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

Percentage Accurate: 96.0% → 99.8%

Time: 6.6s

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: 96.0% 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, 0.1× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;{\left(t \cdot \frac{-z}{x}\right)}^{-1}\\ \mathbf{elif}\;z \cdot t \leq 10^{+275}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (pow (* t (/ (- z) x)) -1.0)
   (if (<= (* z t) 1e+275) (/ x (- y (* z t))) (/ (/ (- x) t) z))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = pow((t * (-z / x)), -1.0);
	} else if ((z * t) <= 1e+275) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-x / t) / z;
	}
	return tmp;
}

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.pow((t * (-z / x)), -1.0);
	} else if ((z * t) <= 1e+275) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-x / t) / z;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = math.pow((t * (-z / x)), -1.0)
	elif (z * t) <= 1e+275:
		tmp = x / (y - (z * t))
	else:
		tmp = (-x / t) / z
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(t * Float64(Float64(-z) / x)) ^ -1.0;
	elseif (Float64(z * t) <= 1e+275)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(Float64(-x) / t) / z);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = (t * (-z / x)) ^ -1.0;
	elseif ((z * t) <= 1e+275)
		tmp = x / (y - (z * t));
	else
		tmp = (-x / t) / z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[Power[N[(t * N[((-z) / x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+275], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 72.3%

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

      1. clear-num 72.3%

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

      2. inv-pow 72.3%

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

   3. Applied egg-rr 72.3%

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

   4. Taylor expanded in y around 0 72.3%

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

      1. mul-1-neg 72.3%

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

      2. *-commutative 72.3%

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

      3. associate-*l/ 99.9%

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

      4. distribute-rgt-neg-in 99.9%

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

   6. Simplified 99.9%

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

   if -inf.0 < (*.f64 z t) < 9.9999999999999996e274

   1. Initial program 99.9%

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

   if 9.9999999999999996e274 < (*.f64 z t)

   1. Initial program 68.6%

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

      1. clear-num 68.6%

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

      2. inv-pow 68.6%

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

   3. Applied egg-rr 68.6%

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

   4. Taylor expanded in y around 0 68.6%

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

      1. associate-*r/ 68.6%

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

      2. associate-/r* 99.9%

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

      3. neg-mul-1 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;{\left(t \cdot \frac{-z}{x}\right)}^{-1}\\ \mathbf{elif}\;z \cdot t \leq 10^{+275}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternative 2: 75.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-42} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{-115}\right) \land \left(z \cdot t \leq 2 \cdot 10^{-63} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+41}\right)\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -1e-42)
         (and (not (<= (* z t) 2e-115))
              (or (<= (* z t) 2e-63) (not (<= (* z t) 2e+41)))))
   (/ (- x) (* z t))
   (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -1e-42) || (!((z * t) <= 2e-115) && (((z * t) <= 2e-63) || !((z * t) <= 2e+41)))) {
		tmp = -x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
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-42)) .or. (.not. ((z * t) <= 2d-115)) .and. ((z * t) <= 2d-63) .or. (.not. ((z * t) <= 2d+41))) then
        tmp = -x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -1e-42) || (!((z * t) <= 2e-115) && (((z * t) <= 2e-63) || !((z * t) <= 2e+41)))) {
		tmp = -x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -1e-42) or (not ((z * t) <= 2e-115) and (((z * t) <= 2e-63) or not ((z * t) <= 2e+41))):
		tmp = -x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -1e-42) || (!(Float64(z * t) <= 2e-115) && ((Float64(z * t) <= 2e-63) || !(Float64(z * t) <= 2e+41))))
		tmp = Float64(Float64(-x) / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -1e-42) || (~(((z * t) <= 2e-115)) && (((z * t) <= 2e-63) || ~(((z * t) <= 2e+41)))))
		tmp = -x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -1e-42], And[N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e-115]], $MachinePrecision], Or[LessEqual[N[(z * t), $MachinePrecision], 2e-63], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+41]], $MachinePrecision]]]], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.00000000000000004e-42 or 2.0000000000000001e-115 < (*.f64 z t) < 2.00000000000000013e-63 or 2.00000000000000001e41 < (*.f64 z t)

   1. Initial program 92.3%

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

   2. Taylor expanded in y around 0 74.4%

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

      1. associate-*r/ 74.4%

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

      2. neg-mul-1 74.4%

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

   4. Simplified 74.4%

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

   if -1.00000000000000004e-42 < (*.f64 z t) < 2.0000000000000001e-115 or 2.00000000000000013e-63 < (*.f64 z t) < 2.00000000000000001e41

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 86.0%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-42} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{-115}\right) \land \left(z \cdot t \leq 2 \cdot 10^{-63} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+41}\right)\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3: 77.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{-x}{t}}{z}\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-63}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (/ (- x) t) z)))
   (if (<= (* z t) -5e-37)
     t_1
     (if (<= (* z t) 2e-115)
       (/ x y)
       (if (<= (* z t) 2e-63)
         (/ (- x) (* z t))
         (if (<= (* z t) 2e+41) (/ x y) t_1))))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (-x / t) / z;
	double tmp;
	if ((z * t) <= -5e-37) {
		tmp = t_1;
	} else if ((z * t) <= 2e-115) {
		tmp = x / y;
	} else if ((z * t) <= 2e-63) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 2e+41) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
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) <= (-5d-37)) then
        tmp = t_1
    else if ((z * t) <= 2d-115) then
        tmp = x / y
    else if ((z * t) <= 2d-63) then
        tmp = -x / (z * t)
    else if ((z * t) <= 2d+41) then
        tmp = x / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = (-x / t) / z;
	double tmp;
	if ((z * t) <= -5e-37) {
		tmp = t_1;
	} else if ((z * t) <= 2e-115) {
		tmp = x / y;
	} else if ((z * t) <= 2e-63) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 2e+41) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	t_1 = (-x / t) / z
	tmp = 0
	if (z * t) <= -5e-37:
		tmp = t_1
	elif (z * t) <= 2e-115:
		tmp = x / y
	elif (z * t) <= 2e-63:
		tmp = -x / (z * t)
	elif (z * t) <= 2e+41:
		tmp = x / y
	else:
		tmp = t_1
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(-x) / t) / z)
	tmp = 0.0
	if (Float64(z * t) <= -5e-37)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e-115)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 2e-63)
		tmp = Float64(Float64(-x) / Float64(z * t));
	elseif (Float64(z * t) <= 2e+41)
		tmp = Float64(x / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = (-x / t) / z;
	tmp = 0.0;
	if ((z * t) <= -5e-37)
		tmp = t_1;
	elseif ((z * t) <= 2e-115)
		tmp = x / y;
	elseif ((z * t) <= 2e-63)
		tmp = -x / (z * t);
	elseif ((z * t) <= 2e+41)
		tmp = x / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -5e-37], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e-115], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-63], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+41], N[(x / y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -4.9999999999999997e-37 or 2.00000000000000001e41 < (*.f64 z t)

   1. Initial program 91.5%

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

      1. clear-num 91.0%

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

      2. inv-pow 91.0%

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

   3. Applied egg-rr 91.0%

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

   4. Taylor expanded in y around 0 73.9%

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

      1. associate-*r/ 73.9%

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

      2. associate-/r* 79.6%

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

      3. neg-mul-1 79.6%

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

   6. Simplified 79.6%

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

   if -4.9999999999999997e-37 < (*.f64 z t) < 2.0000000000000001e-115 or 2.00000000000000013e-63 < (*.f64 z t) < 2.00000000000000001e41

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 85.4%

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

   if 2.0000000000000001e-115 < (*.f64 z t) < 2.00000000000000013e-63

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 77.4%

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

      1. associate-*r/ 77.4%

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

      2. neg-mul-1 77.4%

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

   4. Simplified 77.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-63}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternative 4: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 10^{+275}\right):\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) (- INFINITY)) (not (<= (* z t) 1e+275)))
   (/ (/ (- x) t) z)
   (/ x (- y (* z t)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -((double) INFINITY)) || !((z * t) <= 1e+275)) {
		tmp = (-x / t) / z;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -Double.POSITIVE_INFINITY) || !((z * t) <= 1e+275)) {
		tmp = (-x / t) / z;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -math.inf) or not ((z * t) <= 1e+275):
		tmp = (-x / t) / z
	else:
		tmp = x / (y - (z * t))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= Float64(-Inf)) || !(Float64(z * t) <= 1e+275))
		tmp = Float64(Float64(Float64(-x) / t) / z);
	else
		tmp = Float64(x / Float64(y - Float64(z * t)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -Inf) || ~(((z * t) <= 1e+275)))
		tmp = (-x / t) / z;
	else
		tmp = x / (y - (z * t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+275]], $MachinePrecision]], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 9.9999999999999996e274 < (*.f64 z t)

   1. Initial program 70.4%

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

      1. clear-num 70.4%

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

      2. inv-pow 70.4%

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

   3. Applied egg-rr 70.4%

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

   4. Taylor expanded in y around 0 70.4%

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

      1. associate-*r/ 70.4%

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

      2. associate-/r* 99.8%

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

      3. neg-mul-1 99.8%

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

   6. Simplified 99.8%

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

   if -inf.0 < (*.f64 z t) < 9.9999999999999996e274

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 10^{+275}\right):\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

Alternative 5: 62.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+187} \lor \neg \left(z \cdot t \leq 10^{+136}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -2e+187) (not (<= (* z t) 1e+136)))
   (/ x (* z t))
   (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -2e+187) || !((z * t) <= 1e+136)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
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) <= (-2d+187)) .or. (.not. ((z * t) <= 1d+136))) then
        tmp = x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -2e+187) || !((z * t) <= 1e+136)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -2e+187) or not ((z * t) <= 1e+136):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -2e+187) || !(Float64(z * t) <= 1e+136))
		tmp = Float64(x / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -2e+187) || ~(((z * t) <= 1e+136)))
		tmp = x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -2e+187], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+136]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.99999999999999981e187 or 1.00000000000000006e136 < (*.f64 z t)

   1. Initial program 84.5%

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

      1. clear-num 84.4%

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

      2. associate-/r/ 84.4%

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

   3. Applied egg-rr 84.4%

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

   4. Taylor expanded in y around 0 82.0%

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

      1. associate-*l/ 82.1%

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

      2. neg-mul-1 82.1%

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

      3. add-sqr-sqrt 36.3%

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

      4. sqrt-unprod 58.2%

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

      5. sqr-neg 58.2%

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

      6. sqrt-unprod 28.2%

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

      7. add-sqr-sqrt 54.4%

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

      8. *-commutative 54.4%

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

   6. Applied egg-rr 54.4%

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

   if -1.99999999999999981e187 < (*.f64 z t) < 1.00000000000000006e136

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 68.2%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+187} \lor \neg \left(z \cdot t \leq 10^{+136}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6: 53.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (/ x y))

C

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

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
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

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

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	return x / y

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	return Float64(x / y)
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp = code(x, y, z, t)
	tmp = x / y;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 95.8%

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

2. Taylor expanded in y around inf 55.0%

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

3. Final simplification 55.0%

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

Developer target: 96.8% 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 2023196 
(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))))