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

Average Error: 10.6 → 3.0

Time: 28.3s

Precision: binary64

Cost: 16464

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\mathsf{fma}\left(z, y, -x\right)}{\mathsf{fma}\left(z, a, -t\right)}\\ t_2 := t - a \cdot z\\ t_3 := \frac{x - y \cdot z}{t_2}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\frac{-z}{1} \cdot \frac{y}{t_2}\\ \mathbf{elif}\;t_3 \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\frac{\left(-\frac{x}{a}\right) + \frac{y \cdot t}{a \cdot a}}{z} + \frac{y}{a}\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\frac{-y}{1} \cdot \frac{z}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (fma z y (- x)) (fma z a (- t))))
        (t_2 (- t (* a z)))
        (t_3 (/ (- x (* y z)) t_2)))
   (if (<= t_3 (- INFINITY))
     (* (/ (- z) 1.0) (/ y t_2))
     (if (<= t_3 -1e-310)
       t_1
       (if (<= t_3 0.0)
         (+ (/ (+ (- (/ x a)) (/ (* y t) (* a a))) z) (/ y a))
         (if (<= t_3 5e+298)
           t_1
           (if (<= t_3 INFINITY) (* (/ (- y) 1.0) (/ z t_2)) (/ y a))))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, y, -x) / fma(z, a, -t);
	double t_2 = t - (a * z);
	double t_3 = (x - (y * z)) / t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (-z / 1.0) * (y / t_2);
	} else if (t_3 <= -1e-310) {
		tmp = t_1;
	} else if (t_3 <= 0.0) {
		tmp = ((-(x / a) + ((y * t) / (a * a))) / z) + (y / a);
	} else if (t_3 <= 5e+298) {
		tmp = t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (-y / 1.0) * (z / t_2);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(fma(z, y, Float64(-x)) / fma(z, a, Float64(-t)))
	t_2 = Float64(t - Float64(a * z))
	t_3 = Float64(Float64(x - Float64(y * z)) / t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-z) / 1.0) * Float64(y / t_2));
	elseif (t_3 <= -1e-310)
		tmp = t_1;
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(-Float64(x / a)) + Float64(Float64(y * t) / Float64(a * a))) / z) + Float64(y / a));
	elseif (t_3 <= 5e+298)
		tmp = t_1;
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(Float64(-y) / 1.0) * Float64(z / t_2));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * y + (-x)), $MachinePrecision] / N[(z * a + (-t)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[((-z) / 1.0), $MachinePrecision] * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -1e-310], t$95$1, If[LessEqual[t$95$3, 0.0], N[(N[(N[((-N[(x / a), $MachinePrecision]) + N[(N[(y * t), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+298], t$95$1, If[LessEqual[t$95$3, Infinity], N[(N[((-y) / 1.0), $MachinePrecision] * N[(z / t$95$2), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]]]]

Target

Original	10.6
Target	1.7
Herbie	3.0

\[\begin{array}{l} \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array} \]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   3. Taylor expanded in y around inf 64.0

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

   4. Applied egg-rr 0.2

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -9.999999999999969e-311 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 5.0000000000000003e298

   1. Initial program 0.2

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

   2. Simplified 0.2

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

   if -9.999999999999969e-311 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 24.5

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

   2. Taylor expanded in z around -inf 19.1

      \[\leadsto \color{blue}{-1 \cdot \frac{\frac{x}{a} - \frac{y \cdot t}{{a}^{2}}}{z} + \frac{y}{a}} \]

   3. Simplified 19.1

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

   if 5.0000000000000003e298 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 58.2

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

   2. Simplified 58.2

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

   3. Taylor expanded in y around inf 61.8

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

   4. Applied egg-rr 4.0

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

   if +inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 64.0

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

   2. Taylor expanded in z around inf 0

      \[\leadsto \color{blue}{\frac{y}{a}} \]
3. Recombined 5 regimes into one program.

Alternatives

Alternative 1
Error	3.0
Cost	4628

\[\begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{x - y \cdot z}{t_1}\\ t_3 := t - z \cdot a\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{-z}{1} \cdot \frac{y}{t_1}\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{t_3} - \frac{z \cdot y}{t_3}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{\left(-\frac{x}{a}\right) + \frac{y \cdot t}{a \cdot a}}{z} + \frac{y}{a}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\frac{-y}{1} \cdot \frac{z}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 2
Error	3.8
Cost	3084

\[\begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{-y}{1} \cdot \frac{z}{t_1}\\ t_3 := \frac{x - y \cdot z}{t_1}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 3
Error	3.8
Cost	3084

\[\begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{x - y \cdot z}{t_1}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{-z}{1} \cdot \frac{y}{t_1}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq \infty:\\ \;\;\;\;\frac{-y}{1} \cdot \frac{z}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 4
Error	5.3
Cost	2248

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - a \cdot z}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 5
Error	37.2
Cost	1836

\[\begin{array}{l} t_1 := -\frac{x}{a \cdot z}\\ t_2 := \frac{y \cdot z}{-t}\\ \mathbf{if}\;a \leq -3.6 \cdot 10^{+259}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -8.2 \cdot 10^{+138}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;a \leq -7.5 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -3.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;a \leq -1.35 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;a \leq -8.6 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.46 \cdot 10^{-62}:\\ \;\;\;\;\frac{z}{-t} \cdot y\\ \mathbf{elif}\;a \leq -4.4 \cdot 10^{-259}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-276}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-87}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-14}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{+27}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 6
Error	25.9
Cost	1564

\[\begin{array}{l} t_1 := \left(-\frac{y}{t} \cdot z\right) + \frac{x}{t}\\ \mathbf{if}\;y \leq -1.42 \cdot 10^{+184}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5.7 \cdot 10^{+134}:\\ \;\;\;\;\frac{-y}{1} \cdot \frac{-1}{a}\\ \mathbf{elif}\;y \leq -9.8 \cdot 10^{+104}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{+26}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{elif}\;y \leq 1.14 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{t} - \frac{y \cdot z}{t}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+81}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	26.3
Cost	1504

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t}\\ \mathbf{if}\;y \leq -4.5 \cdot 10^{+184}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{+134}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+29}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{elif}\;y \leq 1.14 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+86}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{+223}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 8
Error	26.2
Cost	1504

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t}\\ \mathbf{if}\;y \leq -1.12 \cdot 10^{+184}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5.7 \cdot 10^{+134}:\\ \;\;\;\;\frac{-y}{1} \cdot \frac{-1}{a}\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{+104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-64}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+83}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+225}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 9
Error	26.3
Cost	1504

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t}\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{+184}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5.7 \cdot 10^{+134}:\\ \;\;\;\;\frac{-y}{1} \cdot \frac{-1}{a}\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{+26}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{elif}\;y \leq 1.14 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{t} - \frac{y \cdot z}{t}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+87}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+225}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 10
Error	23.1
Cost	1372

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t}\\ t_2 := \frac{y \cdot z - x}{a \cdot z}\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{+108}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{-27}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-265}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.05 \cdot 10^{-59}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 11
Error	22.8
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{t - a \cdot z}\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{+108}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{y \cdot z}{-t}\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+113}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 12
Error	29.8
Cost	648

\[\begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+108}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -7.6 \cdot 10^{-28}:\\ \;\;\;\;-\frac{x}{a \cdot z}\\ \mathbf{elif}\;z \leq 1.56 \cdot 10^{+35}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 13
Error	29.5
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-27}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 6.3 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 14
Error	41.7
Cost	192

\[\frac{x}{t} \]

Reproduce

herbie shell --seed 2023010 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

  (/ (- x (* y z)) (- t (* a z))))