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

Percentage Accurate: 95.8% → 97.8%

Time: 8.9s

Alternatives: 7

Speedup: 1.0×

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: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+292}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \end{array} \]

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+292) (/ (/ (- 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+292) {
		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+292)) 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+292) {
		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+292:
		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+292)
		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+292)
		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+292], 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) < -1e292

   1. Initial program 69.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -1e292 < (*.f64 z t)

   1. Initial program 98.3%

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

Alternative 2: 63.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+212} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{x}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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 (<= (* z t) -1e+212) (not (<= (* z t) 2e+69))) (/ 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 (((z * t) <= -1e+212) || !((z * t) <= 2e+69)) {
		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 (((z * t) <= (-1d+212)) .or. (.not. ((z * t) <= 2d+69))) 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 (((z * t) <= -1e+212) || !((z * t) <= 2e+69)) {
		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 ((z * t) <= -1e+212) or not ((z * t) <= 2e+69):
		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 ((Float64(z * t) <= -1e+212) || !(Float64(z * t) <= 2e+69))
		tmp = Float64(x / Float64(t * 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 (((z * t) <= -1e+212) || ~(((z * t) <= 2e+69)))
		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[N[(z * t), $MachinePrecision], -1e+212], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+69]], $MachinePrecision]], N[(x / N[(t * z), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.9999999999999991e211 or 2.0000000000000001e69 < (*.f64 z t)

   1. Initial program 85.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/r/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 29.2

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 29.2

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

   4. Applied rewrites 29.2%

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

   6. Applied rewrites 66.5%

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

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 61.5

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

   9. Applied rewrites 61.5%

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

   if -9.9999999999999991e211 < (*.f64 z t) < 2.0000000000000001e69

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 71.3

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

   5. Applied rewrites 71.3%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+212} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{x}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.6× speedup

Math

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

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) (- INFINITY)) (/ (/ (- x) t) z) (/ 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) <= -((double) INFINITY)) {
		tmp = (-x / t) / z;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

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) <= -Double.POSITIVE_INFINITY) {
		tmp = (-x / t) / z;
	} 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) <= -math.inf:
		tmp = (-x / t) / z
	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) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-x) / t) / z);
	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) <= -Inf)
		tmp = (-x / t) / z;
	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], (-Infinity)], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0

   1. Initial program 64.1%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   if -inf.0 < (*.f64 z t)

   1. Initial program 98.3%

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

Alternative 4: 77.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-62} \lor \neg \left(y \leq 2.9 \cdot 10^{-108}\right):\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(-t\right) \cdot z}\\ \end{array} \end{array} \]

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 (or (<= y -4.8e-62) (not (<= y 2.9e-108)))
   (/ x (fma t z 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 ((y <= -4.8e-62) || !(y <= 2.9e-108)) {
		tmp = x / fma(t, z, 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 ((y <= -4.8e-62) || !(y <= 2.9e-108))
		tmp = Float64(x / fma(t, z, y));
	else
		tmp = Float64(x / Float64(Float64(-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[Or[LessEqual[y, -4.8e-62], N[Not[LessEqual[y, 2.9e-108]], $MachinePrecision]], N[(x / N[(t * z + y), $MachinePrecision]), $MachinePrecision], N[(x / N[((-t) * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.79999999999999967e-62 or 2.9000000000000001e-108 < y

   1. Initial program 96.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/r/ N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 54.0

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 54.0

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

   4. Applied rewrites 54.0%

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

   6. Applied rewrites 84.7%

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

   if -4.79999999999999967e-62 < y < 2.9000000000000001e-108

   1. Initial program 93.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 76.1

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

   5. Applied rewrites 76.1%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-62} \lor \neg \left(y \leq 2.9 \cdot 10^{-108}\right):\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(t, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(-t\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.8% accurate, 1.0× speedup

Math

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

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 (* z t))))

C

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

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 - (z * t))
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 - (z * t));
}

Python

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

Julia

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

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = x / (y - (z * t));
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 / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.5%

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

Alternative 6: 65.6% accurate, 1.1× speedup

Math

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

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 (fma t z y)))

C

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

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(x / fma(t, z, 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 / N[(t * z + y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.5%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. associate-/r/ N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. pow2 N/A

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

   15. lower-pow.f64 55.6

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 55.6

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

4. Applied rewrites 55.6%

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

6. Applied rewrites 68.5%

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

Alternative 7: 54.9% accurate, 1.7× 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 95.5%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{x}{y}} \]
4. Step-by-step derivation

   1. lower-/.f64 56.6

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

5. Applied rewrites 56.6%

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

Developer Target 1: 96.6% accurate, 0.4× 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 2024342 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -161819597360704900000000000000000000000000000000000) (/ 1 (- (/ y x) (* (/ z x) t))) (if (< x 213783064348764440000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ x (- y (* z t))) (/ 1 (- (/ y x) (* (/ z x) t))))))

  (/ x (- y (* z t))))