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

Percentage Accurate: 95.8% → 97.7%

Time: 7.6s

Alternatives: 7

Speedup: 0.6×

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: 95.8% 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, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+232}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \end{array} \end{array} \]

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) 1e+232) (/ x (fma z (- t) y)) (/ (- (/ x t)) z)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 1e+232) {
		tmp = x / fma(z, -t, y);
	} else {
		tmp = -(x / t) / z;
	}
	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) <= 1e+232)
		tmp = Float64(x / fma(z, Float64(-t), y));
	else
		tmp = Float64(Float64(-Float64(x / t)) / z);
	end
	return 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], 1e+232], N[(x / N[(z * (-t) + y), $MachinePrecision]), $MachinePrecision], N[((-N[(x / t), $MachinePrecision]) / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 1.00000000000000006e232

   1. Initial program 99.1%

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

      1. cancel-sign-sub-inv 99.1%

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

      2. +-commutative 99.1%

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

      3. distribute-lft-neg-out 99.1%

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

      4. distribute-rgt-neg-out 99.1%

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

      5. fma-def 99.1%

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

   3. Simplified 99.1%

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

   if 1.00000000000000006e232 < (*.f64 z t)

   1. Initial program 71.4%

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

   2. Taylor expanded in y around 0 71.4%

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

      1. mul-1-neg 71.4%

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

      2. associate-/r* 99.7%

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

      3. distribute-neg-frac 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+232}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \end{array} \]

Alternative 2: 68.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{-x}{z \cdot t}\\ \mathbf{if}\;z \leq -1.12 \cdot 10^{+235}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{+214}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{+105}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -0.031:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \end{array} \end{array} \]

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 (<= z -1.12e+235)
     t_1
     (if (<= z -1.05e+214)
       (/ x y)
       (if (<= z -3.5e+131)
         t_1
         (if (<= z -1.5e+105)
           (/ x y)
           (if (<= z -0.031)
             t_1
             (if (<= z 4.8e-73) (/ x y) (/ (- (/ x t)) z)))))))))

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 (z <= -1.12e+235) {
		tmp = t_1;
	} else if (z <= -1.05e+214) {
		tmp = x / y;
	} else if (z <= -3.5e+131) {
		tmp = t_1;
	} else if (z <= -1.5e+105) {
		tmp = x / y;
	} else if (z <= -0.031) {
		tmp = t_1;
	} else if (z <= 4.8e-73) {
		tmp = x / y;
	} else {
		tmp = -(x / t) / z;
	}
	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 (z <= (-1.12d+235)) then
        tmp = t_1
    else if (z <= (-1.05d+214)) then
        tmp = x / y
    else if (z <= (-3.5d+131)) then
        tmp = t_1
    else if (z <= (-1.5d+105)) then
        tmp = x / y
    else if (z <= (-0.031d0)) then
        tmp = t_1
    else if (z <= 4.8d-73) then
        tmp = x / y
    else
        tmp = -(x / t) / z
    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 (z <= -1.12e+235) {
		tmp = t_1;
	} else if (z <= -1.05e+214) {
		tmp = x / y;
	} else if (z <= -3.5e+131) {
		tmp = t_1;
	} else if (z <= -1.5e+105) {
		tmp = x / y;
	} else if (z <= -0.031) {
		tmp = t_1;
	} else if (z <= 4.8e-73) {
		tmp = x / y;
	} else {
		tmp = -(x / t) / z;
	}
	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 z <= -1.12e+235:
		tmp = t_1
	elif z <= -1.05e+214:
		tmp = x / y
	elif z <= -3.5e+131:
		tmp = t_1
	elif z <= -1.5e+105:
		tmp = x / y
	elif z <= -0.031:
		tmp = t_1
	elif z <= 4.8e-73:
		tmp = x / y
	else:
		tmp = -(x / t) / z
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(-x) / Float64(z * t))
	tmp = 0.0
	if (z <= -1.12e+235)
		tmp = t_1;
	elseif (z <= -1.05e+214)
		tmp = Float64(x / y);
	elseif (z <= -3.5e+131)
		tmp = t_1;
	elseif (z <= -1.5e+105)
		tmp = Float64(x / y);
	elseif (z <= -0.031)
		tmp = t_1;
	elseif (z <= 4.8e-73)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(-Float64(x / t)) / z);
	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 (z <= -1.12e+235)
		tmp = t_1;
	elseif (z <= -1.05e+214)
		tmp = x / y;
	elseif (z <= -3.5e+131)
		tmp = t_1;
	elseif (z <= -1.5e+105)
		tmp = x / y;
	elseif (z <= -0.031)
		tmp = t_1;
	elseif (z <= 4.8e-73)
		tmp = x / y;
	else
		tmp = -(x / t) / z;
	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[z, -1.12e+235], t$95$1, If[LessEqual[z, -1.05e+214], N[(x / y), $MachinePrecision], If[LessEqual[z, -3.5e+131], t$95$1, If[LessEqual[z, -1.5e+105], N[(x / y), $MachinePrecision], If[LessEqual[z, -0.031], t$95$1, If[LessEqual[z, 4.8e-73], N[(x / y), $MachinePrecision], N[((-N[(x / t), $MachinePrecision]) / z), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1199999999999999e235 or -1.05e214 < z < -3.4999999999999999e131 or -1.5e105 < z < -0.031

   1. Initial program 93.1%

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

   2. Taylor expanded in y around 0 74.8%

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

      1. associate-*r/ 74.8%

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

      2. neg-mul-1 74.8%

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

   4. Simplified 74.8%

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

   if -1.1199999999999999e235 < z < -1.05e214 or -3.4999999999999999e131 < z < -1.5e105 or -0.031 < z < 4.80000000000000011e-73

   1. Initial program 99.2%

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

   2. Taylor expanded in y around inf 79.1%

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

   if 4.80000000000000011e-73 < z

   1. Initial program 95.3%

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

   2. Taylor expanded in y around 0 63.3%

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

      1. mul-1-neg 63.3%

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

      2. associate-/r* 64.8%

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

      3. distribute-neg-frac 64.8%

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

   4. Simplified 64.8%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.12 \cdot 10^{+235}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{+214}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{+105}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -0.031:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \end{array} \]

Alternative 3: 69.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-65} \lor \neg \left(y \leq 2.5 \cdot 10^{-147}\right) \land \left(y \leq 4.8 \cdot 10^{-43} \lor \neg \left(y \leq 1.2\right)\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \end{array} \]

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 (or (<= y -4.2e-65)
         (and (not (<= y 2.5e-147)) (or (<= y 4.8e-43) (not (<= y 1.2)))))
   (/ x y)
   (/ (- x) (* z t))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4.2e-65) || (!(y <= 2.5e-147) && ((y <= 4.8e-43) || !(y <= 1.2)))) {
		tmp = x / y;
	} else {
		tmp = -x / (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 ((y <= (-4.2d-65)) .or. (.not. (y <= 2.5d-147)) .and. (y <= 4.8d-43) .or. (.not. (y <= 1.2d0))) then
        tmp = x / y
    else
        tmp = -x / (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 ((y <= -4.2e-65) || (!(y <= 2.5e-147) && ((y <= 4.8e-43) || !(y <= 1.2)))) {
		tmp = x / y;
	} else {
		tmp = -x / (z * t);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (y <= -4.2e-65) or (not (y <= 2.5e-147) and ((y <= 4.8e-43) or not (y <= 1.2))):
		tmp = x / y
	else:
		tmp = -x / (z * t)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -4.2e-65) || (!(y <= 2.5e-147) && ((y <= 4.8e-43) || !(y <= 1.2))))
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(-x) / 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 ((y <= -4.2e-65) || (~((y <= 2.5e-147)) && ((y <= 4.8e-43) || ~((y <= 1.2)))))
		tmp = x / y;
	else
		tmp = -x / (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[Or[LessEqual[y, -4.2e-65], And[N[Not[LessEqual[y, 2.5e-147]], $MachinePrecision], Or[LessEqual[y, 4.8e-43], N[Not[LessEqual[y, 1.2]], $MachinePrecision]]]], N[(x / y), $MachinePrecision], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.20000000000000006e-65 or 2.50000000000000007e-147 < y < 4.8000000000000004e-43 or 1.19999999999999996 < y

   1. Initial program 98.2%

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

   2. Taylor expanded in y around inf 75.8%

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

   if -4.20000000000000006e-65 < y < 2.50000000000000007e-147 or 4.8000000000000004e-43 < y < 1.19999999999999996

   1. Initial program 94.1%

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

   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%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-65} \lor \neg \left(y \leq 2.5 \cdot 10^{-147}\right) \land \left(y \leq 4.8 \cdot 10^{-43} \lor \neg \left(y \leq 1.2\right)\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \]

Alternative 4: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+232}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \end{array} \end{array} \]

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) 1e+232) (/ x (- y (* z t))) (/ (- (/ x t)) z)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 1e+232) {
		tmp = x / (y - (z * t));
	} else {
		tmp = -(x / t) / z;
	}
	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) <= 1d+232) then
        tmp = x / (y - (z * t))
    else
        tmp = -(x / t) / z
    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) <= 1e+232) {
		tmp = x / (y - (z * t));
	} else {
		tmp = -(x / t) / z;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= 1e+232:
		tmp = x / (y - (z * t))
	else:
		tmp = -(x / t) / z
	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) <= 1e+232)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(-Float64(x / t)) / z);
	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) <= 1e+232)
		tmp = x / (y - (z * t));
	else
		tmp = -(x / t) / z;
	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], 1e+232], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(x / t), $MachinePrecision]) / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 1.00000000000000006e232

   1. Initial program 99.1%

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

   if 1.00000000000000006e232 < (*.f64 z t)

   1. Initial program 71.4%

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

   2. Taylor expanded in y around 0 71.4%

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

      1. mul-1-neg 71.4%

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

      2. associate-/r* 99.7%

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

      3. distribute-neg-frac 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+232}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \end{array} \]

Alternative 5: 56.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{+237}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \end{array} \]

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 (<= t 1.1e+237) (/ x y) (/ x (* z t))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.1e+237) {
		tmp = x / y;
	} else {
		tmp = x / (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 (t <= 1.1d+237) then
        tmp = x / y
    else
        tmp = x / (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 (t <= 1.1e+237) {
		tmp = x / y;
	} else {
		tmp = x / (z * t);
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 1.1e+237:
		tmp = x / y
	else:
		tmp = x / (z * t)
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.1e+237)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / 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 (t <= 1.1e+237)
		tmp = x / y;
	else
		tmp = x / (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[t, 1.1e+237], N[(x / y), $MachinePrecision], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.1e237

   1. Initial program 96.8%

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

   2. Taylor expanded in y around inf 59.5%

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

   if 1.1e237 < t

   1. Initial program 95.0%

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

   2. Taylor expanded in y around 0 94.8%

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

      1. mul-1-neg 94.8%

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

      2. associate-/r* 88.8%

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

      3. distribute-neg-frac 88.8%

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

   4. Simplified 88.8%

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

      1. expm1-log1p-u 82.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\frac{x}{t}}{z}\right)\right)} \]

      2. expm1-udef 66.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\frac{x}{t}}{z}\right)} - 1} \]

      3. add-sqr-sqrt 54.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-\frac{x}{t}} \cdot \sqrt{-\frac{x}{t}}}}{z}\right)} - 1 \]

      4. sqrt-unprod 60.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-\frac{x}{t}\right) \cdot \left(-\frac{x}{t}\right)}}}{z}\right)} - 1 \]

      5. sqr-neg 60.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{\frac{x}{t} \cdot \frac{x}{t}}}}{z}\right)} - 1 \]

      6. sqrt-unprod 36.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\frac{x}{t}} \cdot \sqrt{\frac{x}{t}}}}{z}\right)} - 1 \]

      7. add-sqr-sqrt 55.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{x}{t}}}{z}\right)} - 1 \]

   6. Applied egg-rr 55.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{x}{t}}{z}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 55.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{x}{t}}{z}\right)\right)} \]

      2. expm1-log1p 55.1%

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

      3. associate-/r* 55.0%

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

   8. Simplified 55.0%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{+237}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \]

Alternative 6: 57.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 1.38 \cdot 10^{+197}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \end{array} \]

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 (<= t 1.38e+197) (/ x y) (/ (/ x t) z)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.38e+197) {
		tmp = x / y;
	} else {
		tmp = (x / t) / z;
	}
	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 (t <= 1.38d+197) then
        tmp = x / y
    else
        tmp = (x / t) / z
    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 (t <= 1.38e+197) {
		tmp = x / y;
	} else {
		tmp = (x / t) / z;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 1.38e+197:
		tmp = x / y
	else:
		tmp = (x / t) / z
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.38e+197)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / t) / z);
	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 (t <= 1.38e+197)
		tmp = x / y;
	else
		tmp = (x / t) / z;
	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[t, 1.38e+197], N[(x / y), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.38e197

   1. Initial program 96.7%

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

   2. Taylor expanded in y around inf 59.3%

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

   if 1.38e197 < t

   1. Initial program 96.4%

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

   2. Taylor expanded in y around 0 82.7%

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

      1. mul-1-neg 82.7%

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

      2. associate-/r* 78.1%

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

      3. distribute-neg-frac 78.1%

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

   4. Simplified 78.1%

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

      1. expm1-log1p-u 73.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\frac{x}{t}}{z}\right)\right)} \]

      2. expm1-udef 62.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\frac{x}{t}}{z}\right)} - 1} \]

      3. add-sqr-sqrt 53.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-\frac{x}{t}} \cdot \sqrt{-\frac{x}{t}}}}{z}\right)} - 1 \]

      4. sqrt-unprod 58.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-\frac{x}{t}\right) \cdot \left(-\frac{x}{t}\right)}}}{z}\right)} - 1 \]

      5. sqr-neg 58.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{\frac{x}{t} \cdot \frac{x}{t}}}}{z}\right)} - 1 \]

      6. sqrt-unprod 40.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\frac{x}{t}} \cdot \sqrt{\frac{x}{t}}}}{z}\right)} - 1 \]

      7. add-sqr-sqrt 54.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{x}{t}}}{z}\right)} - 1 \]

   6. Applied egg-rr 54.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{x}{t}}{z}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 54.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{x}{t}}{z}\right)\right)} \]

      2. expm1-log1p 54.3%

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

      3. associate-/r* 50.3%

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

   8. Simplified 50.3%

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

      1. frac-2neg 50.3%

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

      2. div-inv 50.3%

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

      3. distribute-lft-neg-in 50.3%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 55.0%

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

      6. sqr-neg 55.0%

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

      7. sqrt-unprod 82.5%

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

      8. add-sqr-sqrt 82.6%

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

   10. Applied egg-rr 82.6%

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

       1. un-div-inv 82.7%

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

       2. associate-/r* 78.1%

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

       3. add-sqr-sqrt 45.9%

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

       4. sqrt-unprod 49.6%

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

       5. sqr-neg 49.6%

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

       6. sqrt-unprod 24.1%

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

       7. add-sqr-sqrt 54.3%

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

   12. Applied egg-rr 54.3%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.38 \cdot 10^{+197}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 7: 54.2% 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 96.7%

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

2. Taylor expanded in y around inf 56.8%

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

3. Final simplification 56.8%

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

Developer target: 96.1% 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 2023332 
(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))))