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

Percentage Accurate: 96.0% → 98.1%

Time: 7.7s

Alternatives: 6

Speedup: 0.5×

Specification

Math

\[\begin{array}{l} \\ \frac{x}{y - z \cdot t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ x (- y (* z t))))

C

double code(double x, double y, double z, double t) {
	return x / (y - (z * t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x / (y - (z * t))
end function

Java

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

Python

def code(x, y, z, t):
	return x / (y - (z * t))

Julia

function code(x, y, z, t)
	return Float64(x / Float64(y - Float64(z * t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x / (y - (z * t));
end

Wolfram

code[x_, y_, z_, t_] := N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y - z \cdot t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ x (- y (* z t))))

C

double code(double x, double y, double z, double t) {
	return x / (y - (z * t));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x / (y - (z * t))
end function

Java

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

Python

def code(x, y, z, t):
	return x / (y - (z * t))

Julia

function code(x, y, z, t)
	return Float64(x / Float64(y - Float64(z * t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x / (y - (z * t));
end

Wolfram

code[x_, y_, z_, t_] := N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{z}}{t}\\ \end{array} \end{array} \]

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) 5e+305) (/ x (fma z (- t) 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 ((z * t) <= 5e+305) {
		tmp = x / fma(z, -t, 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 (Float64(z * t) <= 5e+305)
		tmp = Float64(x / fma(z, Float64(-t), y));
	else
		tmp = Float64(-Float64(Float64(x / z) / t));
	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], 5e+305], N[(x / N[(z * (-t) + y), $MachinePrecision]), $MachinePrecision], (-N[(N[(x / z), $MachinePrecision] / t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 5.00000000000000009e305

   1. Initial program 98.2%

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

      1. cancel-sign-sub-inv 98.2%

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

      2. +-commutative 98.2%

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

      3. distribute-lft-neg-out 98.2%

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

      4. distribute-rgt-neg-out 98.2%

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

      5. fma-define 98.2%

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

   3. Simplified 98.2%

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

   if 5.00000000000000009e305 < (*.f64 z t)

   1. Initial program 75.5%

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

   3. Taylor expanded in y around 0 75.5%

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

      1. +-commutative 75.5%

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

      2. mul-1-neg 75.5%

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

      3. unsub-neg 75.5%

         \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{{t}^{2} \cdot {z}^{2}} - \frac{x}{t \cdot z}} \]

      4. associate-*r/ 75.5%

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

      5. associate-*r* 75.5%

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

      6. neg-mul-1 75.5%

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

      7. *-commutative 75.5%

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

      8. unpow2 75.5%

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

      9. unpow2 75.5%

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

      10. swap-sqr 75.5%

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

      11. unpow2 75.5%

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

      12. *-commutative 75.5%

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

      13. *-commutative 75.5%

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

      14. associate-/r* 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 75.5%

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

      1. associate-*r/ 75.5%

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

      2. times-frac 99.9%

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

      3. associate-*l/ 99.9%

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

      4. mul-1-neg 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{-6} \lor \neg \left(y \leq 10^{-21}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \end{array} \]

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 -7.4e-6) (not (<= y 1e-21))) (/ 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 <= -7.4e-6) || !(y <= 1e-21)) {
		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 <= (-7.4d-6)) .or. (.not. (y <= 1d-21))) 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 <= -7.4e-6) || !(y <= 1e-21)) {
		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 <= -7.4e-6) or not (y <= 1e-21):
		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 <= -7.4e-6) || !(y <= 1e-21))
		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 <= -7.4e-6) || ~((y <= 1e-21)))
		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, -7.4e-6], N[Not[LessEqual[y, 1e-21]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.4000000000000003e-6 or 9.99999999999999908e-22 < y

   1. Initial program 96.9%

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

   3. Taylor expanded in y around inf 81.5%

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

   if -7.4000000000000003e-6 < y < 9.99999999999999908e-22

   1. Initial program 95.4%

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

   3. Taylor expanded in y around 0 72.9%

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

      1. associate-*r/ 72.9%

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

      2. neg-mul-1 72.9%

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

   5. Simplified 72.9%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{-6} \lor \neg \left(y \leq 10^{-21}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-35} \lor \neg \left(t \leq 2.8 \cdot 10^{+47}\right):\\ \;\;\;\;-\frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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 (<= t -7.2e-35) (not (<= t 2.8e+47))) (- (/ (/ 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 <= -7.2e-35) || !(t <= 2.8e+47)) {
		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 <= (-7.2d-35)) .or. (.not. (t <= 2.8d+47))) 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 <= -7.2e-35) || !(t <= 2.8e+47)) {
		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 <= -7.2e-35) or not (t <= 2.8e+47):
		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 <= -7.2e-35) || !(t <= 2.8e+47))
		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 ((t <= -7.2e-35) || ~((t <= 2.8e+47)))
		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, -7.2e-35], N[Not[LessEqual[t, 2.8e+47]], $MachinePrecision]], (-N[(N[(x / z), $MachinePrecision] / t), $MachinePrecision]), N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.20000000000000038e-35 or 2.79999999999999988e47 < t

   1. Initial program 92.0%

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

   3. Taylor expanded in y around 0 52.1%

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

      1. +-commutative 52.1%

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

      2. mul-1-neg 52.1%

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

      3. unsub-neg 52.1%

         \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{{t}^{2} \cdot {z}^{2}} - \frac{x}{t \cdot z}} \]

      4. associate-*r/ 52.1%

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

      5. associate-*r* 52.1%

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

      6. neg-mul-1 52.1%

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

      7. *-commutative 52.1%

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

      8. unpow2 52.1%

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

      9. unpow2 52.1%

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

      10. swap-sqr 57.9%

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

      11. unpow2 57.9%

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

      12. *-commutative 57.9%

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

      13. *-commutative 57.9%

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

      14. associate-/r* 63.0%

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

   5. Simplified 63.0%

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

   6. Taylor expanded in y around 0 62.4%

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

      1. associate-*r/ 62.4%

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

      2. times-frac 68.3%

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

      3. associate-*l/ 68.3%

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

      4. mul-1-neg 68.3%

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

   8. Simplified 68.3%

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

   if -7.20000000000000038e-35 < t < 2.79999999999999988e47

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 72.3%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-35} \lor \neg \left(t \leq 2.8 \cdot 10^{+47}\right):\\ \;\;\;\;-\frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{-35}:\\ \;\;\;\;-\frac{\frac{x}{z}}{t}\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \end{array} \]

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 -2.15e-35)
   (- (/ (/ x z) t))
   (if (<= t 6.1e+34) (/ 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 <= -2.15e-35) {
		tmp = -((x / z) / t);
	} else if (t <= 6.1e+34) {
		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 <= (-2.15d-35)) then
        tmp = -((x / z) / t)
    else if (t <= 6.1d+34) 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 <= -2.15e-35) {
		tmp = -((x / z) / t);
	} else if (t <= 6.1e+34) {
		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 <= -2.15e-35:
		tmp = -((x / z) / t)
	elif t <= 6.1e+34:
		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 <= -2.15e-35)
		tmp = Float64(-Float64(Float64(x / z) / t));
	elseif (t <= 6.1e+34)
		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)
	tmp = 0.0;
	if (t <= -2.15e-35)
		tmp = -((x / z) / t);
	elseif (t <= 6.1e+34)
		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, -2.15e-35], (-N[(N[(x / z), $MachinePrecision] / t), $MachinePrecision]), If[LessEqual[t, 6.1e+34], N[(x / y), $MachinePrecision], (-N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if t < -2.1500000000000001e-35

   1. Initial program 92.7%

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

   3. Taylor expanded in y around 0 44.1%

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

      1. +-commutative 44.1%

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

      2. mul-1-neg 44.1%

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

      3. unsub-neg 44.1%

         \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{{t}^{2} \cdot {z}^{2}} - \frac{x}{t \cdot z}} \]

      4. associate-*r/ 44.1%

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

      5. associate-*r* 44.1%

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

      6. neg-mul-1 44.1%

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

      7. *-commutative 44.1%

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

      8. unpow2 44.1%

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

      9. unpow2 44.1%

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

      10. swap-sqr 47.3%

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

      11. unpow2 47.3%

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

      12. *-commutative 47.3%

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

      13. *-commutative 47.3%

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

      14. associate-/r* 50.5%

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

   5. Simplified 50.5%

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

   6. Taylor expanded in y around 0 54.3%

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

      1. associate-*r/ 54.3%

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

      2. times-frac 59.4%

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

      3. associate-*l/ 59.4%

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

      4. mul-1-neg 59.4%

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

   8. Simplified 59.4%

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

   if -2.1500000000000001e-35 < t < 6.09999999999999996e34

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 73.2%

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

   if 6.09999999999999996e34 < t

   1. Initial program 91.7%

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

   3. Taylor expanded in y around 0 70.9%

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

      1. associate-*r/ 70.9%

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

      2. neg-mul-1 70.9%

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

      3. associate-/r* 82.8%

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

   5. Simplified 82.8%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{-35}:\\ \;\;\;\;-\frac{\frac{x}{z}}{t}\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{z}}{t}\\ \end{array} \end{array} \]

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) 5e+305) (/ 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) <= 5e+305) {
		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) <= 5d+305) 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) <= 5e+305) {
		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) <= 5e+305:
		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) <= 5e+305)
		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) <= 5e+305)
		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], 5e+305], 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) < 5.00000000000000009e305

   1. Initial program 98.2%

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

   if 5.00000000000000009e305 < (*.f64 z t)

   1. Initial program 75.5%

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

   3. Taylor expanded in y around 0 75.5%

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

      1. +-commutative 75.5%

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

      2. mul-1-neg 75.5%

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

      3. unsub-neg 75.5%

         \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot y}{{t}^{2} \cdot {z}^{2}} - \frac{x}{t \cdot z}} \]

      4. associate-*r/ 75.5%

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

      5. associate-*r* 75.5%

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

      6. neg-mul-1 75.5%

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

      7. *-commutative 75.5%

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

      8. unpow2 75.5%

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

      9. unpow2 75.5%

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

      10. swap-sqr 75.5%

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

      11. unpow2 75.5%

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

      12. *-commutative 75.5%

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

      13. *-commutative 75.5%

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

      14. associate-/r* 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 75.5%

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

      1. associate-*r/ 75.5%

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

      2. times-frac 99.9%

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

      3. associate-*l/ 99.9%

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

      4. mul-1-neg 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.3% 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 96.2%

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

3. Taylor expanded in y around inf 55.8%

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

4. Final simplification 55.8%

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

Developer target: 96.3% 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 2024034 
(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))))