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

Percentage Accurate: 95.7% → 99.8%

Time: 7.8s

Alternatives: 6

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+265}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (/ (/ (- x) t) z)
   (if (<= (* z t) 5e+265) (/ x (- y (* z t))) (/ (/ (- x) z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = (-x / t) / z;
	} else if ((z * t) <= 5e+265) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-x / z) / t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = (-x / t) / z;
	} else if ((z * t) <= 5e+265) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-x / z) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = (-x / t) / z
	elif (z * t) <= 5e+265:
		tmp = x / (y - (z * t))
	else:
		tmp = (-x / z) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-x) / t) / z);
	elseif (Float64(z * t) <= 5e+265)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(Float64(-x) / z) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = (-x / t) / z;
	elseif ((z * t) <= 5e+265)
		tmp = x / (y - (z * t));
	else
		tmp = (-x / z) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+265], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -inf.0

   1. Initial program 73.2%

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

   2. Taylor expanded in y around 0 73.2%

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

      1. associate-*r/ 73.2%

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

      2. neg-mul-1 73.2%

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

   4. Simplified 73.2%

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

      1. neg-mul-1 73.2%

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

      2. *-commutative 73.2%

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

      3. times-frac 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.9%

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

      2. mul-1-neg 99.9%

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

   8. Simplified 99.9%

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

   if -inf.0 < (*.f64 z t) < 5.0000000000000002e265

   1. Initial program 99.8%

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

   if 5.0000000000000002e265 < (*.f64 z t)

   1. Initial program 65.5%

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

   2. Taylor expanded in y around 0 65.5%

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

      1. associate-*r/ 65.5%

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

      2. neg-mul-1 65.5%

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

   4. Simplified 65.5%

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

      1. neg-mul-1 65.5%

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

      2. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. mul-1-neg 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+265}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternative 2: 78.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{-x}{t}}{z}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq -4 \cdot 10^{-34}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-131}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (/ (- x) t) z)))
   (if (<= (* z t) -1e+52)
     t_1
     (if (<= (* z t) -4e-34)
       (/ x y)
       (if (<= (* z t) -1e-131)
         (/ (- x) (* z t))
         (if (<= (* z t) 2e-11) (/ x y) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (-x / t) / z;
	double tmp;
	if ((z * t) <= -1e+52) {
		tmp = t_1;
	} else if ((z * t) <= -4e-34) {
		tmp = x / y;
	} else if ((z * t) <= -1e-131) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 2e-11) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-x / t) / z
    if ((z * t) <= (-1d+52)) then
        tmp = t_1
    else if ((z * t) <= (-4d-34)) then
        tmp = x / y
    else if ((z * t) <= (-1d-131)) then
        tmp = -x / (z * t)
    else if ((z * t) <= 2d-11) then
        tmp = x / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (-x / t) / z;
	double tmp;
	if ((z * t) <= -1e+52) {
		tmp = t_1;
	} else if ((z * t) <= -4e-34) {
		tmp = x / y;
	} else if ((z * t) <= -1e-131) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 2e-11) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (-x / t) / z
	tmp = 0
	if (z * t) <= -1e+52:
		tmp = t_1
	elif (z * t) <= -4e-34:
		tmp = x / y
	elif (z * t) <= -1e-131:
		tmp = -x / (z * t)
	elif (z * t) <= 2e-11:
		tmp = x / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(-x) / t) / z)
	tmp = 0.0
	if (Float64(z * t) <= -1e+52)
		tmp = t_1;
	elseif (Float64(z * t) <= -4e-34)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= -1e-131)
		tmp = Float64(Float64(-x) / Float64(z * t));
	elseif (Float64(z * t) <= 2e-11)
		tmp = Float64(x / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (-x / t) / z;
	tmp = 0.0;
	if ((z * t) <= -1e+52)
		tmp = t_1;
	elseif ((z * t) <= -4e-34)
		tmp = x / y;
	elseif ((z * t) <= -1e-131)
		tmp = -x / (z * t);
	elseif ((z * t) <= 2e-11)
		tmp = x / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+52], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], -4e-34], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -1e-131], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-11], N[(x / y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -9.9999999999999999e51 or 1.99999999999999988e-11 < (*.f64 z t)

   1. Initial program 88.9%

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

   2. Taylor expanded in y around 0 73.4%

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

      1. associate-*r/ 73.4%

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

      2. neg-mul-1 73.4%

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

   4. Simplified 73.4%

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

      1. neg-mul-1 73.4%

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

      2. *-commutative 73.4%

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

      3. times-frac 80.4%

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

   6. Applied egg-rr 80.4%

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

      1. associate-*l/ 80.4%

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

      2. mul-1-neg 80.4%

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

   8. Simplified 80.4%

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

   if -9.9999999999999999e51 < (*.f64 z t) < -3.99999999999999971e-34 or -9.9999999999999999e-132 < (*.f64 z t) < 1.99999999999999988e-11

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 84.0%

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

   if -3.99999999999999971e-34 < (*.f64 z t) < -9.9999999999999999e-132

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0 68.4%

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

      1. associate-*r/ 68.4%

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

      2. neg-mul-1 68.4%

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

   4. Simplified 68.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq -4 \cdot 10^{-34}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-131}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternative 3: 78.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{-x}{z}}{t}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq -4 \cdot 10^{-34}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-131}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (/ (- x) z) t)))
   (if (<= (* z t) -1e+40)
     t_1
     (if (<= (* z t) -4e-34)
       (/ x y)
       (if (<= (* z t) -1e-131)
         (/ (- x) (* z t))
         (if (<= (* z t) 2e-11) (/ x y) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (-x / z) / t;
	double tmp;
	if ((z * t) <= -1e+40) {
		tmp = t_1;
	} else if ((z * t) <= -4e-34) {
		tmp = x / y;
	} else if ((z * t) <= -1e-131) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 2e-11) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-x / z) / t
    if ((z * t) <= (-1d+40)) then
        tmp = t_1
    else if ((z * t) <= (-4d-34)) then
        tmp = x / y
    else if ((z * t) <= (-1d-131)) then
        tmp = -x / (z * t)
    else if ((z * t) <= 2d-11) then
        tmp = x / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (-x / z) / t;
	double tmp;
	if ((z * t) <= -1e+40) {
		tmp = t_1;
	} else if ((z * t) <= -4e-34) {
		tmp = x / y;
	} else if ((z * t) <= -1e-131) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 2e-11) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (-x / z) / t
	tmp = 0
	if (z * t) <= -1e+40:
		tmp = t_1
	elif (z * t) <= -4e-34:
		tmp = x / y
	elif (z * t) <= -1e-131:
		tmp = -x / (z * t)
	elif (z * t) <= 2e-11:
		tmp = x / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(-x) / z) / t)
	tmp = 0.0
	if (Float64(z * t) <= -1e+40)
		tmp = t_1;
	elseif (Float64(z * t) <= -4e-34)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= -1e-131)
		tmp = Float64(Float64(-x) / Float64(z * t));
	elseif (Float64(z * t) <= 2e-11)
		tmp = Float64(x / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (-x / z) / t;
	tmp = 0.0;
	if ((z * t) <= -1e+40)
		tmp = t_1;
	elseif ((z * t) <= -4e-34)
		tmp = x / y;
	elseif ((z * t) <= -1e-131)
		tmp = -x / (z * t);
	elseif ((z * t) <= 2e-11)
		tmp = x / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+40], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], -4e-34], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -1e-131], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-11], N[(x / y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.00000000000000003e40 or 1.99999999999999988e-11 < (*.f64 z t)

   1. Initial program 89.2%

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

   2. Taylor expanded in y around 0 72.6%

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

      1. associate-*r/ 72.6%

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

      2. neg-mul-1 72.6%

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

   4. Simplified 72.6%

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

      1. neg-mul-1 72.6%

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

      2. times-frac 81.9%

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

   6. Applied egg-rr 81.9%

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

      1. associate-*l/ 81.9%

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

      2. mul-1-neg 81.9%

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

   8. Simplified 81.9%

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

   if -1.00000000000000003e40 < (*.f64 z t) < -3.99999999999999971e-34 or -9.9999999999999999e-132 < (*.f64 z t) < 1.99999999999999988e-11

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 84.5%

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

   if -3.99999999999999971e-34 < (*.f64 z t) < -9.9999999999999999e-132

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0 68.4%

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

      1. associate-*r/ 68.4%

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

      2. neg-mul-1 68.4%

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

   4. Simplified 68.4%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq -4 \cdot 10^{-34}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-131}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternative 4: 62.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+113} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+265}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -1e+113) (not (<= (* z t) 5e+265)))
   (/ x (* z t))
   (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -1e+113) || !((z * t) <= 5e+265)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * t) <= (-1d+113)) .or. (.not. ((z * t) <= 5d+265))) then
        tmp = x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -1e+113) || !((z * t) <= 5e+265)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -1e+113) or not ((z * t) <= 5e+265):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -1e+113) || !(Float64(z * t) <= 5e+265))
		tmp = Float64(x / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -1e+113) || ~(((z * t) <= 5e+265)))
		tmp = x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -1e+113], N[Not[LessEqual[N[(z * t), $MachinePrecision], 5e+265]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1e113 or 5.0000000000000002e265 < (*.f64 z t)

   1. Initial program 79.6%

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

   2. Taylor expanded in y around 0 76.8%

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

      1. associate-*r/ 76.8%

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

      2. neg-mul-1 76.8%

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

   4. Simplified 76.8%

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

      1. neg-mul-1 76.8%

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

      2. times-frac 96.8%

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

   6. Applied egg-rr 96.8%

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

      1. *-commutative 96.8%

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

   8. Simplified 96.8%

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

      1. frac-times 76.8%

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

      2. *-commutative 76.8%

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

      3. neg-mul-1 76.8%

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

      4. add-sqr-sqrt 40.6%

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

      5. sqrt-unprod 62.5%

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

      6. sqr-neg 62.5%

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

      7. sqrt-unprod 27.3%

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

      8. add-sqr-sqrt 57.7%

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

   10. Applied egg-rr 57.7%

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

   if -1e113 < (*.f64 z t) < 5.0000000000000002e265

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 63.1%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+113} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+265}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 72.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+37}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5.5e+37) (/ x y) (if (<= y 3.8e-38) (/ (- x) (* z t)) (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.5e+37) {
		tmp = x / y;
	} else if (y <= 3.8e-38) {
		tmp = -x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-5.5d+37)) then
        tmp = x / y
    else if (y <= 3.8d-38) then
        tmp = -x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.5e+37) {
		tmp = x / y;
	} else if (y <= 3.8e-38) {
		tmp = -x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -5.5e+37:
		tmp = x / y
	elif y <= 3.8e-38:
		tmp = -x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5.5e+37)
		tmp = Float64(x / y);
	elseif (y <= 3.8e-38)
		tmp = Float64(Float64(-x) / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -5.5e+37)
		tmp = x / y;
	elseif (y <= 3.8e-38)
		tmp = -x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5.5e+37], N[(x / y), $MachinePrecision], If[LessEqual[y, 3.8e-38], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.50000000000000016e37 or 3.8e-38 < y

   1. Initial program 93.8%

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

   2. Taylor expanded in y around inf 77.4%

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

   if -5.50000000000000016e37 < y < 3.8e-38

   1. Initial program 95.8%

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

   2. Taylor expanded in y around 0 75.2%

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

      1. associate-*r/ 75.2%

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

      2. neg-mul-1 75.2%

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

   4. Simplified 75.2%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+37}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6: 54.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ x y))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 94.7%

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

2. Taylor expanded in y around inf 51.8%

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

3. Final simplification 51.8%

   \[\leadsto \frac{x}{y} \]

Developer target: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{if}\;x < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x < 2.1378306434876444 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (- (/ y x) (* (/ z x) t)))))
   (if (< x -1.618195973607049e+50)
     t_1
     (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / ((y / x) - ((z / x) * t))
    if (x < (-1.618195973607049d+50)) then
        tmp = t_1
    else if (x < 2.1378306434876444d+131) then
        tmp = x / (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 t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 / ((y / x) - ((z / x) * t))
	tmp = 0
	if x < -1.618195973607049e+50:
		tmp = t_1
	elif x < 2.1378306434876444e+131:
		tmp = x / (y - (z * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(Float64(y / x) - Float64(Float64(z / x) * t)))
	tmp = 0.0
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 / ((y / x) - ((z / x) * t));
	tmp = 0.0;
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = x / (y - (z * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 / N[(N[(y / x), $MachinePrecision] - N[(N[(z / x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[x, -1.618195973607049e+50], t$95$1, If[Less[x, 2.1378306434876444e+131], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2023290 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< x -1.618195973607049e+50) (/ 1.0 (- (/ y x) (* (/ z x) t))) (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) (/ 1.0 (- (/ y x) (* (/ z x) t)))))

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