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

Percentage Accurate: 95.8% → 99.7%

Time: 13.5s

Precision: binary64

Cost: 7304

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+231}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{t}}{\frac{z}{x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (/ (/ (- x) t) z)
   (if (<= (* z t) 5e+231) (/ x (fma (- z) t y)) (/ (/ -1.0 t) (/ z x)))))

C

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

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = (-x / t) / z;
	} else if ((z * t) <= 5e+231) {
		tmp = x / fma(-z, t, y);
	} else {
		tmp = (-1.0 / t) / (z / x);
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-x) / t) / z);
	elseif (Float64(z * t) <= 5e+231)
		tmp = Float64(x / fma(Float64(-z), t, y));
	else
		tmp = Float64(Float64(-1.0 / t) / Float64(z / x));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+231], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / t), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision]]]

Target

Original	95.8%
Target	96.3%
Herbie	99.7%

\[\begin{array}{l} \mathbf{if}\;x < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{elif}\;x < 2.1378306434876444 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 75.4%

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

   2. Applied egg-rr 75.4%

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

      [Start] 75.4%	\[ \frac{x}{y - z \cdot t} \]
      clear-num [=>] 75.4%	\[ \color{blue}{\frac{1}{\frac{y - z \cdot t}{x}}} \]
      associate-/r/ [=>] 75.4%	\[ \color{blue}{\frac{1}{y - z \cdot t} \cdot x} \]

   3. Taylor expanded in y around 0 75.4%

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

   4. Simplified 100.0%

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

      [Start] 75.4%	\[ -1 \cdot \frac{x}{t \cdot z} \]
      mul-1-neg [=>] 75.4%	\[ \color{blue}{-\frac{x}{t \cdot z}} \]
      associate-/r* [=>] 100.0%	\[ -\color{blue}{\frac{\frac{x}{t}}{z}} \]

   if -inf.0 < (*.f64 z t) < 5.00000000000000028e231

   1. Initial program 99.9%

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

   2. Applied egg-rr 99.9%

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

      [Start] 99.9%	\[ \frac{x}{y - z \cdot t} \]
      sub-neg [=>] 99.9%	\[ \frac{x}{\color{blue}{y + \left(-z \cdot t\right)}} \]
      +-commutative [=>] 99.9%	\[ \frac{x}{\color{blue}{\left(-z \cdot t\right) + y}} \]
      distribute-lft-neg-in [=>] 99.9%	\[ \frac{x}{\color{blue}{\left(-z\right) \cdot t} + y} \]
      fma-def [=>] 99.9%	\[ \frac{x}{\color{blue}{\mathsf{fma}\left(-z, t, y\right)}} \]

   if 5.00000000000000028e231 < (*.f64 z t)

   1. Initial program 78.3%

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

   2. Applied egg-rr 52.4%

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

      [Start] 78.3%	\[ \frac{x}{y - z \cdot t} \]
      add-sqr-sqrt [=>] 52.5%	\[ \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{y - z \cdot t} \]
      *-un-lft-identity [=>] 52.5%	\[ \frac{\sqrt{x} \cdot \sqrt{x}}{\color{blue}{1 \cdot \left(y - z \cdot t\right)}} \]
      times-frac [=>] 52.4%	\[ \color{blue}{\frac{\sqrt{x}}{1} \cdot \frac{\sqrt{x}}{y - z \cdot t}} \]

   3. Taylor expanded in y around 0 52.4%

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

   4. Simplified 52.4%

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

      [Start] 52.4%	\[ \frac{\sqrt{x}}{1} \cdot \frac{\sqrt{x}}{-1 \cdot \left(t \cdot z\right)} \]
      mul-1-neg [=>] 52.4%	\[ \frac{\sqrt{x}}{1} \cdot \frac{\sqrt{x}}{\color{blue}{-t \cdot z}} \]
      *-commutative [<=] 52.4%	\[ \frac{\sqrt{x}}{1} \cdot \frac{\sqrt{x}}{-\color{blue}{z \cdot t}} \]
      distribute-lft-neg-in [=>] 52.4%	\[ \frac{\sqrt{x}}{1} \cdot \frac{\sqrt{x}}{\color{blue}{\left(-z\right) \cdot t}} \]

   5. Applied egg-rr 99.8%

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

      [Start] 52.4%	\[ \frac{\sqrt{x}}{1} \cdot \frac{\sqrt{x}}{\left(-z\right) \cdot t} \]
      frac-times [=>] 52.5%	\[ \color{blue}{\frac{\sqrt{x} \cdot \sqrt{x}}{1 \cdot \left(\left(-z\right) \cdot t\right)}} \]
      add-sqr-sqrt [<=] 78.3%	\[ \frac{\color{blue}{x}}{1 \cdot \left(\left(-z\right) \cdot t\right)} \]
      *-un-lft-identity [<=] 78.3%	\[ \frac{x}{\color{blue}{\left(-z\right) \cdot t}} \]
      clear-num [=>] 78.4%	\[ \color{blue}{\frac{1}{\frac{\left(-z\right) \cdot t}{x}}} \]
      metadata-eval [<=] 78.4%	\[ \frac{\color{blue}{--1}}{\frac{\left(-z\right) \cdot t}{x}} \]
      distribute-lft-neg-out [=>] 78.4%	\[ \frac{--1}{\frac{\color{blue}{-z \cdot t}}{x}} \]
      *-commutative [<=] 78.4%	\[ \frac{--1}{\frac{-\color{blue}{t \cdot z}}{x}} \]
      distribute-neg-frac [<=] 78.4%	\[ \frac{--1}{\color{blue}{-\frac{t \cdot z}{x}}} \]
      *-commutative [=>] 78.4%	\[ \frac{--1}{-\frac{\color{blue}{z \cdot t}}{x}} \]
      associate-*l/ [<=] 97.2%	\[ \frac{--1}{-\color{blue}{\frac{z}{x} \cdot t}} \]
      frac-2neg [<=] 97.2%	\[ \color{blue}{\frac{-1}{\frac{z}{x} \cdot t}} \]
      *-commutative [=>] 97.2%	\[ \frac{-1}{\color{blue}{t \cdot \frac{z}{x}}} \]
      associate-/r* [=>] 99.8%	\[ \color{blue}{\frac{\frac{-1}{t}}{\frac{z}{x}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+231}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{t}}{\frac{z}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+231}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{t}}{\frac{z}{x}}\\ \end{array} \]

Alternative 2
Accuracy	99.7%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 4 \cdot 10^{+230}\right):\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

Alternative 3
Accuracy	99.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+231}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{t}}{\frac{z}{x}}\\ \end{array} \]

Alternative 4
Accuracy	70.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-61} \lor \neg \left(t \leq 1.4 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{-1}{t \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5
Accuracy	71.2%
Cost	649

\[\begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-61} \lor \neg \left(t \leq 2.2 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6
Accuracy	56.6%
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{+78} \lor \neg \left(t \leq 1.05 \cdot 10^{+176}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7
Accuracy	56.8%
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+76}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+176}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 8
Accuracy	54.7%
Cost	192

\[\frac{x}{y} \]

Reproduce

herbie shell --seed 2023271 
(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))))