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

Percentage Accurate: 96.1% → 97.7%

Time: 15.8s

Alternatives: 5

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: 96.1% 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: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+277}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-1}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -2e+277) (* (/ x z) (/ -1.0 t)) (/ x (- y (* z t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e+277) {
		tmp = (x / z) * (-1.0 / t);
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Fortran

NOTE: x, y, 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+277)) then
        tmp = (x / z) * ((-1.0d0) / t)
    else
        tmp = x / (y - (z * t))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e+277) {
		tmp = (x / z) * (-1.0 / t);
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -2e+277:
		tmp = (x / z) * (-1.0 / t)
	else:
		tmp = x / (y - (z * t))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -2e+277)
		tmp = Float64(Float64(x / z) * Float64(-1.0 / t));
	else
		tmp = Float64(x / Float64(y - Float64(z * t)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -2e+277)
		tmp = (x / z) * (-1.0 / t);
	else
		tmp = x / (y - (z * t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+277], N[(N[(x / z), $MachinePrecision] * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.00000000000000001e277

   1. Initial program 70.3%

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

   3. Taylor expanded in t around -inf 99.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{x}{z} + \frac{x \cdot y}{t \cdot {z}^{2}}}{t}} \]
   4. Step-by-step derivation

      1. div-inv 99.7%

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

      2. +-commutative 99.7%

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

      3. times-frac 99.7%

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

      4. fma-define 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in t around inf 99.7%

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

   if -2.00000000000000001e277 < (*.f64 z t)

   1. Initial program 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+277}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-1}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x}{z}}{-t}\\ \mathbf{if}\;y \leq -1.12 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-171}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-49}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (/ x z) (- t))))
   (if (<= y -1.12e+32)
     (/ x y)
     (if (<= y -5.5e-115)
       t_1
       (if (<= y -2.5e-171) (/ x y) (if (<= y 4e-49) t_1 (/ 1.0 (/ y x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (x / z) / -t;
	double tmp;
	if (y <= -1.12e+32) {
		tmp = x / y;
	} else if (y <= -5.5e-115) {
		tmp = t_1;
	} else if (y <= -2.5e-171) {
		tmp = x / y;
	} else if (y <= 4e-49) {
		tmp = t_1;
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Fortran

NOTE: x, y, 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 / z) / -t
    if (y <= (-1.12d+32)) then
        tmp = x / y
    else if (y <= (-5.5d-115)) then
        tmp = t_1
    else if (y <= (-2.5d-171)) then
        tmp = x / y
    else if (y <= 4d-49) then
        tmp = t_1
    else
        tmp = 1.0d0 / (y / x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = (x / z) / -t;
	double tmp;
	if (y <= -1.12e+32) {
		tmp = x / y;
	} else if (y <= -5.5e-115) {
		tmp = t_1;
	} else if (y <= -2.5e-171) {
		tmp = x / y;
	} else if (y <= 4e-49) {
		tmp = t_1;
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = (x / z) / -t
	tmp = 0
	if y <= -1.12e+32:
		tmp = x / y
	elif y <= -5.5e-115:
		tmp = t_1
	elif y <= -2.5e-171:
		tmp = x / y
	elif y <= 4e-49:
		tmp = t_1
	else:
		tmp = 1.0 / (y / x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(x / z) / Float64(-t))
	tmp = 0.0
	if (y <= -1.12e+32)
		tmp = Float64(x / y);
	elseif (y <= -5.5e-115)
		tmp = t_1;
	elseif (y <= -2.5e-171)
		tmp = Float64(x / y);
	elseif (y <= 4e-49)
		tmp = t_1;
	else
		tmp = Float64(1.0 / Float64(y / x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = (x / z) / -t;
	tmp = 0.0;
	if (y <= -1.12e+32)
		tmp = x / y;
	elseif (y <= -5.5e-115)
		tmp = t_1;
	elseif (y <= -2.5e-171)
		tmp = x / y;
	elseif (y <= 4e-49)
		tmp = t_1;
	else
		tmp = 1.0 / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, 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 / z), $MachinePrecision] / (-t)), $MachinePrecision]}, If[LessEqual[y, -1.12e+32], N[(x / y), $MachinePrecision], If[LessEqual[y, -5.5e-115], t$95$1, If[LessEqual[y, -2.5e-171], N[(x / y), $MachinePrecision], If[LessEqual[y, 4e-49], t$95$1, N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.12000000000000007e32 or -5.50000000000000028e-115 < y < -2.49999999999999996e-171

   1. Initial program 99.0%

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

   3. Taylor expanded in y around inf 86.8%

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

   if -1.12000000000000007e32 < y < -5.50000000000000028e-115 or -2.49999999999999996e-171 < y < 3.99999999999999975e-49

   1. Initial program 95.4%

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

   3. Taylor expanded in t around -inf 69.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{x}{z} + \frac{x \cdot y}{t \cdot {z}^{2}}}{t}} \]
   4. Step-by-step derivation

      1. div-inv 69.6%

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

      2. +-commutative 69.6%

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

      3. times-frac 64.8%

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

      4. fma-define 64.8%

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

   5. Applied egg-rr 64.8%

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

   6. Taylor expanded in t around inf 77.9%

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

      1. un-div-inv 78.1%

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

   8. Applied egg-rr 78.1%

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

   if 3.99999999999999975e-49 < y

   1. Initial program 98.9%

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

      1. add-cube-cbrt 97.4%

         \[\leadsto \frac{x}{\color{blue}{\left(\sqrt[3]{y - z \cdot t} \cdot \sqrt[3]{y - z \cdot t}\right) \cdot \sqrt[3]{y - z \cdot t}}} \]

      2. pow3 97.5%

         \[\leadsto \frac{x}{\color{blue}{{\left(\sqrt[3]{y - z \cdot t}\right)}^{3}}} \]

   4. Applied egg-rr 97.5%

      \[\leadsto \frac{x}{\color{blue}{{\left(\sqrt[3]{y - z \cdot t}\right)}^{3}}} \]
   5. Step-by-step derivation

      1. rem-cube-cbrt 98.9%

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

      2. clear-num 98.2%

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

      3. inv-pow 98.2%

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

      4. *-commutative 98.2%

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

   6. Applied egg-rr 98.2%

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

      1. unpow-1 98.2%

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

      2. *-commutative 98.2%

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

   8. Simplified 98.2%

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

   9. Taylor expanded in y around inf 88.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{y}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-171}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-49}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{z \cdot \left(-t\right)}\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-171}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-49}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* z (- t)))))
   (if (<= y -1.2e+32)
     (/ x y)
     (if (<= y -8.5e-115)
       t_1
       (if (<= y -2.5e-171) (/ x y) (if (<= y 3e-49) t_1 (/ 1.0 (/ y x))))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / (z * -t);
	double tmp;
	if (y <= -1.2e+32) {
		tmp = x / y;
	} else if (y <= -8.5e-115) {
		tmp = t_1;
	} else if (y <= -2.5e-171) {
		tmp = x / y;
	} else if (y <= 3e-49) {
		tmp = t_1;
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Fortran

NOTE: x, y, 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 / (z * -t)
    if (y <= (-1.2d+32)) then
        tmp = x / y
    else if (y <= (-8.5d-115)) then
        tmp = t_1
    else if (y <= (-2.5d-171)) then
        tmp = x / y
    else if (y <= 3d-49) then
        tmp = t_1
    else
        tmp = 1.0d0 / (y / x)
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = x / (z * -t);
	double tmp;
	if (y <= -1.2e+32) {
		tmp = x / y;
	} else if (y <= -8.5e-115) {
		tmp = t_1;
	} else if (y <= -2.5e-171) {
		tmp = x / y;
	} else if (y <= 3e-49) {
		tmp = t_1;
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / (z * -t)
	tmp = 0
	if y <= -1.2e+32:
		tmp = x / y
	elif y <= -8.5e-115:
		tmp = t_1
	elif y <= -2.5e-171:
		tmp = x / y
	elif y <= 3e-49:
		tmp = t_1
	else:
		tmp = 1.0 / (y / x)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(z * Float64(-t)))
	tmp = 0.0
	if (y <= -1.2e+32)
		tmp = Float64(x / y);
	elseif (y <= -8.5e-115)
		tmp = t_1;
	elseif (y <= -2.5e-171)
		tmp = Float64(x / y);
	elseif (y <= 3e-49)
		tmp = t_1;
	else
		tmp = Float64(1.0 / Float64(y / x));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / (z * -t);
	tmp = 0.0;
	if (y <= -1.2e+32)
		tmp = x / y;
	elseif (y <= -8.5e-115)
		tmp = t_1;
	elseif (y <= -2.5e-171)
		tmp = x / y;
	elseif (y <= 3e-49)
		tmp = t_1;
	else
		tmp = 1.0 / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(z * (-t)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.2e+32], N[(x / y), $MachinePrecision], If[LessEqual[y, -8.5e-115], t$95$1, If[LessEqual[y, -2.5e-171], N[(x / y), $MachinePrecision], If[LessEqual[y, 3e-49], t$95$1, N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.19999999999999996e32 or -8.49999999999999953e-115 < y < -2.49999999999999996e-171

   1. Initial program 99.0%

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

   3. Taylor expanded in y around inf 86.8%

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

   if -1.19999999999999996e32 < y < -8.49999999999999953e-115 or -2.49999999999999996e-171 < y < 3e-49

   1. Initial program 95.4%

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

   3. Taylor expanded in y around 0 77.3%

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

      1. associate-*r/ 77.3%

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

      2. neg-mul-1 77.3%

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

   5. Simplified 77.3%

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

   if 3e-49 < y

   1. Initial program 98.9%

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

      1. add-cube-cbrt 97.4%

         \[\leadsto \frac{x}{\color{blue}{\left(\sqrt[3]{y - z \cdot t} \cdot \sqrt[3]{y - z \cdot t}\right) \cdot \sqrt[3]{y - z \cdot t}}} \]

      2. pow3 97.5%

         \[\leadsto \frac{x}{\color{blue}{{\left(\sqrt[3]{y - z \cdot t}\right)}^{3}}} \]

   4. Applied egg-rr 97.5%

      \[\leadsto \frac{x}{\color{blue}{{\left(\sqrt[3]{y - z \cdot t}\right)}^{3}}} \]
   5. Step-by-step derivation

      1. rem-cube-cbrt 98.9%

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

      2. clear-num 98.2%

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

      3. inv-pow 98.2%

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

      4. *-commutative 98.2%

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

   6. Applied egg-rr 98.2%

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

      1. unpow-1 98.2%

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

      2. *-commutative 98.2%

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

   8. Simplified 98.2%

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

   9. Taylor expanded in y around inf 88.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{y}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{x}{z \cdot \left(-t\right)}\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{-171}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-49}:\\ \;\;\;\;\frac{x}{z \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+277}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -2e+277) (/ (/ x z) (- t)) (/ x (- y (* z t)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e+277) {
		tmp = (x / z) / -t;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Fortran

NOTE: x, y, 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+277)) then
        tmp = (x / z) / -t
    else
        tmp = x / (y - (z * t))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e+277) {
		tmp = (x / z) / -t;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -2e+277:
		tmp = (x / z) / -t
	else:
		tmp = x / (y - (z * t))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -2e+277)
		tmp = Float64(Float64(x / z) / Float64(-t));
	else
		tmp = Float64(x / Float64(y - Float64(z * t)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -2e+277)
		tmp = (x / z) / -t;
	else
		tmp = x / (y - (z * t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+277], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.00000000000000001e277

   1. Initial program 70.3%

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

   3. Taylor expanded in t around -inf 99.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{x}{z} + \frac{x \cdot y}{t \cdot {z}^{2}}}{t}} \]
   4. Step-by-step derivation

      1. div-inv 99.7%

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

      2. +-commutative 99.7%

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

      3. times-frac 99.7%

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

      4. fma-define 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in t around inf 99.7%

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

      1. un-div-inv 99.7%

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

   8. Applied egg-rr 99.7%

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

   if -2.00000000000000001e277 < (*.f64 z t)

   1. Initial program 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+277}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x, y, 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 97.5%

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

3. Taylor expanded in y around inf 62.5%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
4. Add Preprocessing

Developer target: 96.1% accurate, 0.3× 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 2024111 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (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))))