

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

       4. clear-num 81.3%

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

   12. Applied egg-rr 81.3%

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

       1. unpow-1 81.3%

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

   14. Simplified 81.3%

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

   if -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 2e292

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

   if 2e292 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 35.1%

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

      1. *-commutative 35.1%

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

   3. Simplified 35.1%

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

   5. Taylor expanded in x around 0 35.1%

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

   6. Taylor expanded in t around 0 81.6%

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

      1. +-commutative 81.6%

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

      2. mul-1-neg 81.6%

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

      3. sub-neg 81.6%

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

      4. *-commutative 81.6%

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

      5. associate-/r* 81.6%

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

      6. div-sub 81.6%

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

   8. Simplified 81.6%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -5 \cdot 10^{-305}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y}{a} + \frac{-1}{z \cdot \frac{a}{x}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-305}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{y}{a} + \frac{-1}{z \cdot \frac{a}{x}}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* y z)) (- t (* z a)))))
   (if (<= t_1 -5e-305)
     t_1
     (if (<= t_1 0.0)
       (+ (/ y a) (/ -1.0 (* z (/ a x))))
       (if (<= t_1 2e+292) t_1 (/ (- y (/ x z)) a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if (t_1 <= -5e-305) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (y / a) + (-1.0 / (z * (a / x)));
	} else if (t_1 <= 2e+292) {
		tmp = t_1;
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - (y * z)) / (t - (z * a))
    if (t_1 <= (-5d-305)) then
        tmp = t_1
    else if (t_1 <= 0.0d0) then
        tmp = (y / a) + ((-1.0d0) / (z * (a / x)))
    else if (t_1 <= 2d+292) then
        tmp = t_1
    else
        tmp = (y - (x / z)) / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if (t_1 <= -5e-305) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (y / a) + (-1.0 / (z * (a / x)));
	} else if (t_1 <= 2e+292) {
		tmp = t_1;
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - (y * z)) / (t - (z * a))
	tmp = 0
	if t_1 <= -5e-305:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = (y / a) + (-1.0 / (z * (a / x)))
	elif t_1 <= 2e+292:
		tmp = t_1
	else:
		tmp = (y - (x / z)) / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (t_1 <= -5e-305)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(y / a) + Float64(-1.0 / Float64(z * Float64(a / x))));
	elseif (t_1 <= 2e+292)
		tmp = t_1;
	else
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x - (y * z)) / (t - (z * a));
	tmp = 0.0;
	if (t_1 <= -5e-305)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = (y / a) + (-1.0 / (z * (a / x)));
	elseif (t_1 <= 2e+292)
		tmp = t_1;
	else
		tmp = (y - (x / z)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-305], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(y / a), $MachinePrecision] + N[(-1.0 / N[(z * N[(a / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+292], t$95$1, N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -4.99999999999999985e-305 or -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 2e292

   1. Initial program 97.9%

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

   if -4.99999999999999985e-305 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 52.6%

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

      1. *-commutative 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in z around inf 65.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{x}{a \cdot z} + \frac{y}{a}\right) - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}} \]
   6. Step-by-step derivation

      1. +-commutative 65.7%

         \[\leadsto \color{blue}{\left(\frac{y}{a} + -1 \cdot \frac{x}{a \cdot z}\right)} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z} \]

      2. associate--l+ 65.7%

         \[\leadsto \color{blue}{\frac{y}{a} + \left(-1 \cdot \frac{x}{a \cdot z} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}\right)} \]

      3. associate-/r* 81.0%

         \[\leadsto \frac{y}{a} + \left(-1 \cdot \color{blue}{\frac{\frac{x}{a}}{z}} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}\right) \]

      4. associate-*r/ 81.0%

         \[\leadsto \frac{y}{a} + \left(\color{blue}{\frac{-1 \cdot \frac{x}{a}}{z}} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}\right) \]

      5. associate-/r* 81.0%

         \[\leadsto \frac{y}{a} + \left(\frac{-1 \cdot \frac{x}{a}}{z} - -1 \cdot \color{blue}{\frac{\frac{t \cdot y}{{a}^{2}}}{z}}\right) \]

      6. associate-*r/ 81.0%

         \[\leadsto \frac{y}{a} + \left(\frac{-1 \cdot \frac{x}{a}}{z} - \color{blue}{\frac{-1 \cdot \frac{t \cdot y}{{a}^{2}}}{z}}\right) \]

      7. div-sub 81.0%

         \[\leadsto \frac{y}{a} + \color{blue}{\frac{-1 \cdot \frac{x}{a} - -1 \cdot \frac{t \cdot y}{{a}^{2}}}{z}} \]

      8. distribute-lft-out-- 81.0%

         \[\leadsto \frac{y}{a} + \frac{\color{blue}{-1 \cdot \left(\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}\right)}}{z} \]

      9. associate-*r/ 81.0%

         \[\leadsto \frac{y}{a} + \color{blue}{-1 \cdot \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}} \]

      10. mul-1-neg 81.0%

          \[\leadsto \frac{y}{a} + \color{blue}{\left(-\frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}\right)} \]

      11. unsub-neg 81.0%

          \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}} \]

   7. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{t}{{a}^{2}} \cdot y}{z}} \]

   8. Taylor expanded in x around inf 65.9%

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

      1. associate-/r* 81.2%

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

   10. Simplified 81.2%

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

       1. clear-num 81.2%

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

       2. inv-pow 81.2%

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

       3. div-inv 81.3%

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

       4. clear-num 81.3%

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

   12. Applied egg-rr 81.3%

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

       1. unpow-1 81.3%

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

   14. Simplified 81.3%

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

   if 2e292 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 35.1%

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

      1. *-commutative 35.1%

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

   3. Simplified 35.1%

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

   5. Taylor expanded in x around 0 35.1%

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

   6. Taylor expanded in t around 0 81.6%

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

      1. +-commutative 81.6%

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

      2. mul-1-neg 81.6%

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

      3. sub-neg 81.6%

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

      4. *-commutative 81.6%

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

      5. associate-/r* 81.6%

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

      6. div-sub 81.6%

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

   8. Simplified 81.6%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -5 \cdot 10^{-305}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y}{a} + \frac{-1}{z \cdot \frac{a}{x}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -1.65 \cdot 10^{+253}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{+190}:\\ \;\;\;\;\frac{y}{\frac{t - z \cdot a}{-z}}\\ \mathbf{elif}\;z \leq -2.65 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{t} - \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -1.65e+253)
     t_1
     (if (<= z -1.05e+190)
       (/ y (/ (- t (* z a)) (- z)))
       (if (<= z -2.65e+85)
         t_1
         (if (<= z -1.05e+55)
           (/ (- x (* y z)) t)
           (if (<= z -1.25e-71)
             (- (/ y a) (/ (/ x a) z))
             (if (<= z 2.2e+60) (- (/ x t) (/ (* y z) 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 <= -1.65e+253) {
		tmp = t_1;
	} else if (z <= -1.05e+190) {
		tmp = y / ((t - (z * a)) / -z);
	} else if (z <= -2.65e+85) {
		tmp = t_1;
	} else if (z <= -1.05e+55) {
		tmp = (x - (y * z)) / t;
	} else if (z <= -1.25e-71) {
		tmp = (y / a) - ((x / a) / z);
	} else if (z <= 2.2e+60) {
		tmp = (x / t) - ((y * z) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y - (x / z)) / a
    if (z <= (-1.65d+253)) then
        tmp = t_1
    else if (z <= (-1.05d+190)) then
        tmp = y / ((t - (z * a)) / -z)
    else if (z <= (-2.65d+85)) then
        tmp = t_1
    else if (z <= (-1.05d+55)) then
        tmp = (x - (y * z)) / t
    else if (z <= (-1.25d-71)) then
        tmp = (y / a) - ((x / a) / z)
    else if (z <= 2.2d+60) then
        tmp = (x / t) - ((y * z) / t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -1.65e+253) {
		tmp = t_1;
	} else if (z <= -1.05e+190) {
		tmp = y / ((t - (z * a)) / -z);
	} else if (z <= -2.65e+85) {
		tmp = t_1;
	} else if (z <= -1.05e+55) {
		tmp = (x - (y * z)) / t;
	} else if (z <= -1.25e-71) {
		tmp = (y / a) - ((x / a) / z);
	} else if (z <= 2.2e+60) {
		tmp = (x / t) - ((y * z) / 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 <= -1.65e+253:
		tmp = t_1
	elif z <= -1.05e+190:
		tmp = y / ((t - (z * a)) / -z)
	elif z <= -2.65e+85:
		tmp = t_1
	elif z <= -1.05e+55:
		tmp = (x - (y * z)) / t
	elif z <= -1.25e-71:
		tmp = (y / a) - ((x / a) / z)
	elif z <= 2.2e+60:
		tmp = (x / t) - ((y * z) / 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 <= -1.65e+253)
		tmp = t_1;
	elseif (z <= -1.05e+190)
		tmp = Float64(y / Float64(Float64(t - Float64(z * a)) / Float64(-z)));
	elseif (z <= -2.65e+85)
		tmp = t_1;
	elseif (z <= -1.05e+55)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	elseif (z <= -1.25e-71)
		tmp = Float64(Float64(y / a) - Float64(Float64(x / a) / z));
	elseif (z <= 2.2e+60)
		tmp = Float64(Float64(x / t) - Float64(Float64(y * z) / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - (x / z)) / a;
	tmp = 0.0;
	if (z <= -1.65e+253)
		tmp = t_1;
	elseif (z <= -1.05e+190)
		tmp = y / ((t - (z * a)) / -z);
	elseif (z <= -2.65e+85)
		tmp = t_1;
	elseif (z <= -1.05e+55)
		tmp = (x - (y * z)) / t;
	elseif (z <= -1.25e-71)
		tmp = (y / a) - ((x / a) / z);
	elseif (z <= 2.2e+60)
		tmp = (x / t) - ((y * z) / 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, -1.65e+253], t$95$1, If[LessEqual[z, -1.05e+190], N[(y / N[(N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -2.65e+85], t$95$1, If[LessEqual[z, -1.05e+55], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, -1.25e-71], N[(N[(y / a), $MachinePrecision] - N[(N[(x / a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.2e+60], N[(N[(x / t), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -1.65e253 or -1.05e190 < z < -2.65e85 or 2.19999999999999996e60 < z

   1. Initial program 66.1%

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

      1. *-commutative 66.1%

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

   3. Simplified 66.1%

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

   5. Taylor expanded in x around 0 66.1%

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

   6. Taylor expanded in t around 0 82.2%

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

      1. +-commutative 82.2%

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

      2. mul-1-neg 82.2%

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

      3. sub-neg 82.2%

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

      4. *-commutative 82.2%

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

      5. associate-/r* 86.2%

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

      6. div-sub 86.2%

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

   8. Simplified 86.2%

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

   if -1.65e253 < z < -1.05e190

   1. Initial program 74.0%

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

      1. *-commutative 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in x around 0 69.0%

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

      1. associate-*r/ 69.0%

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

      2. mul-1-neg 69.0%

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

      3. distribute-rgt-neg-out 69.0%

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

      4. associate-/l* 86.1%

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

      5. *-commutative 86.1%

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

   7. Simplified 86.1%

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

   if -2.65e85 < z < -1.05e55

   1. Initial program 86.3%

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

      1. *-commutative 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in t around inf 80.4%

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

   if -1.05e55 < z < -1.24999999999999999e-71

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in z around inf 52.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{x}{a \cdot z} + \frac{y}{a}\right) - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}} \]
   6. Step-by-step derivation

      1. +-commutative 52.4%

         \[\leadsto \color{blue}{\left(\frac{y}{a} + -1 \cdot \frac{x}{a \cdot z}\right)} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z} \]

      2. associate--l+ 52.4%

         \[\leadsto \color{blue}{\frac{y}{a} + \left(-1 \cdot \frac{x}{a \cdot z} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}\right)} \]

      3. associate-/r* 52.5%

         \[\leadsto \frac{y}{a} + \left(-1 \cdot \color{blue}{\frac{\frac{x}{a}}{z}} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}\right) \]

      4. associate-*r/ 52.5%

         \[\leadsto \frac{y}{a} + \left(\color{blue}{\frac{-1 \cdot \frac{x}{a}}{z}} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}\right) \]

      5. associate-/r* 52.5%

         \[\leadsto \frac{y}{a} + \left(\frac{-1 \cdot \frac{x}{a}}{z} - -1 \cdot \color{blue}{\frac{\frac{t \cdot y}{{a}^{2}}}{z}}\right) \]

      6. associate-*r/ 52.5%

         \[\leadsto \frac{y}{a} + \left(\frac{-1 \cdot \frac{x}{a}}{z} - \color{blue}{\frac{-1 \cdot \frac{t \cdot y}{{a}^{2}}}{z}}\right) \]

      7. div-sub 52.5%

         \[\leadsto \frac{y}{a} + \color{blue}{\frac{-1 \cdot \frac{x}{a} - -1 \cdot \frac{t \cdot y}{{a}^{2}}}{z}} \]

      8. distribute-lft-out-- 52.5%

         \[\leadsto \frac{y}{a} + \frac{\color{blue}{-1 \cdot \left(\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}\right)}}{z} \]

      9. associate-*r/ 52.5%

         \[\leadsto \frac{y}{a} + \color{blue}{-1 \cdot \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}} \]

      10. mul-1-neg 52.5%

          \[\leadsto \frac{y}{a} + \color{blue}{\left(-\frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}\right)} \]

      11. unsub-neg 52.5%

          \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}} \]

   7. Simplified 52.7%

      \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{t}{{a}^{2}} \cdot y}{z}} \]

   8. Taylor expanded in x around inf 67.7%

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

      1. associate-/r* 67.9%

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

   10. Simplified 67.9%

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

   if -1.24999999999999999e-71 < z < 2.19999999999999996e60

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

   6. Taylor expanded in a around 0 77.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{t} + \frac{x}{t}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+253}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{+190}:\\ \;\;\;\;\frac{y}{\frac{t - z \cdot a}{-z}}\\ \mathbf{elif}\;z \leq -2.65 \cdot 10^{+85}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{t} - \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ t_2 := \frac{x - y \cdot z}{t}\\ \mathbf{if}\;z \leq -5.7 \cdot 10^{+253}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -5 \cdot 10^{+187}:\\ \;\;\;\;\frac{y}{\frac{t - z \cdot a}{-z}}\\ \mathbf{elif}\;z \leq -3.05 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -9.6 \cdot 10^{+54}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+55}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)) (t_2 (/ (- x (* y z)) t)))
   (if (<= z -5.7e+253)
     t_1
     (if (<= z -5e+187)
       (/ y (/ (- t (* z a)) (- z)))
       (if (<= z -3.05e+85)
         t_1
         (if (<= z -9.6e+54)
           t_2
           (if (<= z -1.25e-71)
             (- (/ y a) (/ (/ x a) z))
             (if (<= z 2.3e+55) t_2 t_1))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double t_2 = (x - (y * z)) / t;
	double tmp;
	if (z <= -5.7e+253) {
		tmp = t_1;
	} else if (z <= -5e+187) {
		tmp = y / ((t - (z * a)) / -z);
	} else if (z <= -3.05e+85) {
		tmp = t_1;
	} else if (z <= -9.6e+54) {
		tmp = t_2;
	} else if (z <= -1.25e-71) {
		tmp = (y / a) - ((x / a) / z);
	} else if (z <= 2.3e+55) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = (y - (x / z)) / a
    t_2 = (x - (y * z)) / t
    if (z <= (-5.7d+253)) then
        tmp = t_1
    else if (z <= (-5d+187)) then
        tmp = y / ((t - (z * a)) / -z)
    else if (z <= (-3.05d+85)) then
        tmp = t_1
    else if (z <= (-9.6d+54)) then
        tmp = t_2
    else if (z <= (-1.25d-71)) then
        tmp = (y / a) - ((x / a) / z)
    else if (z <= 2.3d+55) then
        tmp = t_2
    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 t_2 = (x - (y * z)) / t;
	double tmp;
	if (z <= -5.7e+253) {
		tmp = t_1;
	} else if (z <= -5e+187) {
		tmp = y / ((t - (z * a)) / -z);
	} else if (z <= -3.05e+85) {
		tmp = t_1;
	} else if (z <= -9.6e+54) {
		tmp = t_2;
	} else if (z <= -1.25e-71) {
		tmp = (y / a) - ((x / a) / z);
	} else if (z <= 2.3e+55) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - (x / z)) / a
	t_2 = (x - (y * z)) / t
	tmp = 0
	if z <= -5.7e+253:
		tmp = t_1
	elif z <= -5e+187:
		tmp = y / ((t - (z * a)) / -z)
	elif z <= -3.05e+85:
		tmp = t_1
	elif z <= -9.6e+54:
		tmp = t_2
	elif z <= -1.25e-71:
		tmp = (y / a) - ((x / a) / z)
	elif z <= 2.3e+55:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - Float64(x / z)) / a)
	t_2 = Float64(Float64(x - Float64(y * z)) / t)
	tmp = 0.0
	if (z <= -5.7e+253)
		tmp = t_1;
	elseif (z <= -5e+187)
		tmp = Float64(y / Float64(Float64(t - Float64(z * a)) / Float64(-z)));
	elseif (z <= -3.05e+85)
		tmp = t_1;
	elseif (z <= -9.6e+54)
		tmp = t_2;
	elseif (z <= -1.25e-71)
		tmp = Float64(Float64(y / a) - Float64(Float64(x / a) / z));
	elseif (z <= 2.3e+55)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - (x / z)) / a;
	t_2 = (x - (y * z)) / t;
	tmp = 0.0;
	if (z <= -5.7e+253)
		tmp = t_1;
	elseif (z <= -5e+187)
		tmp = y / ((t - (z * a)) / -z);
	elseif (z <= -3.05e+85)
		tmp = t_1;
	elseif (z <= -9.6e+54)
		tmp = t_2;
	elseif (z <= -1.25e-71)
		tmp = (y / a) - ((x / a) / z);
	elseif (z <= 2.3e+55)
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[z, -5.7e+253], t$95$1, If[LessEqual[z, -5e+187], N[(y / N[(N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -3.05e+85], t$95$1, If[LessEqual[z, -9.6e+54], t$95$2, If[LessEqual[z, -1.25e-71], N[(N[(y / a), $MachinePrecision] - N[(N[(x / a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.3e+55], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -5.70000000000000016e253 or -5.0000000000000001e187 < z < -3.04999999999999991e85 or 2.29999999999999987e55 < z

   1. Initial program 66.1%

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

      1. *-commutative 66.1%

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

   3. Simplified 66.1%

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

   5. Taylor expanded in x around 0 66.1%

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

   6. Taylor expanded in t around 0 82.2%

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

      1. +-commutative 82.2%

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

      2. mul-1-neg 82.2%

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

      3. sub-neg 82.2%

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

      4. *-commutative 82.2%

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

      5. associate-/r* 86.2%

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

      6. div-sub 86.2%

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

   8. Simplified 86.2%

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

   if -5.70000000000000016e253 < z < -5.0000000000000001e187

   1. Initial program 74.0%

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

      1. *-commutative 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in x around 0 69.0%

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

      1. associate-*r/ 69.0%

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

      2. mul-1-neg 69.0%

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

      3. distribute-rgt-neg-out 69.0%

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

      4. associate-/l* 86.1%

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

      5. *-commutative 86.1%

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

   7. Simplified 86.1%

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

   if -3.04999999999999991e85 < z < -9.59999999999999993e54 or -1.24999999999999999e-71 < z < 2.29999999999999987e55

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t around inf 77.8%

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

   if -9.59999999999999993e54 < z < -1.24999999999999999e-71

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in z around inf 52.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{x}{a \cdot z} + \frac{y}{a}\right) - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}} \]
   6. Step-by-step derivation

      1. +-commutative 52.4%

         \[\leadsto \color{blue}{\left(\frac{y}{a} + -1 \cdot \frac{x}{a \cdot z}\right)} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z} \]

      2. associate--l+ 52.4%

         \[\leadsto \color{blue}{\frac{y}{a} + \left(-1 \cdot \frac{x}{a \cdot z} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}\right)} \]

      3. associate-/r* 52.5%

         \[\leadsto \frac{y}{a} + \left(-1 \cdot \color{blue}{\frac{\frac{x}{a}}{z}} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}\right) \]

      4. associate-*r/ 52.5%

         \[\leadsto \frac{y}{a} + \left(\color{blue}{\frac{-1 \cdot \frac{x}{a}}{z}} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}\right) \]

      5. associate-/r* 52.5%

         \[\leadsto \frac{y}{a} + \left(\frac{-1 \cdot \frac{x}{a}}{z} - -1 \cdot \color{blue}{\frac{\frac{t \cdot y}{{a}^{2}}}{z}}\right) \]

      6. associate-*r/ 52.5%

         \[\leadsto \frac{y}{a} + \left(\frac{-1 \cdot \frac{x}{a}}{z} - \color{blue}{\frac{-1 \cdot \frac{t \cdot y}{{a}^{2}}}{z}}\right) \]

      7. div-sub 52.5%

         \[\leadsto \frac{y}{a} + \color{blue}{\frac{-1 \cdot \frac{x}{a} - -1 \cdot \frac{t \cdot y}{{a}^{2}}}{z}} \]

      8. distribute-lft-out-- 52.5%

         \[\leadsto \frac{y}{a} + \frac{\color{blue}{-1 \cdot \left(\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}\right)}}{z} \]

      9. associate-*r/ 52.5%

         \[\leadsto \frac{y}{a} + \color{blue}{-1 \cdot \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}} \]

      10. mul-1-neg 52.5%

          \[\leadsto \frac{y}{a} + \color{blue}{\left(-\frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}\right)} \]

      11. unsub-neg 52.5%

          \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}} \]

   7. Simplified 52.7%

      \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{t}{{a}^{2}} \cdot y}{z}} \]

   8. Taylor expanded in x around inf 67.7%

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

      1. associate-/r* 67.9%

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

   10. Simplified 67.9%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.7 \cdot 10^{+253}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq -5 \cdot 10^{+187}:\\ \;\;\;\;\frac{y}{\frac{t - z \cdot a}{-z}}\\ \mathbf{elif}\;z \leq -3.05 \cdot 10^{+85}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq -9.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{+85} \lor \neg \left(z \leq -1.16 \cdot 10^{+55} \lor \neg \left(z \leq -7.5 \cdot 10^{-72}\right) \land z \leq 2.8 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.85e+85)
         (not
          (or (<= z -1.16e+55) (and (not (<= z -7.5e-72)) (<= z 2.8e+55)))))
   (/ (- y (/ x z)) a)
   (/ (- x (* y z)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.85e+85) || !((z <= -1.16e+55) || (!(z <= -7.5e-72) && (z <= 2.8e+55)))) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (y * z)) / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-2.85d+85)) .or. (.not. (z <= (-1.16d+55)) .or. (.not. (z <= (-7.5d-72))) .and. (z <= 2.8d+55))) then
        tmp = (y - (x / z)) / a
    else
        tmp = (x - (y * z)) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.85e+85) || !((z <= -1.16e+55) || (!(z <= -7.5e-72) && (z <= 2.8e+55)))) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (y * z)) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -2.85e+85) or not ((z <= -1.16e+55) or (not (z <= -7.5e-72) and (z <= 2.8e+55))):
		tmp = (y - (x / z)) / a
	else:
		tmp = (x - (y * z)) / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.85e+85) || !((z <= -1.16e+55) || (!(z <= -7.5e-72) && (z <= 2.8e+55))))
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	else
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -2.85e+85) || ~(((z <= -1.16e+55) || (~((z <= -7.5e-72)) && (z <= 2.8e+55)))))
		tmp = (y - (x / z)) / a;
	else
		tmp = (x - (y * z)) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.85e+85], N[Not[Or[LessEqual[z, -1.16e+55], And[N[Not[LessEqual[z, -7.5e-72]], $MachinePrecision], LessEqual[z, 2.8e+55]]]], $MachinePrecision]], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8500000000000001e85 or -1.1599999999999999e55 < z < -7.5000000000000004e-72 or 2.8000000000000001e55 < z

   1. Initial program 74.7%

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

      1. *-commutative 74.7%

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

   3. Simplified 74.7%

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

   5. Taylor expanded in x around 0 74.7%

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

   6. Taylor expanded in t around 0 73.8%

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

      1. +-commutative 73.8%

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

      2. mul-1-neg 73.8%

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

      3. sub-neg 73.8%

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

      4. *-commutative 73.8%

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

      5. associate-/r* 77.3%

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

      6. div-sub 77.3%

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

   8. Simplified 77.3%

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

   if -2.8500000000000001e85 < z < -1.1599999999999999e55 or -7.5000000000000004e-72 < z < 2.8000000000000001e55

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t around inf 77.8%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{+85} \lor \neg \left(z \leq -1.16 \cdot 10^{+55} \lor \neg \left(z \leq -7.5 \cdot 10^{-72}\right) \land z \leq 2.8 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y \cdot z}{t}\\ t_2 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{+85}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.16 \cdot 10^{+55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -9.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* y z)) t)) (t_2 (/ (- y (/ x z)) a)))
   (if (<= z -2.6e+85)
     t_2
     (if (<= z -1.16e+55)
       t_1
       (if (<= z -9.2e-72)
         (- (/ y a) (/ (/ x a) z))
         (if (<= z 1.3e+59) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (y * z)) / t;
	double t_2 = (y - (x / z)) / a;
	double tmp;
	if (z <= -2.6e+85) {
		tmp = t_2;
	} else if (z <= -1.16e+55) {
		tmp = t_1;
	} else if (z <= -9.2e-72) {
		tmp = (y / a) - ((x / a) / z);
	} else if (z <= 1.3e+59) {
		tmp = 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 = (x - (y * z)) / t
    t_2 = (y - (x / z)) / a
    if (z <= (-2.6d+85)) then
        tmp = t_2
    else if (z <= (-1.16d+55)) then
        tmp = t_1
    else if (z <= (-9.2d-72)) then
        tmp = (y / a) - ((x / a) / z)
    else if (z <= 1.3d+59) then
        tmp = 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 = (x - (y * z)) / t;
	double t_2 = (y - (x / z)) / a;
	double tmp;
	if (z <= -2.6e+85) {
		tmp = t_2;
	} else if (z <= -1.16e+55) {
		tmp = t_1;
	} else if (z <= -9.2e-72) {
		tmp = (y / a) - ((x / a) / z);
	} else if (z <= 1.3e+59) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - (y * z)) / t
	t_2 = (y - (x / z)) / a
	tmp = 0
	if z <= -2.6e+85:
		tmp = t_2
	elif z <= -1.16e+55:
		tmp = t_1
	elif z <= -9.2e-72:
		tmp = (y / a) - ((x / a) / z)
	elif z <= 1.3e+59:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - Float64(y * z)) / t)
	t_2 = Float64(Float64(y - Float64(x / z)) / a)
	tmp = 0.0
	if (z <= -2.6e+85)
		tmp = t_2;
	elseif (z <= -1.16e+55)
		tmp = t_1;
	elseif (z <= -9.2e-72)
		tmp = Float64(Float64(y / a) - Float64(Float64(x / a) / z));
	elseif (z <= 1.3e+59)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x - (y * z)) / t;
	t_2 = (y - (x / z)) / a;
	tmp = 0.0;
	if (z <= -2.6e+85)
		tmp = t_2;
	elseif (z <= -1.16e+55)
		tmp = t_1;
	elseif (z <= -9.2e-72)
		tmp = (y / a) - ((x / a) / z);
	elseif (z <= 1.3e+59)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -2.6e+85], t$95$2, If[LessEqual[z, -1.16e+55], t$95$1, If[LessEqual[z, -9.2e-72], N[(N[(y / a), $MachinePrecision] - N[(N[(x / a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.3e+59], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.60000000000000011e85 or 1.3e59 < z

   1. Initial program 67.1%

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

      1. *-commutative 67.1%

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

   3. Simplified 67.1%

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

   5. Taylor expanded in x around 0 67.1%

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

   6. Taylor expanded in t around 0 75.6%

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

      1. +-commutative 75.6%

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

      2. mul-1-neg 75.6%

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

      3. sub-neg 75.6%

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

      4. *-commutative 75.6%

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

      5. associate-/r* 80.2%

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

      6. div-sub 80.2%

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

   8. Simplified 80.2%

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

   if -2.60000000000000011e85 < z < -1.1599999999999999e55 or -9.19999999999999978e-72 < z < 1.3e59

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in t around inf 77.8%

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

   if -1.1599999999999999e55 < z < -9.19999999999999978e-72

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in z around inf 52.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{x}{a \cdot z} + \frac{y}{a}\right) - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}} \]
   6. Step-by-step derivation

      1. +-commutative 52.4%

         \[\leadsto \color{blue}{\left(\frac{y}{a} + -1 \cdot \frac{x}{a \cdot z}\right)} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z} \]

      2. associate--l+ 52.4%

         \[\leadsto \color{blue}{\frac{y}{a} + \left(-1 \cdot \frac{x}{a \cdot z} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}\right)} \]

      3. associate-/r* 52.5%

         \[\leadsto \frac{y}{a} + \left(-1 \cdot \color{blue}{\frac{\frac{x}{a}}{z}} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}\right) \]

      4. associate-*r/ 52.5%

         \[\leadsto \frac{y}{a} + \left(\color{blue}{\frac{-1 \cdot \frac{x}{a}}{z}} - -1 \cdot \frac{t \cdot y}{{a}^{2} \cdot z}\right) \]

      5. associate-/r* 52.5%

         \[\leadsto \frac{y}{a} + \left(\frac{-1 \cdot \frac{x}{a}}{z} - -1 \cdot \color{blue}{\frac{\frac{t \cdot y}{{a}^{2}}}{z}}\right) \]

      6. associate-*r/ 52.5%

         \[\leadsto \frac{y}{a} + \left(\frac{-1 \cdot \frac{x}{a}}{z} - \color{blue}{\frac{-1 \cdot \frac{t \cdot y}{{a}^{2}}}{z}}\right) \]

      7. div-sub 52.5%

         \[\leadsto \frac{y}{a} + \color{blue}{\frac{-1 \cdot \frac{x}{a} - -1 \cdot \frac{t \cdot y}{{a}^{2}}}{z}} \]

      8. distribute-lft-out-- 52.5%

         \[\leadsto \frac{y}{a} + \frac{\color{blue}{-1 \cdot \left(\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}\right)}}{z} \]

      9. associate-*r/ 52.5%

         \[\leadsto \frac{y}{a} + \color{blue}{-1 \cdot \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}} \]

      10. mul-1-neg 52.5%

          \[\leadsto \frac{y}{a} + \color{blue}{\left(-\frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}\right)} \]

      11. unsub-neg 52.5%

          \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{t \cdot y}{{a}^{2}}}{z}} \]

   7. Simplified 52.7%

      \[\leadsto \color{blue}{\frac{y}{a} - \frac{\frac{x}{a} - \frac{t}{{a}^{2}} \cdot y}{z}} \]

   8. Taylor expanded in x around inf 67.7%

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

      1. associate-/r* 67.9%

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

   10. Simplified 67.9%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+85}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq -1.16 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq -9.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+59}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-205}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-166}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{+66}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.2e-71)
   (/ y a)
   (if (<= z 1.95e-205)
     (/ x t)
     (if (<= z 2.2e-166)
       (/ (- x) (* z a))
       (if (<= z 2.65e+66) (/ x t) (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.2e-71) {
		tmp = y / a;
	} else if (z <= 1.95e-205) {
		tmp = x / t;
	} else if (z <= 2.2e-166) {
		tmp = -x / (z * a);
	} else if (z <= 2.65e+66) {
		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 <= (-1.2d-71)) then
        tmp = y / a
    else if (z <= 1.95d-205) then
        tmp = x / t
    else if (z <= 2.2d-166) then
        tmp = -x / (z * a)
    else if (z <= 2.65d+66) 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 <= -1.2e-71) {
		tmp = y / a;
	} else if (z <= 1.95e-205) {
		tmp = x / t;
	} else if (z <= 2.2e-166) {
		tmp = -x / (z * a);
	} else if (z <= 2.65e+66) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.2e-71:
		tmp = y / a
	elif z <= 1.95e-205:
		tmp = x / t
	elif z <= 2.2e-166:
		tmp = -x / (z * a)
	elif z <= 2.65e+66:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.2e-71)
		tmp = Float64(y / a);
	elseif (z <= 1.95e-205)
		tmp = Float64(x / t);
	elseif (z <= 2.2e-166)
		tmp = Float64(Float64(-x) / Float64(z * a));
	elseif (z <= 2.65e+66)
		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 <= -1.2e-71)
		tmp = y / a;
	elseif (z <= 1.95e-205)
		tmp = x / t;
	elseif (z <= 2.2e-166)
		tmp = -x / (z * a);
	elseif (z <= 2.65e+66)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.2e-71], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.95e-205], N[(x / t), $MachinePrecision], If[LessEqual[z, 2.2e-166], N[((-x) / N[(z * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.65e+66], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.2e-71 or 2.6499999999999998e66 < z

   1. Initial program 74.9%

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

      1. *-commutative 74.9%

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

   3. Simplified 74.9%

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

   5. Taylor expanded in z around inf 54.5%

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

   if -1.2e-71 < z < 1.95000000000000009e-205 or 2.2000000000000001e-166 < z < 2.6499999999999998e66

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 63.1%

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

   if 1.95000000000000009e-205 < z < 2.2000000000000001e-166

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 78.3%

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

      1. associate-*r/ 78.3%

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

      2. neg-mul-1 78.3%

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

      3. neg-sub0 78.3%

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

      4. sub-neg 78.3%

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

      5. distribute-rgt-neg-out 78.3%

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

      6. +-commutative 78.3%

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

      7. associate--r+ 78.3%

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

      8. neg-sub0 78.3%

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

      9. distribute-rgt-neg-out 78.3%

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

      10. remove-double-neg 78.3%

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

      11. *-commutative 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in y around 0 67.7%

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

      1. neg-mul-1 67.7%

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

   10. Simplified 67.7%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-205}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-166}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{+66}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+163}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+160}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -9e+163)
   (/ y a)
   (if (<= y 6e-84)
     (/ x (- t (* z a)))
     (if (<= y 2.2e+160) (/ (- x (* y z)) t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -9e+163) {
		tmp = y / a;
	} else if (y <= 6e-84) {
		tmp = x / (t - (z * a));
	} else if (y <= 2.2e+160) {
		tmp = (x - (y * z)) / 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 (y <= (-9d+163)) then
        tmp = y / a
    else if (y <= 6d-84) then
        tmp = x / (t - (z * a))
    else if (y <= 2.2d+160) then
        tmp = (x - (y * z)) / 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 (y <= -9e+163) {
		tmp = y / a;
	} else if (y <= 6e-84) {
		tmp = x / (t - (z * a));
	} else if (y <= 2.2e+160) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -9e+163:
		tmp = y / a
	elif y <= 6e-84:
		tmp = x / (t - (z * a))
	elif y <= 2.2e+160:
		tmp = (x - (y * z)) / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -9e+163)
		tmp = Float64(y / a);
	elseif (y <= 6e-84)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (y <= 2.2e+160)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -9e+163)
		tmp = y / a;
	elseif (y <= 6e-84)
		tmp = x / (t - (z * a));
	elseif (y <= 2.2e+160)
		tmp = (x - (y * z)) / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -9e+163], N[(y / a), $MachinePrecision], If[LessEqual[y, 6e-84], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e+160], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.99999999999999976e163 or 2.19999999999999992e160 < y

   1. Initial program 69.5%

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

      1. *-commutative 69.5%

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

   3. Simplified 69.5%

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

   5. Taylor expanded in z around inf 61.6%

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

   if -8.99999999999999976e163 < y < 6.0000000000000002e-84

   1. Initial program 94.0%

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

      1. *-commutative 94.0%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in x around inf 74.1%

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

      1. *-commutative 74.1%

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

   7. Simplified 74.1%

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

   if 6.0000000000000002e-84 < y < 2.19999999999999992e160

   1. Initial program 89.7%

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

      1. *-commutative 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in t around inf 62.3%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+163}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+160}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+159} \lor \neg \left(y \leq 10^{+125}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= y -1.15e+159) (not (<= y 1e+125))) (/ y a) (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -1.15e+159) || !(y <= 1e+125)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y <= (-1.15d+159)) .or. (.not. (y <= 1d+125))) then
        tmp = y / a
    else
        tmp = x / (t - (z * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -1.15e+159) || !(y <= 1e+125)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (y <= -1.15e+159) or not (y <= 1e+125):
		tmp = y / a
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((y <= -1.15e+159) || !(y <= 1e+125))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / Float64(t - Float64(z * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((y <= -1.15e+159) || ~((y <= 1e+125)))
		tmp = y / a;
	else
		tmp = x / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[y, -1.15e+159], N[Not[LessEqual[y, 1e+125]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.14999999999999998e159 or 9.9999999999999992e124 < y

   1. Initial program 70.6%

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

      1. *-commutative 70.6%

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

   3. Simplified 70.6%

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

   5. Taylor expanded in z around inf 57.1%

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

   if -1.14999999999999998e159 < y < 9.9999999999999992e124

   1. Initial program 93.9%

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

      1. *-commutative 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in x around inf 70.1%

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

      1. *-commutative 70.1%

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

   7. Simplified 70.1%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+159} \lor \neg \left(y \leq 10^{+125}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-72} \lor \neg \left(z \leq 5.3 \cdot 10^{+66}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -7.2e-72) (not (<= z 5.3e+66))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -7.2e-72) || !(z <= 5.3e+66)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-7.2d-72)) .or. (.not. (z <= 5.3d+66))) then
        tmp = y / a
    else
        tmp = x / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -7.2e-72) || !(z <= 5.3e+66)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -7.2e-72) or not (z <= 5.3e+66):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -7.2e-72) || !(z <= 5.3e+66))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -7.2e-72) || ~((z <= 5.3e+66)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -7.2e-72], N[Not[LessEqual[z, 5.3e+66]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.2e-72 or 5.2999999999999997e66 < z

   1. Initial program 74.9%

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

      1. *-commutative 74.9%

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

   3. Simplified 74.9%

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

   5. Taylor expanded in z around inf 54.5%

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

   if -7.2e-72 < z < 5.2999999999999997e66

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 60.1%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-72} \lor \neg \left(z \leq 5.3 \cdot 10^{+66}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.6% 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 88.2%

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

   1. *-commutative 88.2%

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

3. Simplified 88.2%

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

5. Taylor expanded in z around 0 37.3%

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

6. Final simplification 37.3%

   \[\leadsto \frac{x}{t} \]
7. Add Preprocessing

Developer target: 97.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{x}{t\_1} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z))) (t_2 (- (/ x t_1) (/ y (- (/ t z) a)))))
   (if (< z -32113435955957344.0)
     t_2
     (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 t_1)) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = (x / t_1) - (y / ((t / z) - a));
	double tmp;
	if (z < -32113435955957344.0) {
		tmp = t_2;
	} else if (z < 3.5139522372978296e-86) {
		tmp = (x - (y * z)) * (1.0 / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t - (a * z)
    t_2 = (x / t_1) - (y / ((t / z) - a))
    if (z < (-32113435955957344.0d0)) then
        tmp = t_2
    else if (z < 3.5139522372978296d-86) then
        tmp = (x - (y * z)) * (1.0d0 / t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = (x / t_1) - (y / ((t / z) - a));
	double tmp;
	if (z < -32113435955957344.0) {
		tmp = t_2;
	} else if (z < 3.5139522372978296e-86) {
		tmp = (x - (y * z)) * (1.0 / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (a * z)
	t_2 = (x / t_1) - (y / ((t / z) - a))
	tmp = 0
	if z < -32113435955957344.0:
		tmp = t_2
	elif z < 3.5139522372978296e-86:
		tmp = (x - (y * z)) * (1.0 / t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	t_2 = Float64(Float64(x / t_1) - Float64(y / Float64(Float64(t / z) - a)))
	tmp = 0.0
	if (z < -32113435955957344.0)
		tmp = t_2;
	elseif (z < 3.5139522372978296e-86)
		tmp = Float64(Float64(x - Float64(y * z)) * Float64(1.0 / t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (a * z);
	t_2 = (x / t_1) - (y / ((t / z) - a));
	tmp = 0.0;
	if (z < -32113435955957344.0)
		tmp = t_2;
	elseif (z < 3.5139522372978296e-86)
		tmp = (x - (y * z)) * (1.0 / t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / t$95$1), $MachinePrecision] - N[(y / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -32113435955957344.0], t$95$2, If[Less[z, 3.5139522372978296e-86], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024031 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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