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

Average Error: 2.8 → 0.1

Time: 9.3s

Precision: binary64

Cost: 7304

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+287}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -5e+287)
   (/ (/ (- x) z) t)
   (if (<= (* z t) 5e+299) (/ x (fma (- z) t y)) (/ (/ (- x) t) z))))

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) <= -5e+287) {
		tmp = (-x / z) / t;
	} else if ((z * t) <= 5e+299) {
		tmp = x / fma(-z, t, y);
	} else {
		tmp = (-x / t) / z;
	}
	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) <= -5e+287)
		tmp = Float64(Float64(Float64(-x) / z) / t);
	elseif (Float64(z * t) <= 5e+299)
		tmp = Float64(x / fma(Float64(-z), t, y));
	else
		tmp = Float64(Float64(Float64(-x) / t) / z);
	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], -5e+287], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+299], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision]]]

Target

Original	2.8
Target	1.7
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5e287

   1. Initial program 18.7

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

   2. Applied egg-rr 18.7

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

   3. Taylor expanded in y around 0 18.7

      \[\leadsto \color{blue}{-1 \cdot \left({1}^{0.3333333333333333} \cdot \frac{x}{t \cdot z}\right)} \]

   4. Simplified 0.1

      \[\leadsto \color{blue}{\frac{\frac{-x}{t}}{z}} \]
      Proof
      (/.f64 (/.f64 (neg.f64 x) t) z): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 x t))) z): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (/.f64 x t))) z): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (/.f64 x t) z))): 0 points increase in error, 0 points decrease in error
      (*.f64 -1 (Rewrite<= associate-/r*_binary64 (/.f64 x (*.f64 t z)))): 50 points increase in error, 49 points decrease in error
      (*.f64 -1 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 x (*.f64 t z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 -1 (*.f64 (Rewrite<= pow-base-1_binary64 (pow.f64 1 1/3)) (/.f64 x (*.f64 t z)))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 0.1

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

   6. Applied egg-rr 0.1

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

   if -5e287 < (*.f64 z t) < 5.0000000000000003e299

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

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

   if 5.0000000000000003e299 < (*.f64 z t)

   1. Initial program 19.1

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

   2. Applied egg-rr 19.2

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

   3. Taylor expanded in y around 0 19.1

      \[\leadsto \color{blue}{-1 \cdot \left({1}^{0.3333333333333333} \cdot \frac{x}{t \cdot z}\right)} \]

   4. Simplified 0.1

      \[\leadsto \color{blue}{\frac{\frac{-x}{t}}{z}} \]
      Proof
      (/.f64 (/.f64 (neg.f64 x) t) z): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 x t))) z): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (/.f64 x t))) z): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (/.f64 x t) z))): 0 points increase in error, 0 points decrease in error
      (*.f64 -1 (Rewrite<= associate-/r*_binary64 (/.f64 x (*.f64 t z)))): 50 points increase in error, 49 points decrease in error
      (*.f64 -1 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 x (*.f64 t z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 -1 (*.f64 (Rewrite<= pow-base-1_binary64 (pow.f64 1 1/3)) (/.f64 x (*.f64 t z)))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+287}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+287}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternative 2
Error	19.1
Cost	912

\[\begin{array}{l} t_1 := \frac{\frac{-x}{z}}{t}\\ \mathbf{if}\;t \leq -6.677250519744025 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.6666977515880187 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t \leq 2.0415141205519829 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.0227375745711695 \cdot 10^{+183}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	19.2
Cost	912

\[\begin{array}{l} t_1 := \frac{\frac{-x}{z}}{t}\\ \mathbf{if}\;t \leq -6.677250519744025 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.6666977515880187 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t \leq 2.0415141205519829 \cdot 10^{+152}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;t \leq 6.0227375745711695 \cdot 10^{+183}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	17.3
Cost	648

\[\begin{array}{l} \mathbf{if}\;t \leq -6.677250519744025 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;t \leq 1.6666977515880187 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternative 5
Error	27.0
Cost	584

\[\begin{array}{l} t_1 := \frac{\frac{x}{t}}{z}\\ \mathbf{if}\;t \leq -1.0664552978003736 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.816848631449851 \cdot 10^{+185}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	27.4
Cost	584

\[\begin{array}{l} t_1 := \frac{x}{z \cdot t}\\ \mathbf{if}\;t \leq -1.0664552978003736 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.816848631449851 \cdot 10^{+185}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	30.1
Cost	192

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

Reproduce

herbie shell --seed 2022313 
(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))))