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

Percentage Accurate: 96.2% → 98.1%

Time: 9.4s

Alternatives: 6

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: 96.2% 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, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := t \cdot \left(z \cdot t\right)\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+276}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(y, \frac{y}{t \cdot \left(z \cdot t\_1\right)}, \frac{y}{t\_1}\right), \frac{x}{t}\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \end{array} \end{array} \]

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 (* t (* z t))))
   (if (<= (* z t) -1e+276)
     (/ (fma x (fma y (/ y (* t (* z t_1))) (/ y t_1)) (/ x t)) (- z))
     (/ x (fma (- z) t y)))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = t * (z * t);
	double tmp;
	if ((z * t) <= -1e+276) {
		tmp = fma(x, fma(y, (y / (t * (z * t_1))), (y / t_1)), (x / t)) / -z;
	} else {
		tmp = x / fma(-z, t, y);
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(t * Float64(z * t))
	tmp = 0.0
	if (Float64(z * t) <= -1e+276)
		tmp = Float64(fma(x, fma(y, Float64(y / Float64(t * Float64(z * t_1))), Float64(y / t_1)), Float64(x / t)) / Float64(-z));
	else
		tmp = Float64(x / fma(Float64(-z), t, y));
	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_] := Block[{t$95$1 = N[(t * N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+276], N[(N[(x * N[(y * N[(y / N[(t * N[(z * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(x / t), $MachinePrecision]), $MachinePrecision] / (-z)), $MachinePrecision], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.0000000000000001e276

   1. Initial program 64.4%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{\left(-1 \cdot \frac{x}{t} + -1 \cdot \frac{x \cdot {y}^{2}}{{t}^{3} \cdot {z}^{2}}\right) - \frac{x \cdot y}{{t}^{2} \cdot z}}{z}} \]

   5. Applied rewrites 99.8%

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

   if -1.0000000000000001e276 < (*.f64 z t)

   1. Initial program 98.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 98.8

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

   4. Applied rewrites 98.8%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+276}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(y, \frac{y}{t \cdot \left(z \cdot \left(t \cdot \left(z \cdot t\right)\right)\right)}, \frac{y}{t \cdot \left(z \cdot t\right)}\right), \frac{x}{t}\right)}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{-z \cdot t}\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{-54}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (- (* z t)))))
   (if (<= (* z t) -5e-15) t_1 (if (<= (* z t) 1e-54) (/ 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 / -(z * t);
	double tmp;
	if ((z * t) <= -5e-15) {
		tmp = t_1;
	} else if ((z * t) <= 1e-54) {
		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 / -(z * t)
    if ((z * t) <= (-5d-15)) then
        tmp = t_1
    else if ((z * t) <= 1d-54) 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 / -(z * t);
	double tmp;
	if ((z * t) <= -5e-15) {
		tmp = t_1;
	} else if ((z * t) <= 1e-54) {
		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 / -(z * t)
	tmp = 0
	if (z * t) <= -5e-15:
		tmp = t_1
	elif (z * t) <= 1e-54:
		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(x / Float64(-Float64(z * t)))
	tmp = 0.0
	if (Float64(z * t) <= -5e-15)
		tmp = t_1;
	elseif (Float64(z * t) <= 1e-54)
		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 / -(z * t);
	tmp = 0.0;
	if ((z * t) <= -5e-15)
		tmp = t_1;
	elseif ((z * t) <= 1e-54)
		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[(x / (-N[(z * t), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -5e-15], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e-54], N[(x / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.99999999999999999e-15 or 1e-54 < (*.f64 z t)

   1. Initial program 92.8%

      \[\frac{x}{y - z \cdot t} \]
   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. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 77.8

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

   5. Applied rewrites 77.8%

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

   if -4.99999999999999999e-15 < (*.f64 z t) < 1e-54

   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 88.4

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

   5. Applied rewrites 88.4%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{-z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 10^{-54}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+265}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \end{array} \end{array} \]

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+265) (/ (/ x z) (- t)) (/ x (fma (- z) t 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+265) {
		tmp = (x / z) / -t;
	} else {
		tmp = x / fma(-z, t, 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+265)
		tmp = Float64(Float64(x / z) / Float64(-t));
	else
		tmp = Float64(x / fma(Float64(-z), t, y));
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+265], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.00000000000000007e265

   1. Initial program 67.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lift--.f64 N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{\frac{1}{\color{blue}{\frac{{y}^{3} - {\left(z \cdot t\right)}^{3}}{y \cdot y + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + y \cdot \left(z \cdot t\right)\right)}}}}{\frac{1}{x}} \]

      7. clear-num N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot y + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + y \cdot \left(z \cdot t\right)\right)}{{y}^{3} - {\left(z \cdot t\right)}^{3}}}}{\frac{1}{x}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y \cdot y + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + y \cdot \left(z \cdot t\right)\right)}{{y}^{3} - {\left(z \cdot t\right)}^{3}}}{\frac{1}{x}}} \]

      9. clear-num N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{{y}^{3} - {\left(z \cdot t\right)}^{3}}{y \cdot y + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + y \cdot \left(z \cdot t\right)\right)}}}}{\frac{1}{x}} \]

      10. flip3-- N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 67.3

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

   4. Applied rewrites 67.3%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{\left(-1 \cdot \frac{x}{z} + -1 \cdot \frac{x \cdot {y}^{2}}{{t}^{2} \cdot {z}^{3}}\right) - \frac{x \cdot y}{t \cdot {z}^{2}}}{t}} \]

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{x}{z}}{\mathsf{neg}\left(\color{blue}{t}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

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

   if -1.00000000000000007e265 < (*.f64 z t)

   1. Initial program 98.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 98.8

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

   4. Applied rewrites 98.8%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(-z, t, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 96.2% accurate, 1.0× speedup

Math

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

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 (- z) t y)))

C

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

Julia

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

Derivation

1. Initial program 95.9%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-neg-in N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-neg.f64 95.9

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

4. Applied rewrites 95.9%

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

Alternative 5: 96.2% 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 95.9%

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

Alternative 6: 54.3% 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 95.9%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 51.0

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

5. Applied rewrites 51.0%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
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 2024226 
(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))))