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

Percentage Accurate: 96.1% → 97.9%

Time: 10.9s

Alternatives: 5

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: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+227}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0 - z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) 2e+227) (/ x (fma (- 0.0 z) t y)) (/ -1.0 (* t (/ z x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 2e+227) {
		tmp = x / fma((0.0 - z), t, y);
	} else {
		tmp = -1.0 / (t * (z / x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= 2e+227)
		tmp = Float64(x / fma(Float64(0.0 - z), t, y));
	else
		tmp = Float64(-1.0 / Float64(t * Float64(z / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], 2e+227], N[(x / N[(N[(0.0 - z), $MachinePrecision] * t + y), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(t * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 2.0000000000000002e227

   1. Initial program 99.2%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. neg-sub0 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{0 - z}, t, y\right)} \]

      6. --lowering--.f64 99.2

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{0 - z}, t, y\right)} \]

   4. Applied egg-rr 99.2%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(0 - z, t, y\right)}} \]
   5. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 99.2

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

   6. Applied egg-rr 99.2%

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

   if 2.0000000000000002e227 < (*.f64 z t)

   1. Initial program 82.4%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(z, \mathsf{neg}\left(t\right), y\right)}}{x}} \]

      8. neg-sub0 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(z, \color{blue}{0 - t}, y\right)}{x}} \]

      9. --lowering--.f64 82.6

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(z, \color{blue}{0 - t}, y\right)}{x}} \]

   4. Applied egg-rr 82.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(z, 0 - t, y\right)}{x}}} \]

   5. Taylor expanded in t around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 99.9

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \frac{1}{t \cdot \frac{z}{\color{blue}{0 - x}}} \]

      7. --lowering--.f64 99.9

         \[\leadsto \frac{1}{t \cdot \frac{z}{\color{blue}{0 - x}}} \]

   10. Simplified 99.9%

       \[\leadsto \frac{1}{t \cdot \color{blue}{\frac{z}{0 - x}}} \]
   11. Step-by-step derivation

       1. sub0-neg N/A

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

       2. neg-lowering-neg.f64 99.9

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

   12. Applied egg-rr 99.9%

       \[\leadsto \frac{1}{t \cdot \frac{z}{\color{blue}{-x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+227}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0 - z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t \cdot \frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 5 \cdot 10^{+231}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0 - z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - \frac{x}{z}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) 5e+231) (/ x (fma (- 0.0 z) t y)) (/ (- 0.0 (/ x z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 5e+231) {
		tmp = x / fma((0.0 - z), t, y);
	} else {
		tmp = (0.0 - (x / z)) / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= 5e+231)
		tmp = Float64(x / fma(Float64(0.0 - z), t, y));
	else
		tmp = Float64(Float64(0.0 - Float64(x / z)) / t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], 5e+231], N[(x / N[(N[(0.0 - z), $MachinePrecision] * t + y), $MachinePrecision]), $MachinePrecision], N[(N[(0.0 - N[(x / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 5.00000000000000028e231

   1. Initial program 99.2%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. neg-sub0 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{0 - z}, t, y\right)} \]

      6. --lowering--.f64 99.2

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{0 - z}, t, y\right)} \]

   4. Applied egg-rr 99.2%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(0 - z, t, y\right)}} \]
   5. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 99.2

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

   6. Applied egg-rr 99.2%

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

   if 5.00000000000000028e231 < (*.f64 z t)

   1. Initial program 81.7%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(z, \mathsf{neg}\left(t\right), y\right)}}{x}} \]

      8. neg-sub0 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(z, \color{blue}{0 - t}, y\right)}{x}} \]

      9. --lowering--.f64 81.9

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(z, \color{blue}{0 - t}, y\right)}{x}} \]

   4. Applied egg-rr 81.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(z, 0 - t, y\right)}{x}}} \]

   5. Taylor expanded in t around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 99.9

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \frac{1}{t \cdot \frac{z}{\color{blue}{0 - x}}} \]

      7. --lowering--.f64 99.9

         \[\leadsto \frac{1}{t \cdot \frac{z}{\color{blue}{0 - x}}} \]

   10. Simplified 99.9%

       \[\leadsto \frac{1}{t \cdot \color{blue}{\frac{z}{0 - x}}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{z}{0 - x} \cdot t}} \]

       2. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{\frac{z}{0 - x}}}{t}} \]

       3. clear-num N/A

          \[\leadsto \frac{\color{blue}{\frac{0 - x}{z}}}{t} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{0 - x}{z}}{t}} \]

       5. frac-2neg N/A

          \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(\left(0 - x\right)\right)}{\mathsf{neg}\left(z\right)}}}{t} \]

       6. sub0-neg N/A

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

       7. remove-double-neg N/A

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

       8. /-lowering-/.f64 N/A

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

       9. neg-sub0 N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{0 - z}}}{t} \]

       10. --lowering--.f64 99.9

           \[\leadsto \frac{\frac{x}{\color{blue}{0 - z}}}{t} \]

   12. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{\frac{\frac{x}{0 - z}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 5 \cdot 10^{+231}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0 - z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - \frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\mathsf{fma}\left(0 - z, t, y\right)} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ x (fma (- 0.0 z) t y)))

C

double code(double x, double y, double z, double t) {
	return x / fma((0.0 - z), t, y);
}

Julia

function code(x, y, z, t)
	return Float64(x / fma(Float64(0.0 - z), t, y))
end

Wolfram

code[x_, y_, z_, t_] := N[(x / N[(N[(0.0 - z), $MachinePrecision] * t + y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

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

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. neg-sub0 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{0 - z}, t, y\right)} \]

   6. --lowering--.f64 97.4

      \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{0 - z}, t, y\right)} \]

4. Applied egg-rr 97.4%

   \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(0 - z, t, y\right)}} \]
5. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 97.4

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

6. Applied egg-rr 97.4%

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

7. Final simplification 97.4%

   \[\leadsto \frac{x}{\mathsf{fma}\left(0 - z, t, y\right)} \]
8. Add Preprocessing

Alternative 4: 96.1% 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]

Derivation

1. Initial program 97.4%

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

Alternative 5: 55.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

3. Taylor expanded in y around inf

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

   1. /-lowering-/.f64 51.2

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

5. Simplified 51.2%

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

Developer Target 1: 96.4% 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 2024196 
(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))))