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

Percentage Accurate: 85.6% → 91.5%

Time: 11.9s

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: 85.6% 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: 91.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+140}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+133}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5e+140)
   (/ y (- a (/ t z)))
   (if (<= z 7e+133) (/ (- x (* z y)) (- t (* z a))) (/ (- y (/ x z)) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e+140) {
		tmp = y / (a - (t / z));
	} else if (z <= 7e+133) {
		tmp = (x - (z * y)) / (t - (z * a));
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-5d+140)) then
        tmp = y / (a - (t / z))
    else if (z <= 7d+133) then
        tmp = (x - (z * y)) / (t - (z * a))
    else
        tmp = (y - (x / z)) / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e+140) {
		tmp = y / (a - (t / z));
	} else if (z <= 7e+133) {
		tmp = (x - (z * y)) / (t - (z * a));
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5e+140:
		tmp = y / (a - (t / z))
	elif z <= 7e+133:
		tmp = (x - (z * y)) / (t - (z * a))
	else:
		tmp = (y - (x / z)) / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5e+140)
		tmp = Float64(y / Float64(a - Float64(t / z)));
	elseif (z <= 7e+133)
		tmp = Float64(Float64(x - Float64(z * y)) / Float64(t - Float64(z * a)));
	else
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -5e+140)
		tmp = y / (a - (t / z));
	elseif (z <= 7e+133)
		tmp = (x - (z * y)) / (t - (z * a));
	else
		tmp = (y - (x / z)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5e+140], N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7e+133], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.00000000000000008e140

   1. Initial program 45.5%

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

      1. *-commutative 45.5%

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

   3. Simplified 45.5%

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

   5. Taylor expanded in z around inf 45.5%

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

   6. Taylor expanded in x around 0 89.4%

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

      1. associate-*r/ 89.4%

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

      2. neg-mul-1 89.4%

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

   8. Simplified 89.4%

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

   if -5.00000000000000008e140 < z < 6.9999999999999997e133

   1. Initial program 95.5%

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

   if 6.9999999999999997e133 < z

   1. Initial program 62.1%

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

      1. *-commutative 62.1%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in y around inf 70.8%

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

   7. Simplified 70.8%

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

   8. Taylor expanded in y around 0 62.1%

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

      1. *-commutative 62.1%

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

      2. associate-*r/ 73.9%

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

      3. *-commutative 73.9%

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

      4. fma-neg 73.8%

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

   10. Simplified 73.8%

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

   11. Taylor expanded in a around inf 91.4%

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

       1. mul-1-neg 91.4%

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

       2. unsub-neg 91.4%

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

   13. Simplified 91.4%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+140}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+133}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 69.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - z \cdot y}{t}\\ t_2 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -2.6 \cdot 10^{+91}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-68}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-265}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+130}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* z y)) t)) (t_2 (/ (- y (/ x z)) a)))
   (if (<= z -2.6e+91)
     t_2
     (if (<= z -3.6e+78)
       t_1
       (if (<= z -1.15e-68)
         t_2
         (if (<= z 5e-265)
           t_1
           (if (<= z 1.2e-26)
             (/ x (- t (* z a)))
             (if (<= z 3.6e+130) t_1 t_2))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (z * y)) / t;
	double t_2 = (y - (x / z)) / a;
	double tmp;
	if (z <= -2.6e+91) {
		tmp = t_2;
	} else if (z <= -3.6e+78) {
		tmp = t_1;
	} else if (z <= -1.15e-68) {
		tmp = t_2;
	} else if (z <= 5e-265) {
		tmp = t_1;
	} else if (z <= 1.2e-26) {
		tmp = x / (t - (z * a));
	} else if (z <= 3.6e+130) {
		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 - (z * y)) / t
    t_2 = (y - (x / z)) / a
    if (z <= (-2.6d+91)) then
        tmp = t_2
    else if (z <= (-3.6d+78)) then
        tmp = t_1
    else if (z <= (-1.15d-68)) then
        tmp = t_2
    else if (z <= 5d-265) then
        tmp = t_1
    else if (z <= 1.2d-26) then
        tmp = x / (t - (z * a))
    else if (z <= 3.6d+130) 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 - (z * y)) / t;
	double t_2 = (y - (x / z)) / a;
	double tmp;
	if (z <= -2.6e+91) {
		tmp = t_2;
	} else if (z <= -3.6e+78) {
		tmp = t_1;
	} else if (z <= -1.15e-68) {
		tmp = t_2;
	} else if (z <= 5e-265) {
		tmp = t_1;
	} else if (z <= 1.2e-26) {
		tmp = x / (t - (z * a));
	} else if (z <= 3.6e+130) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - (z * y)) / t
	t_2 = (y - (x / z)) / a
	tmp = 0
	if z <= -2.6e+91:
		tmp = t_2
	elif z <= -3.6e+78:
		tmp = t_1
	elif z <= -1.15e-68:
		tmp = t_2
	elif z <= 5e-265:
		tmp = t_1
	elif z <= 1.2e-26:
		tmp = x / (t - (z * a))
	elif z <= 3.6e+130:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - Float64(z * y)) / t)
	t_2 = Float64(Float64(y - Float64(x / z)) / a)
	tmp = 0.0
	if (z <= -2.6e+91)
		tmp = t_2;
	elseif (z <= -3.6e+78)
		tmp = t_1;
	elseif (z <= -1.15e-68)
		tmp = t_2;
	elseif (z <= 5e-265)
		tmp = t_1;
	elseif (z <= 1.2e-26)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 3.6e+130)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x - (z * y)) / t;
	t_2 = (y - (x / z)) / a;
	tmp = 0.0;
	if (z <= -2.6e+91)
		tmp = t_2;
	elseif (z <= -3.6e+78)
		tmp = t_1;
	elseif (z <= -1.15e-68)
		tmp = t_2;
	elseif (z <= 5e-265)
		tmp = t_1;
	elseif (z <= 1.2e-26)
		tmp = x / (t - (z * a));
	elseif (z <= 3.6e+130)
		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[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -2.6e+91], t$95$2, If[LessEqual[z, -3.6e+78], t$95$1, If[LessEqual[z, -1.15e-68], t$95$2, If[LessEqual[z, 5e-265], t$95$1, If[LessEqual[z, 1.2e-26], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.6e+130], t$95$1, t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.6e91 or -3.6000000000000002e78 < z < -1.14999999999999998e-68 or 3.6000000000000001e130 < z

   1. Initial program 64.8%

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

      1. *-commutative 64.8%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in y around inf 69.5%

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

   7. Simplified 69.5%

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

   8. Taylor expanded in y around 0 64.8%

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

      1. *-commutative 64.8%

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

      2. associate-*r/ 75.9%

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

      3. *-commutative 75.9%

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

      4. fma-neg 75.9%

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

   10. Simplified 75.9%

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

   11. Taylor expanded in a around inf 80.0%

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

       1. mul-1-neg 80.0%

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

       2. unsub-neg 80.0%

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

   13. Simplified 80.0%

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

   if -2.6e91 < z < -3.6000000000000002e78 or -1.14999999999999998e-68 < z < 5.0000000000000001e-265 or 1.2e-26 < z < 3.6000000000000001e130

   1. Initial program 95.1%

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

      1. *-commutative 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in t around inf 82.4%

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

   if 5.0000000000000001e-265 < z < 1.2e-26

   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 inf 81.0%

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

      1. *-commutative 81.0%

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

   7. Simplified 81.0%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+91}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{+78}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-68}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-265}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+130}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a - \frac{t}{z}}\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{-69}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-264}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ y (- a (/ t z)))))
   (if (<= z -4.2e-69)
     t_1
     (if (<= z 1.6e-264)
       (/ (- x (* z y)) t)
       (if (<= z 1.25e-15)
         (/ x (- t (* z a)))
         (if (<= z 3.6e+119) t_1 (/ (- y (/ x z)) a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y / (a - (t / z));
	double tmp;
	if (z <= -4.2e-69) {
		tmp = t_1;
	} else if (z <= 1.6e-264) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 1.25e-15) {
		tmp = x / (t - (z * a));
	} else if (z <= 3.6e+119) {
		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 = y / (a - (t / z))
    if (z <= (-4.2d-69)) then
        tmp = t_1
    else if (z <= 1.6d-264) then
        tmp = (x - (z * y)) / t
    else if (z <= 1.25d-15) then
        tmp = x / (t - (z * a))
    else if (z <= 3.6d+119) 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 = y / (a - (t / z));
	double tmp;
	if (z <= -4.2e-69) {
		tmp = t_1;
	} else if (z <= 1.6e-264) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 1.25e-15) {
		tmp = x / (t - (z * a));
	} else if (z <= 3.6e+119) {
		tmp = t_1;
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y / (a - (t / z))
	tmp = 0
	if z <= -4.2e-69:
		tmp = t_1
	elif z <= 1.6e-264:
		tmp = (x - (z * y)) / t
	elif z <= 1.25e-15:
		tmp = x / (t - (z * a))
	elif z <= 3.6e+119:
		tmp = t_1
	else:
		tmp = (y - (x / z)) / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y / Float64(a - Float64(t / z)))
	tmp = 0.0
	if (z <= -4.2e-69)
		tmp = t_1;
	elseif (z <= 1.6e-264)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 1.25e-15)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 3.6e+119)
		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 = y / (a - (t / z));
	tmp = 0.0;
	if (z <= -4.2e-69)
		tmp = t_1;
	elseif (z <= 1.6e-264)
		tmp = (x - (z * y)) / t;
	elseif (z <= 1.25e-15)
		tmp = x / (t - (z * a));
	elseif (z <= 3.6e+119)
		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[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.2e-69], t$95$1, If[LessEqual[z, 1.6e-264], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 1.25e-15], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.6e+119], t$95$1, N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.1999999999999999e-69 or 1.25e-15 < z < 3.60000000000000001e119

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in z around inf 72.6%

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

   6. Taylor expanded in x around 0 74.0%

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

      1. associate-*r/ 74.0%

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

      2. neg-mul-1 74.0%

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

   8. Simplified 74.0%

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

   if -4.1999999999999999e-69 < z < 1.59999999999999998e-264

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 92.1%

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

   if 1.59999999999999998e-264 < z < 1.25e-15

   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 inf 81.7%

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

      1. *-commutative 81.7%

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

   7. Simplified 81.7%

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

   if 3.60000000000000001e119 < z

   1. Initial program 65.1%

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

      1. *-commutative 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in y around inf 73.1%

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

   7. Simplified 73.1%

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

   8. Taylor expanded in y around 0 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-*r/ 75.9%

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

      3. *-commutative 75.9%

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

      4. fma-neg 75.9%

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

   10. Simplified 75.9%

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

   11. Taylor expanded in a around inf 87.0%

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

       1. mul-1-neg 87.0%

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

       2. unsub-neg 87.0%

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

   13. Simplified 87.0%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-264}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+119}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - z \cdot y}{t}\\ \mathbf{if}\;z \leq -6.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-265}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* z y)) t)))
   (if (<= z -6.2e+92)
     (/ y a)
     (if (<= z 3.8e-265)
       t_1
       (if (<= z 2.3e-23)
         (/ x (- t (* z a)))
         (if (<= z 1.6e+132) t_1 (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (z * y)) / t;
	double tmp;
	if (z <= -6.2e+92) {
		tmp = y / a;
	} else if (z <= 3.8e-265) {
		tmp = t_1;
	} else if (z <= 2.3e-23) {
		tmp = x / (t - (z * a));
	} else if (z <= 1.6e+132) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (x - (z * y)) / t
    if (z <= (-6.2d+92)) then
        tmp = y / a
    else if (z <= 3.8d-265) then
        tmp = t_1
    else if (z <= 2.3d-23) then
        tmp = x / (t - (z * a))
    else if (z <= 1.6d+132) then
        tmp = t_1
    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 t_1 = (x - (z * y)) / t;
	double tmp;
	if (z <= -6.2e+92) {
		tmp = y / a;
	} else if (z <= 3.8e-265) {
		tmp = t_1;
	} else if (z <= 2.3e-23) {
		tmp = x / (t - (z * a));
	} else if (z <= 1.6e+132) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - (z * y)) / t
	tmp = 0
	if z <= -6.2e+92:
		tmp = y / a
	elif z <= 3.8e-265:
		tmp = t_1
	elif z <= 2.3e-23:
		tmp = x / (t - (z * a))
	elif z <= 1.6e+132:
		tmp = t_1
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - Float64(z * y)) / t)
	tmp = 0.0
	if (z <= -6.2e+92)
		tmp = Float64(y / a);
	elseif (z <= 3.8e-265)
		tmp = t_1;
	elseif (z <= 2.3e-23)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 1.6e+132)
		tmp = t_1;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x - (z * y)) / t;
	tmp = 0.0;
	if (z <= -6.2e+92)
		tmp = y / a;
	elseif (z <= 3.8e-265)
		tmp = t_1;
	elseif (z <= 2.3e-23)
		tmp = x / (t - (z * a));
	elseif (z <= 1.6e+132)
		tmp = t_1;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[z, -6.2e+92], N[(y / a), $MachinePrecision], If[LessEqual[z, 3.8e-265], t$95$1, If[LessEqual[z, 2.3e-23], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.6e+132], t$95$1, N[(y / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.2000000000000004e92 or 1.5999999999999999e132 < z

   1. Initial program 57.4%

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

      1. *-commutative 57.4%

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

   3. Simplified 57.4%

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

   5. Taylor expanded in z around inf 67.7%

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

   if -6.2000000000000004e92 < z < 3.7999999999999998e-265 or 2.3000000000000001e-23 < z < 1.5999999999999999e132

   1. Initial program 94.4%

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

      1. *-commutative 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in t around inf 74.1%

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

   if 3.7999999999999998e-265 < z < 2.3000000000000001e-23

   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 inf 81.0%

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

      1. *-commutative 81.0%

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

   7. Simplified 81.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-265}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+132}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+93}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+73}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+105}:\\ \;\;\;\;y \cdot \frac{z}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.5e+93)
   (/ y a)
   (if (<= z 1.9e+73)
     (/ x (- t (* z a)))
     (if (<= z 5.5e+105) (* y (/ z (- t))) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.5e+93) {
		tmp = y / a;
	} else if (z <= 1.9e+73) {
		tmp = x / (t - (z * a));
	} else if (z <= 5.5e+105) {
		tmp = 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 (z <= (-2.5d+93)) then
        tmp = y / a
    else if (z <= 1.9d+73) then
        tmp = x / (t - (z * a))
    else if (z <= 5.5d+105) then
        tmp = 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 (z <= -2.5e+93) {
		tmp = y / a;
	} else if (z <= 1.9e+73) {
		tmp = x / (t - (z * a));
	} else if (z <= 5.5e+105) {
		tmp = y * (z / -t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.5e+93:
		tmp = y / a
	elif z <= 1.9e+73:
		tmp = x / (t - (z * a))
	elif z <= 5.5e+105:
		tmp = y * (z / -t)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.5e+93)
		tmp = Float64(y / a);
	elseif (z <= 1.9e+73)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 5.5e+105)
		tmp = Float64(y * Float64(z / Float64(-t)));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.5e+93)
		tmp = y / a;
	elseif (z <= 1.9e+73)
		tmp = x / (t - (z * a));
	elseif (z <= 5.5e+105)
		tmp = y * (z / -t);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.5e+93], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.9e+73], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.5e+105], N[(y * N[(z / (-t)), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.5000000000000001e93 or 5.49999999999999979e105 < z

   1. Initial program 59.1%

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

      1. *-commutative 59.1%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in z around inf 64.8%

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

   if -2.5000000000000001e93 < z < 1.90000000000000011e73

   1. Initial program 97.4%

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

      1. *-commutative 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in x around inf 69.9%

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

      1. *-commutative 69.9%

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

   7. Simplified 69.9%

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

   if 1.90000000000000011e73 < z < 5.49999999999999979e105

   1. Initial program 76.2%

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

      1. *-commutative 76.2%

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

   3. Simplified 76.2%

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

      1. clear-num 76.0%

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

      2. associate-/r/ 75.8%

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

      3. sub-neg 75.8%

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

      4. +-commutative 75.8%

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

      5. *-commutative 75.8%

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

      6. distribute-rgt-neg-in 75.8%

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

      7. fma-define 75.8%

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

   6. Applied egg-rr 75.8%

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

   7. Taylor expanded in a around 0 75.0%

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

   8. Taylor expanded in x around 0 75.1%

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

      1. mul-1-neg 75.1%

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

      2. distribute-rgt-neg-out 75.1%

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

   10. Simplified 75.1%

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

   11. Taylor expanded in t around 0 75.5%

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

       1. mul-1-neg 75.5%

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

       2. associate-*r/ 87.1%

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

       3. distribute-rgt-neg-in 87.1%

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

       4. distribute-neg-frac2 87.1%

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

   13. Simplified 87.1%

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

Alternative 6: 53.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-68}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{+104}:\\ \;\;\;\;y \cdot \frac{z}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.15e-68)
   (/ y a)
   (if (<= z 3.6e-17) (/ x t) (if (<= z 5.3e+104) (* y (/ z (- t))) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.15e-68) {
		tmp = y / a;
	} else if (z <= 3.6e-17) {
		tmp = x / t;
	} else if (z <= 5.3e+104) {
		tmp = 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 (z <= (-1.15d-68)) then
        tmp = y / a
    else if (z <= 3.6d-17) then
        tmp = x / t
    else if (z <= 5.3d+104) then
        tmp = 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 (z <= -1.15e-68) {
		tmp = y / a;
	} else if (z <= 3.6e-17) {
		tmp = x / t;
	} else if (z <= 5.3e+104) {
		tmp = y * (z / -t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.15e-68:
		tmp = y / a
	elif z <= 3.6e-17:
		tmp = x / t
	elif z <= 5.3e+104:
		tmp = y * (z / -t)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.15e-68)
		tmp = Float64(y / a);
	elseif (z <= 3.6e-17)
		tmp = Float64(x / t);
	elseif (z <= 5.3e+104)
		tmp = Float64(y * Float64(z / Float64(-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 <= -1.15e-68)
		tmp = y / a;
	elseif (z <= 3.6e-17)
		tmp = x / t;
	elseif (z <= 5.3e+104)
		tmp = y * (z / -t);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.15e-68], N[(y / a), $MachinePrecision], If[LessEqual[z, 3.6e-17], N[(x / t), $MachinePrecision], If[LessEqual[z, 5.3e+104], N[(y * N[(z / (-t)), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.14999999999999998e-68 or 5.2999999999999999e104 < z

   1. Initial program 67.3%

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

      1. *-commutative 67.3%

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

   3. Simplified 67.3%

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

   5. Taylor expanded in z around inf 56.8%

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

   if -1.14999999999999998e-68 < z < 3.59999999999999995e-17

   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 62.7%

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

   if 3.59999999999999995e-17 < z < 5.2999999999999999e104

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

   3. Simplified 85.7%

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

      1. clear-num 84.4%

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

      2. associate-/r/ 85.4%

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

      3. sub-neg 85.4%

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

      4. +-commutative 85.4%

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

      5. *-commutative 85.4%

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

      6. distribute-rgt-neg-in 85.4%

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

      7. fma-define 85.4%

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

   6. Applied egg-rr 85.4%

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

   7. Taylor expanded in a around 0 63.2%

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

   8. Taylor expanded in x around 0 52.5%

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

      1. mul-1-neg 52.5%

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

      2. distribute-rgt-neg-out 52.5%

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

   10. Simplified 52.5%

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

   11. Taylor expanded in t around 0 52.8%

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

       1. mul-1-neg 52.8%

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

       2. associate-*r/ 56.1%

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

       3. distribute-rgt-neg-in 56.1%

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

       4. distribute-neg-frac2 56.1%

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

   13. Simplified 56.1%

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

Alternative 7: 54.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-68} \lor \neg \left(z \leq 1.06 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.15e-68) (not (<= z 1.06e-7))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.15e-68) || !(z <= 1.06e-7)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-1.15d-68)) .or. (.not. (z <= 1.06d-7))) then
        tmp = y / a
    else
        tmp = x / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.15e-68) || !(z <= 1.06e-7)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.15e-68) or not (z <= 1.06e-7):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.15e-68) || !(z <= 1.06e-7))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -1.15e-68) || ~((z <= 1.06e-7)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.15e-68], N[Not[LessEqual[z, 1.06e-7]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.14999999999999998e-68 or 1.06e-7 < z

   1. Initial program 70.1%

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

      1. *-commutative 70.1%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in z around inf 52.1%

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

   if -1.14999999999999998e-68 < z < 1.06e-7

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 62.0%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-68} \lor \neg \left(z \leq 1.06 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.1% 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 83.6%

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

   1. *-commutative 83.6%

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

3. Simplified 83.6%

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

5. Taylor expanded in z around 0 36.0%

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

Developer target: 97.3% 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 2024102 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :alt
  (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))))