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

Average Accuracy: 95.8% → 99.4%

Time: 13.9s

Precision: binary64

Cost: 7304

\[ \begin{array}{c}[z, t] = \mathsf{sort}([z, t])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+181}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

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) 2e+181) (/ x (fma (- z) t y)) (/ (/ (- x) z) t))))

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) <= 2e+181) {
		tmp = x / fma(-z, t, y);
	} else {
		tmp = (-x / z) / t;
	}
	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) <= 2e+181)
		tmp = Float64(x / fma(Float64(-z), t, y));
	else
		tmp = Float64(Float64(Float64(-x) / z) / t);
	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], 2e+181], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision]]]

Target

Original	95.8%
Target	97.4%
Herbie	99.4%

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.6%

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

   2. Applied egg-rr 69.6%

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

      [Start] 69.6	\[ \frac{x}{y - z \cdot t} \]
      clear-num [=>] 69.6	\[ \color{blue}{\frac{1}{\frac{y - z \cdot t}{x}}} \]
      inv-pow [=>] 69.6	\[ \color{blue}{{\left(\frac{y - z \cdot t}{x}\right)}^{-1}} \]

   3. Applied egg-rr 69.6%

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

      [Start] 69.6	\[ {\left(\frac{y - z \cdot t}{x}\right)}^{-1} \]
      unpow-1 [=>] 69.6	\[ \color{blue}{\frac{1}{\frac{y - z \cdot t}{x}}} \]
      clear-num [<=] 69.6	\[ \color{blue}{\frac{x}{y - z \cdot t}} \]
      add-sqr-sqrt [=>] 69.6	\[ \frac{x}{\color{blue}{\sqrt{y - z \cdot t} \cdot \sqrt{y - z \cdot t}}} \]
      associate-/r* [=>] 69.6	\[ \color{blue}{\frac{\frac{x}{\sqrt{y - z \cdot t}}}{\sqrt{y - z \cdot t}}} \]

   4. Taylor expanded in y around 0 69.6%

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

   5. Simplified 99.9%

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

      [Start] 69.6	\[ -1 \cdot \frac{x}{t \cdot z} \]
      mul-1-neg [=>] 69.6	\[ \color{blue}{-\frac{x}{t \cdot z}} \]
      associate-/r* [=>] 99.9	\[ -\color{blue}{\frac{\frac{x}{t}}{z}} \]
      distribute-neg-frac [=>] 99.9	\[ \color{blue}{\frac{-\frac{x}{t}}{z}} \]

   if -inf.0 < (*.f64 z t) < 1.9999999999999998e181

   1. Initial program 99.8%

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

   2. Applied egg-rr 99.8%

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

      [Start] 99.8	\[ \frac{x}{y - z \cdot t} \]
      sub-neg [=>] 99.8	\[ \frac{x}{\color{blue}{y + \left(-z \cdot t\right)}} \]
      +-commutative [=>] 99.8	\[ \frac{x}{\color{blue}{\left(-z \cdot t\right) + y}} \]
      distribute-lft-neg-in [=>] 99.8	\[ \frac{x}{\color{blue}{\left(-z\right) \cdot t} + y} \]
      fma-def [=>] 99.8	\[ \frac{x}{\color{blue}{\mathsf{fma}\left(-z, t, y\right)}} \]

   if 1.9999999999999998e181 < (*.f64 z t)

   1. Initial program 84.2%

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

   2. Applied egg-rr 84.2%

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

      [Start] 84.2	\[ \frac{x}{y - z \cdot t} \]
      sub-neg [=>] 84.2	\[ \frac{x}{\color{blue}{y + \left(-z \cdot t\right)}} \]
      +-commutative [=>] 84.2	\[ \frac{x}{\color{blue}{\left(-z \cdot t\right) + y}} \]
      distribute-lft-neg-in [=>] 84.2	\[ \frac{x}{\color{blue}{\left(-z\right) \cdot t} + y} \]
      fma-def [=>] 84.2	\[ \frac{x}{\color{blue}{\mathsf{fma}\left(-z, t, y\right)}} \]

   3. Taylor expanded in z around inf 81.1%

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

   4. Simplified 96.4%

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

      [Start] 81.1	\[ -1 \cdot \frac{x}{t \cdot z} \]
      associate-*r/ [=>] 81.1	\[ \color{blue}{\frac{-1 \cdot x}{t \cdot z}} \]
      neg-mul-1 [<=] 81.1	\[ \frac{\color{blue}{-x}}{t \cdot z} \]
      associate-/l/ [<=] 96.4	\[ \color{blue}{\frac{\frac{-x}{z}}{t}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+181}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	79.8%
Cost	1164

\[\begin{array}{l} t_1 := \frac{\frac{-x}{t}}{z}\\ \mathbf{if}\;z \cdot t \leq -9 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 10^{-15}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+199}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	79.8%
Cost	1164

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -9 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 10^{-15}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+181}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+181}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternative 4
Accuracy	76.9%
Cost	905

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-7} \lor \neg \left(z \cdot t \leq 10^{-15}\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5
Accuracy	62.1%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -4.7 \cdot 10^{+109} \lor \neg \left(z \cdot t \leq 2.2 \cdot 10^{+118}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6
Accuracy	61.9%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+96} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+107}\right):\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7
Accuracy	52.7%
Cost	192

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

Reproduce

herbie shell --seed 2023146 
(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))))