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

Percentage Accurate: 95.9% → 97.6%

Time: 8.6s

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: 95.9% 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: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+217}:\\ \;\;\;\;-\frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \end{array} \]

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) -1e+217) (- (/ (/ 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) <= -1e+217) {
		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) <= (-1d+217)) 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) <= -1e+217) {
		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) <= -1e+217:
		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) <= -1e+217)
		tmp = Float64(-Float64(Float64(x / z) / 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) <= -1e+217)
		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], -1e+217], (-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) < -9.9999999999999996e216

   1. Initial program 80.5%

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

   3. Taylor expanded in y around 0 80.5%

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

      1. mul-1-neg 80.5%

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

      2. associate-/r* 99.9%

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

      3. distribute-neg-frac2 99.9%

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

   5. Simplified 99.9%

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

      1. distribute-frac-neg2 99.9%

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

      2. add-sqr-sqrt 46.0%

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

      3. sqrt-unprod 75.8%

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

      4. sqr-neg 75.8%

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

      5. sqrt-unprod 34.3%

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

      6. add-sqr-sqrt 63.2%

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

      7. associate-/l/ 63.5%

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

      8. associate-/r* 63.2%

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

      9. add-sqr-sqrt 34.3%

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

      10. sqrt-unprod 75.7%

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

      11. sqr-neg 75.7%

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

      12. sqrt-unprod 46.0%

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

      13. add-sqr-sqrt 99.7%

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

   7. Applied egg-rr 99.7%

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

   if -9.9999999999999996e216 < (*.f64 z t)

   1. Initial program 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+217}:\\ \;\;\;\;-\frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x}{t}}{-z}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-103}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+31}:\\ \;\;\;\;-\frac{\frac{x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+72}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 t) (- z))))
   (if (<= (* z t) -1e-8)
     t_1
     (if (<= (* z t) 2e-103)
       (/ x y)
       (if (<= (* z t) 4e+31)
         (- (/ (/ x z) t))
         (if (<= (* z t) 1e+72) (/ x y) t_1))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (x / t) / -z;
	double tmp;
	if ((z * t) <= -1e-8) {
		tmp = t_1;
	} else if ((z * t) <= 2e-103) {
		tmp = x / y;
	} else if ((z * t) <= 4e+31) {
		tmp = -((x / z) / t);
	} else if ((z * t) <= 1e+72) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	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 / t) / -z
    if ((z * t) <= (-1d-8)) then
        tmp = t_1
    else if ((z * t) <= 2d-103) then
        tmp = x / y
    else if ((z * t) <= 4d+31) then
        tmp = -((x / z) / t)
    else if ((z * t) <= 1d+72) then
        tmp = x / y
    else
        tmp = t_1
    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 / t) / -z;
	double tmp;
	if ((z * t) <= -1e-8) {
		tmp = t_1;
	} else if ((z * t) <= 2e-103) {
		tmp = x / y;
	} else if ((z * t) <= 4e+31) {
		tmp = -((x / z) / t);
	} else if ((z * t) <= 1e+72) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = (x / t) / -z
	tmp = 0
	if (z * t) <= -1e-8:
		tmp = t_1
	elif (z * t) <= 2e-103:
		tmp = x / y
	elif (z * t) <= 4e+31:
		tmp = -((x / z) / t)
	elif (z * t) <= 1e+72:
		tmp = x / y
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(x / t) / Float64(-z))
	tmp = 0.0
	if (Float64(z * t) <= -1e-8)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e-103)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 4e+31)
		tmp = Float64(-Float64(Float64(x / z) / t));
	elseif (Float64(z * t) <= 1e+72)
		tmp = Float64(x / y);
	else
		tmp = t_1;
	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 / t) / -z;
	tmp = 0.0;
	if ((z * t) <= -1e-8)
		tmp = t_1;
	elseif ((z * t) <= 2e-103)
		tmp = x / y;
	elseif ((z * t) <= 4e+31)
		tmp = -((x / z) / t);
	elseif ((z * t) <= 1e+72)
		tmp = x / y;
	else
		tmp = t_1;
	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 / t), $MachinePrecision] / (-z)), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e-8], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e-103], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 4e+31], (-N[(N[(x / z), $MachinePrecision] / t), $MachinePrecision]), If[LessEqual[N[(z * t), $MachinePrecision], 1e+72], N[(x / y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1e-8 or 9.99999999999999944e71 < (*.f64 z t)

   1. Initial program 92.0%

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

   3. Taylor expanded in y around 0 80.4%

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

      1. mul-1-neg 80.4%

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

      2. associate-/r* 85.6%

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

      3. distribute-neg-frac2 85.6%

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

   5. Simplified 85.6%

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

   if -1e-8 < (*.f64 z t) < 1.99999999999999992e-103 or 3.9999999999999999e31 < (*.f64 z t) < 9.99999999999999944e71

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 82.6%

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

   if 1.99999999999999992e-103 < (*.f64 z t) < 3.9999999999999999e31

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 61.8%

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

      1. mul-1-neg 61.8%

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

      2. associate-/r* 54.6%

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

      3. distribute-neg-frac2 54.6%

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

   5. Simplified 54.6%

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

      1. distribute-frac-neg2 54.6%

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

      2. add-sqr-sqrt 15.1%

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

      3. sqrt-unprod 12.3%

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

      4. sqr-neg 12.3%

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

      5. sqrt-unprod 8.1%

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

      6. add-sqr-sqrt 9.4%

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

      7. associate-/l/ 2.8%

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

      8. associate-/r* 2.9%

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

      9. add-sqr-sqrt 1.7%

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

      10. sqrt-unprod 9.6%

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

      11. sqr-neg 9.6%

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

      12. sqrt-unprod 24.7%

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

      13. add-sqr-sqrt 49.6%

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

   7. Applied egg-rr 49.6%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-103}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+31}:\\ \;\;\;\;-\frac{\frac{x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+72}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+201} \lor \neg \left(z \cdot t \leq 10^{+156}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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 (<= (* z t) -5e+201) (not (<= (* z t) 1e+156)))
   (/ 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 (((z * t) <= -5e+201) || !((z * t) <= 1e+156)) {
		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 (((z * t) <= (-5d+201)) .or. (.not. ((z * t) <= 1d+156))) 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 (((z * t) <= -5e+201) || !((z * t) <= 1e+156)) {
		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 ((z * t) <= -5e+201) or not ((z * t) <= 1e+156):
		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 ((Float64(z * t) <= -5e+201) || !(Float64(z * t) <= 1e+156))
		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 (((z * t) <= -5e+201) || ~(((z * t) <= 1e+156)))
		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[N[(z * t), $MachinePrecision], -5e+201], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+156]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.9999999999999995e201 or 9.9999999999999998e155 < (*.f64 z t)

   1. Initial program 86.3%

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

   3. Taylor expanded in y around 0 84.9%

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

      1. mul-1-neg 84.9%

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

      2. associate-/r* 98.4%

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

      3. distribute-neg-frac2 98.4%

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

   5. Simplified 98.4%

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

      1. *-un-lft-identity 98.4%

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

      2. associate-/l/ 84.9%

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

      3. associate-/r* 98.3%

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

      4. add-sqr-sqrt 49.1%

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

      5. sqrt-unprod 74.3%

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

      6. sqr-neg 74.3%

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

      7. sqrt-unprod 32.3%

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

      8. add-sqr-sqrt 61.9%

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

   7. Applied egg-rr 61.9%

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

      1. *-lft-identity 61.9%

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

      2. associate-/l/ 62.1%

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

   9. Simplified 62.1%

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

   if -4.9999999999999995e201 < (*.f64 z t) < 9.9999999999999998e155

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf 63.2%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+201} \lor \neg \left(z \cdot t \leq 10^{+156}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+28} \lor \neg \left(z \leq 6.8 \cdot 10^{-50}\right):\\ \;\;\;\;-\frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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 (<= z -1.3e+28) (not (<= z 6.8e-50))) (- (/ (/ 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 ((z <= -1.3e+28) || !(z <= 6.8e-50)) {
		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 ((z <= (-1.3d+28)) .or. (.not. (z <= 6.8d-50))) 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 ((z <= -1.3e+28) || !(z <= 6.8e-50)) {
		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 (z <= -1.3e+28) or not (z <= 6.8e-50):
		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 ((z <= -1.3e+28) || !(z <= 6.8e-50))
		tmp = Float64(-Float64(Float64(x / 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 ((z <= -1.3e+28) || ~((z <= 6.8e-50)))
		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[z, -1.3e+28], N[Not[LessEqual[z, 6.8e-50]], $MachinePrecision]], (-N[(N[(x / z), $MachinePrecision] / t), $MachinePrecision]), N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3000000000000001e28 or 6.80000000000000029e-50 < z

   1. Initial program 93.4%

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

   3. Taylor expanded in y around 0 68.1%

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

      1. mul-1-neg 68.1%

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

      2. associate-/r* 73.0%

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

      3. distribute-neg-frac2 73.0%

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

   5. Simplified 73.0%

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

      1. distribute-frac-neg2 73.0%

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

      2. add-sqr-sqrt 38.1%

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

      3. sqrt-unprod 47.8%

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

      4. sqr-neg 47.8%

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

      5. sqrt-unprod 15.2%

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

      6. add-sqr-sqrt 33.9%

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

      7. associate-/l/ 34.1%

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

      8. associate-/r* 36.2%

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

      9. add-sqr-sqrt 15.3%

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

      10. sqrt-unprod 47.8%

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

      11. sqr-neg 47.8%

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

      12. sqrt-unprod 38.8%

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

      13. add-sqr-sqrt 71.8%

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

   7. Applied egg-rr 71.8%

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

   if -1.3000000000000001e28 < z < 6.80000000000000029e-50

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 69.8%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+28} \lor \neg \left(z \leq 6.8 \cdot 10^{-50}\right):\\ \;\;\;\;-\frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.0% 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 96.5%

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

3. Taylor expanded in y around inf 53.0%

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

4. Final simplification 53.0%

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

Developer target: 96.5% 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 2024047 
(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))))