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

Percentage Accurate: 95.8% → 97.9%

Time: 10.5s

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: 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: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+287}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{x}{z}}{t}\\ \end{array} \end{array} \]

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) 1e+287) (/ x (- y (* z t))) (/ (- (/ x 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) <= 1e+287) {
		tmp = x / (y - (z * t));
	} 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 ((z * t) <= 1d+287) then
        tmp = x / (y - (z * t))
    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 ((z * t) <= 1e+287) {
		tmp = x / (y - (z * t));
	} 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 (z * t) <= 1e+287:
		tmp = x / (y - (z * t))
	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 (Float64(z * t) <= 1e+287)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(-Float64(x / 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) <= 1e+287)
		tmp = x / (y - (z * t));
	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[N[(z * t), $MachinePrecision], 1e+287], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(x / z), $MachinePrecision]) / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 1.0000000000000001e287

   1. Initial program 99.1%

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

   if 1.0000000000000001e287 < (*.f64 z t)

   1. Initial program 62.3%

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

   3. Taylor expanded in y around 0 62.3%

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

      1. associate-*r/ 62.3%

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

      2. neg-mul-1 62.3%

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

      3. *-commutative 62.3%

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

      4. associate-/r* 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+287}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-70} \lor \neg \left(t \leq 2.1 \cdot 10^{+59}\right):\\ \;\;\;\;-\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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 (<= t -1.05e-70) (not (<= t 2.1e+59))) (- (/ x (* z t))) (/ x y)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.05e-70) || !(t <= 2.1e+59)) {
		tmp = -(x / (z * t));
	} else {
		tmp = x / y;
	}
	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 <= (-1.05d-70)) .or. (.not. (t <= 2.1d+59))) then
        tmp = -(x / (z * t))
    else
        tmp = x / y
    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 <= -1.05e-70) || !(t <= 2.1e+59)) {
		tmp = -(x / (z * t));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (t <= -1.05e-70) or not (t <= 2.1e+59):
		tmp = -(x / (z * t))
	else:
		tmp = x / y
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.05e-70) || !(t <= 2.1e+59))
		tmp = Float64(-Float64(x / Float64(z * t)));
	else
		tmp = Float64(x / y);
	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 <= -1.05e-70) || ~((t <= 2.1e+59)))
		tmp = -(x / (z * t));
	else
		tmp = x / y;
	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[t, -1.05e-70], N[Not[LessEqual[t, 2.1e+59]], $MachinePrecision]], (-N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.0500000000000001e-70 or 2.09999999999999984e59 < t

   1. Initial program 93.9%

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

   3. Taylor expanded in y around 0 72.2%

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

      1. associate-*r/ 72.2%

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

      2. neg-mul-1 72.2%

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

   5. Simplified 72.2%

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

   if -1.0500000000000001e-70 < t < 2.09999999999999984e59

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 77.5%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-70} \lor \neg \left(t \leq 2.1 \cdot 10^{+59}\right):\\ \;\;\;\;-\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-67} \lor \neg \left(t \leq 3.4 \cdot 10^{+104}\right):\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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 (<= t -2.5e-67) (not (<= t 3.4e+104))) (/ (/ x t) (- z)) (/ x y)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.5e-67) || !(t <= 3.4e+104)) {
		tmp = (x / t) / -z;
	} else {
		tmp = x / y;
	}
	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 <= (-2.5d-67)) .or. (.not. (t <= 3.4d+104))) then
        tmp = (x / t) / -z
    else
        tmp = x / y
    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 <= -2.5e-67) || !(t <= 3.4e+104)) {
		tmp = (x / t) / -z;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (t <= -2.5e-67) or not (t <= 3.4e+104):
		tmp = (x / t) / -z
	else:
		tmp = x / y
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -2.5e-67) || !(t <= 3.4e+104))
		tmp = Float64(Float64(x / t) / Float64(-z));
	else
		tmp = Float64(x / y);
	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 <= -2.5e-67) || ~((t <= 3.4e+104)))
		tmp = (x / t) / -z;
	else
		tmp = x / y;
	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[t, -2.5e-67], N[Not[LessEqual[t, 3.4e+104]], $MachinePrecision]], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.4999999999999999e-67 or 3.3999999999999997e104 < t

   1. Initial program 93.3%

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

   3. Taylor expanded in y around 0 72.6%

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

      1. associate-*r/ 72.6%

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

      2. neg-mul-1 72.6%

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

      3. *-commutative 72.6%

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

      4. associate-/r* 76.1%

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

   5. Simplified 76.1%

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

      1. associate-/l/ 72.6%

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

      2. neg-mul-1 72.6%

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

      3. times-frac 76.0%

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

   7. Applied egg-rr 76.0%

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

      1. frac-times 72.6%

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

      2. neg-mul-1 72.6%

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

      3. add-sqr-sqrt 43.0%

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

      4. sqrt-unprod 45.2%

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

      5. sqr-neg 45.2%

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

      6. sqrt-unprod 12.6%

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

      7. add-sqr-sqrt 28.5%

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

      8. associate-/r* 29.3%

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

      9. frac-2neg 29.3%

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

      10. distribute-frac-neg 29.3%

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

      11. add-sqr-sqrt 15.9%

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

      12. sqrt-unprod 38.8%

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

      13. sqr-neg 38.8%

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

      14. sqrt-unprod 32.5%

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

      15. add-sqr-sqrt 76.1%

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

   9. Applied egg-rr 76.1%

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

   if -2.4999999999999999e-67 < t < 3.3999999999999997e104

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 75.2%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-67} \lor \neg \left(t \leq 3.4 \cdot 10^{+104}\right):\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-67} \lor \neg \left(t \leq 8 \cdot 10^{+184}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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 (<= t -2.5e-67) (not (<= t 8e+184))) (/ x (* z t)) (/ x y)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.5e-67) || !(t <= 8e+184)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	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 <= (-2.5d-67)) .or. (.not. (t <= 8d+184))) then
        tmp = x / (z * t)
    else
        tmp = x / y
    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 <= -2.5e-67) || !(t <= 8e+184)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (t <= -2.5e-67) or not (t <= 8e+184):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -2.5e-67) || !(t <= 8e+184))
		tmp = Float64(x / Float64(z * t));
	else
		tmp = Float64(x / y);
	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 <= -2.5e-67) || ~((t <= 8e+184)))
		tmp = x / (z * t);
	else
		tmp = x / y;
	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[t, -2.5e-67], N[Not[LessEqual[t, 8e+184]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.4999999999999999e-67 or 8.00000000000000014e184 < t

   1. Initial program 93.9%

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

   3. Taylor expanded in y around 0 72.5%

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

      1. associate-*r/ 72.5%

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

      2. neg-mul-1 72.5%

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

      3. *-commutative 72.5%

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

      4. associate-/r* 75.7%

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

   5. Simplified 75.7%

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

      1. expm1-log1p-u 68.0%

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

      2. expm1-udef 40.7%

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

      3. div-inv 40.7%

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

      4. div-inv 40.7%

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

      5. add-sqr-sqrt 22.6%

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

      6. sqrt-unprod 31.8%

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

      7. sqr-neg 31.8%

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

      8. sqrt-unprod 14.8%

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

      9. add-sqr-sqrt 31.1%

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

   7. Applied egg-rr 31.1%

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

      1. expm1-def 27.8%

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

      2. expm1-log1p 28.0%

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

      3. associate-/l/ 27.2%

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

   9. Simplified 27.2%

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

   if -2.4999999999999999e-67 < t < 8.00000000000000014e184

   1. Initial program 98.7%

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

   3. Taylor expanded in y around inf 70.0%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-67} \lor \neg \left(t \leq 8 \cdot 10^{+184}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.3% 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 96.8%

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

3. Taylor expanded in y around inf 54.6%

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

4. Final simplification 54.6%

   \[\leadsto \frac{x}{y} \]
5. Add Preprocessing

Developer target: 96.8% 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 2024026 
(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))))