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

Average Accuracy: 95.7% → 99.8%

Time: 12.8s

Precision: binary64

Cost: 7816

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+265}:\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+288}:\\ \;\;\;\;\frac{x}{\left(y + \mathsf{fma}\left(-z, t, z \cdot t\right)\right) - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -2e+265)
   (/ (/ x (- t)) z)
   (if (<= (* z t) 5e+288)
     (/ x (- (+ y (fma (- z) t (* z t))) (* z t)))
     (/ (/ (- 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) <= -2e+265) {
		tmp = (x / -t) / z;
	} else if ((z * t) <= 5e+288) {
		tmp = x / ((y + fma(-z, t, (z * t))) - (z * t));
	} 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) <= -2e+265)
		tmp = Float64(Float64(x / Float64(-t)) / z);
	elseif (Float64(z * t) <= 5e+288)
		tmp = Float64(x / Float64(Float64(y + fma(Float64(-z), t, Float64(z * t))) - Float64(z * t)));
	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], -2e+265], N[(N[(x / (-t)), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+288], N[(x / N[(N[(y + N[((-z) * t + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision]]]

Target

Original	95.7%
Target	97.1%
Herbie	99.8%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.00000000000000013e265

   1. Initial program 75.1%

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

   2. Applied egg-rr 21.8%

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

      [Start] 75.1	\[ \frac{x}{y - z \cdot t} \]
      *-un-lft-identity [=>] 75.1	\[ \frac{x}{\color{blue}{1 \cdot y} - z \cdot t} \]
      prod-diff [=>] 21.8	\[ \frac{x}{\color{blue}{\mathsf{fma}\left(1, y, -t \cdot z\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)}} \]
      *-commutative [<=] 21.8	\[ \frac{x}{\mathsf{fma}\left(1, y, -\color{blue}{z \cdot t}\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)} \]
      fma-neg [<=] 21.8	\[ \frac{x}{\color{blue}{\left(1 \cdot y - z \cdot t\right)} + \mathsf{fma}\left(-t, z, t \cdot z\right)} \]
      prod-diff [=>] 21.8	\[ \frac{x}{\color{blue}{\left(\mathsf{fma}\left(1, y, -t \cdot z\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)\right)} + \mathsf{fma}\left(-t, z, t \cdot z\right)} \]
      *-commutative [<=] 21.8	\[ \frac{x}{\left(\mathsf{fma}\left(1, y, -\color{blue}{z \cdot t}\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)} \]
      fma-neg [<=] 21.8	\[ \frac{x}{\left(\color{blue}{\left(1 \cdot y - z \cdot t\right)} + \mathsf{fma}\left(-t, z, t \cdot z\right)\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)} \]
      *-un-lft-identity [<=] 21.8	\[ \frac{x}{\left(\left(\color{blue}{y} - z \cdot t\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)} \]
      associate-+l+ [=>] 21.8	\[ \frac{x}{\color{blue}{\left(y - z \cdot t\right) + \left(\mathsf{fma}\left(-t, z, t \cdot z\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)\right)}} \]
      *-commutative [<=] 21.8	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\mathsf{fma}\left(-t, z, \color{blue}{z \cdot t}\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)\right)} \]
      *-commutative [<=] 21.8	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\mathsf{fma}\left(-t, z, z \cdot t\right) + \mathsf{fma}\left(-t, z, \color{blue}{z \cdot t}\right)\right)} \]

   3. Simplified 21.8%

      \[\leadsto \frac{x}{\color{blue}{\left(y - t \cdot z\right) + 2 \cdot \mathsf{fma}\left(-t, z, t \cdot z\right)}} \]
      Proof

      [Start] 21.8	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\mathsf{fma}\left(-t, z, z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right)} \]
      associate-+r+ [=>] 21.8	\[ \frac{x}{\color{blue}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)}} \]
      fma-udef [=>] 21.8	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \color{blue}{\left(\left(-t\right) \cdot z + z \cdot t\right)}} \]
      neg-mul-1 [=>] 21.8	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{\left(-1 \cdot t\right)} \cdot z + z \cdot t\right)} \]
      associate-*r* [<=] 21.8	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{-1 \cdot \left(t \cdot z\right)} + z \cdot t\right)} \]
      *-commutative [<=] 21.8	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(-1 \cdot \color{blue}{\left(z \cdot t\right)} + z \cdot t\right)} \]
      mul-1-neg [=>] 21.8	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{\left(-z \cdot t\right)} + z \cdot t\right)} \]
      *-rgt-identity [<=] 21.8	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{\left(-z \cdot t\right) \cdot 1} + z \cdot t\right)} \]
      fma-udef [<=] 21.8	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \color{blue}{\mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)}} \]
      associate-+r+ [<=] 21.8	\[ \frac{x}{\color{blue}{\left(y - z \cdot t\right) + \left(\mathsf{fma}\left(-t, z, z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)}} \]
      fma-udef [=>] 21.8	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\color{blue}{\left(\left(-t\right) \cdot z + z \cdot t\right)} + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)} \]
      distribute-lft-neg-in [<=] 21.8	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\left(\color{blue}{\left(-t \cdot z\right)} + z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)} \]
      *-commutative [<=] 21.8	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\left(\left(-\color{blue}{z \cdot t}\right) + z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)} \]
      associate-+l+ [=>] 21.8	\[ \frac{x}{\left(y - z \cdot t\right) + \color{blue}{\left(\left(-z \cdot t\right) + \left(z \cdot t + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)\right)}} \]
      *-rgt-identity [<=] 21.8	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\color{blue}{\left(-z \cdot t\right) \cdot 1} + \left(z \cdot t + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)\right)} \]
      associate-+l+ [<=] 21.8	\[ \frac{x}{\left(y - z \cdot t\right) + \color{blue}{\left(\left(\left(-z \cdot t\right) \cdot 1 + z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)}} \]

   4. Taylor expanded in y around 0 21.0%

      \[\leadsto \color{blue}{\frac{x}{2 \cdot \left(-1 \cdot \left(t \cdot z\right) + t \cdot z\right) - t \cdot z}} \]

   5. Simplified 99.0%

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

      [Start] 21.0	\[ \frac{x}{2 \cdot \left(-1 \cdot \left(t \cdot z\right) + t \cdot z\right) - t \cdot z} \]
      +-commutative [=>] 21.0	\[ \frac{x}{2 \cdot \color{blue}{\left(t \cdot z + -1 \cdot \left(t \cdot z\right)\right)} - t \cdot z} \]
      mul-1-neg [=>] 21.0	\[ \frac{x}{2 \cdot \left(t \cdot z + \color{blue}{\left(-t \cdot z\right)}\right) - t \cdot z} \]
      sub-neg [<=] 21.0	\[ \frac{x}{2 \cdot \color{blue}{\left(t \cdot z - t \cdot z\right)} - t \cdot z} \]
      +-inverses [=>] 74.3	\[ \frac{x}{2 \cdot \color{blue}{0} - t \cdot z} \]
      metadata-eval [=>] 74.3	\[ \frac{x}{\color{blue}{0} - t \cdot z} \]
      sub0-neg [=>] 74.3	\[ \frac{x}{\color{blue}{-t \cdot z}} \]
      *-commutative [=>] 74.3	\[ \frac{x}{-\color{blue}{z \cdot t}} \]
      distribute-rgt-neg-in [=>] 74.3	\[ \frac{x}{\color{blue}{z \cdot \left(-t\right)}} \]
      neg-sub0 [=>] 74.3	\[ \frac{x}{z \cdot \color{blue}{\left(0 - t\right)}} \]
      metadata-eval [<=] 74.3	\[ \frac{x}{z \cdot \left(\color{blue}{2 \cdot 0} - t\right)} \]
      mul0-lft [<=] 74.3	\[ \frac{x}{z \cdot \left(2 \cdot \color{blue}{\left(0 \cdot t\right)} - t\right)} \]
      metadata-eval [<=] 74.3	\[ \frac{x}{z \cdot \left(2 \cdot \left(\color{blue}{\left(-1 + 1\right)} \cdot t\right) - t\right)} \]
      distribute-lft1-in [<=] 74.3	\[ \frac{x}{z \cdot \left(2 \cdot \color{blue}{\left(-1 \cdot t + t\right)} - t\right)} \]
      associate-/l/ [<=] 99.0	\[ \color{blue}{\frac{\frac{x}{2 \cdot \left(-1 \cdot t + t\right) - t}}{z}} \]

   if -2.00000000000000013e265 < (*.f64 z t) < 5.0000000000000003e288

   1. Initial program 99.8%

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

   2. Applied egg-rr 99.9%

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

      [Start] 99.8	\[ \frac{x}{y - z \cdot t} \]
      *-un-lft-identity [=>] 99.8	\[ \frac{x}{\color{blue}{1 \cdot y} - z \cdot t} \]
      *-commutative [=>] 99.8	\[ \frac{x}{1 \cdot y - \color{blue}{t \cdot z}} \]
      prod-diff [=>] 99.8	\[ \frac{x}{\color{blue}{\mathsf{fma}\left(1, y, -z \cdot t\right) + \mathsf{fma}\left(-z, t, z \cdot t\right)}} \]
      fma-def [<=] 99.8	\[ \frac{x}{\color{blue}{\left(1 \cdot y + \left(-z \cdot t\right)\right)} + \mathsf{fma}\left(-z, t, z \cdot t\right)} \]
      *-un-lft-identity [<=] 99.8	\[ \frac{x}{\left(\color{blue}{y} + \left(-z \cdot t\right)\right) + \mathsf{fma}\left(-z, t, z \cdot t\right)} \]
      +-commutative [=>] 99.8	\[ \frac{x}{\color{blue}{\left(\left(-z \cdot t\right) + y\right)} + \mathsf{fma}\left(-z, t, z \cdot t\right)} \]
      associate-+l+ [=>] 99.9	\[ \frac{x}{\color{blue}{\left(-z \cdot t\right) + \left(y + \mathsf{fma}\left(-z, t, z \cdot t\right)\right)}} \]

   if 5.0000000000000003e288 < (*.f64 z t)

   1. Initial program 71.4%

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

   2. Applied egg-rr 12.7%

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

      [Start] 71.4	\[ \frac{x}{y - z \cdot t} \]
      *-un-lft-identity [=>] 71.4	\[ \frac{x}{\color{blue}{1 \cdot y} - z \cdot t} \]
      prod-diff [=>] 12.7	\[ \frac{x}{\color{blue}{\mathsf{fma}\left(1, y, -t \cdot z\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)}} \]
      *-commutative [<=] 12.7	\[ \frac{x}{\mathsf{fma}\left(1, y, -\color{blue}{z \cdot t}\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)} \]
      fma-neg [<=] 12.7	\[ \frac{x}{\color{blue}{\left(1 \cdot y - z \cdot t\right)} + \mathsf{fma}\left(-t, z, t \cdot z\right)} \]
      prod-diff [=>] 12.7	\[ \frac{x}{\color{blue}{\left(\mathsf{fma}\left(1, y, -t \cdot z\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)\right)} + \mathsf{fma}\left(-t, z, t \cdot z\right)} \]
      *-commutative [<=] 12.7	\[ \frac{x}{\left(\mathsf{fma}\left(1, y, -\color{blue}{z \cdot t}\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)} \]
      fma-neg [<=] 12.7	\[ \frac{x}{\left(\color{blue}{\left(1 \cdot y - z \cdot t\right)} + \mathsf{fma}\left(-t, z, t \cdot z\right)\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)} \]
      *-un-lft-identity [<=] 12.7	\[ \frac{x}{\left(\left(\color{blue}{y} - z \cdot t\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)} \]
      associate-+l+ [=>] 12.7	\[ \frac{x}{\color{blue}{\left(y - z \cdot t\right) + \left(\mathsf{fma}\left(-t, z, t \cdot z\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)\right)}} \]
      *-commutative [<=] 12.7	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\mathsf{fma}\left(-t, z, \color{blue}{z \cdot t}\right) + \mathsf{fma}\left(-t, z, t \cdot z\right)\right)} \]
      *-commutative [<=] 12.7	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\mathsf{fma}\left(-t, z, z \cdot t\right) + \mathsf{fma}\left(-t, z, \color{blue}{z \cdot t}\right)\right)} \]

   3. Simplified 12.7%

      \[\leadsto \frac{x}{\color{blue}{\left(y - t \cdot z\right) + 2 \cdot \mathsf{fma}\left(-t, z, t \cdot z\right)}} \]
      Proof

      [Start] 12.7	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\mathsf{fma}\left(-t, z, z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right)} \]
      associate-+r+ [=>] 12.7	\[ \frac{x}{\color{blue}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)}} \]
      fma-udef [=>] 12.7	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \color{blue}{\left(\left(-t\right) \cdot z + z \cdot t\right)}} \]
      neg-mul-1 [=>] 12.7	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{\left(-1 \cdot t\right)} \cdot z + z \cdot t\right)} \]
      associate-*r* [<=] 12.7	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{-1 \cdot \left(t \cdot z\right)} + z \cdot t\right)} \]
      *-commutative [<=] 12.7	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(-1 \cdot \color{blue}{\left(z \cdot t\right)} + z \cdot t\right)} \]
      mul-1-neg [=>] 12.7	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{\left(-z \cdot t\right)} + z \cdot t\right)} \]
      *-rgt-identity [<=] 12.7	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{\left(-z \cdot t\right) \cdot 1} + z \cdot t\right)} \]
      fma-udef [<=] 12.7	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \color{blue}{\mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)}} \]
      associate-+r+ [<=] 12.7	\[ \frac{x}{\color{blue}{\left(y - z \cdot t\right) + \left(\mathsf{fma}\left(-t, z, z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)}} \]
      fma-udef [=>] 12.7	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\color{blue}{\left(\left(-t\right) \cdot z + z \cdot t\right)} + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)} \]
      distribute-lft-neg-in [<=] 12.7	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\left(\color{blue}{\left(-t \cdot z\right)} + z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)} \]
      *-commutative [<=] 12.7	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\left(\left(-\color{blue}{z \cdot t}\right) + z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)} \]
      associate-+l+ [=>] 12.7	\[ \frac{x}{\left(y - z \cdot t\right) + \color{blue}{\left(\left(-z \cdot t\right) + \left(z \cdot t + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)\right)}} \]
      *-rgt-identity [<=] 12.7	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\color{blue}{\left(-z \cdot t\right) \cdot 1} + \left(z \cdot t + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)\right)} \]
      associate-+l+ [<=] 12.7	\[ \frac{x}{\left(y - z \cdot t\right) + \color{blue}{\left(\left(\left(-z \cdot t\right) \cdot 1 + z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)}} \]

   4. Taylor expanded in t around -inf 71.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{t \cdot \left(2 \cdot \left(-1 \cdot z + z\right) - -1 \cdot z\right)}} \]

   5. Simplified 99.4%

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

      [Start] 71.2	\[ -1 \cdot \frac{x}{t \cdot \left(2 \cdot \left(-1 \cdot z + z\right) - -1 \cdot z\right)} \]
      associate-*r/ [=>] 71.2	\[ \color{blue}{\frac{-1 \cdot x}{t \cdot \left(2 \cdot \left(-1 \cdot z + z\right) - -1 \cdot z\right)}} \]
      neg-mul-1 [<=] 71.2	\[ \frac{\color{blue}{-x}}{t \cdot \left(2 \cdot \left(-1 \cdot z + z\right) - -1 \cdot z\right)} \]
      *-commutative [=>] 71.2	\[ \frac{-x}{\color{blue}{\left(2 \cdot \left(-1 \cdot z + z\right) - -1 \cdot z\right) \cdot t}} \]
      associate-/r* [=>] 99.4	\[ \color{blue}{\frac{\frac{-x}{2 \cdot \left(-1 \cdot z + z\right) - -1 \cdot z}}{t}} \]
      distribute-lft1-in [=>] 99.4	\[ \frac{\frac{-x}{2 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot z\right)} - -1 \cdot z}}{t} \]
      metadata-eval [=>] 99.4	\[ \frac{\frac{-x}{2 \cdot \left(\color{blue}{0} \cdot z\right) - -1 \cdot z}}{t} \]
      mul0-lft [=>] 99.4	\[ \frac{\frac{-x}{2 \cdot \color{blue}{0} - -1 \cdot z}}{t} \]
      metadata-eval [=>] 99.4	\[ \frac{\frac{-x}{\color{blue}{0} - -1 \cdot z}}{t} \]
      neg-sub0 [<=] 99.4	\[ \frac{\frac{-x}{\color{blue}{--1 \cdot z}}}{t} \]
      mul-1-neg [=>] 99.4	\[ \frac{\frac{-x}{-\color{blue}{\left(-z\right)}}}{t} \]
      remove-double-neg [=>] 99.4	\[ \frac{\frac{-x}{\color{blue}{z}}}{t} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+265}:\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+288}:\\ \;\;\;\;\frac{x}{\left(y + \mathsf{fma}\left(-z, t, z \cdot t\right)\right) - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	70.9%
Cost	1177

\[\begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-63}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+37} \lor \neg \left(y \leq 6.8 \cdot 10^{+78}\right) \land y \leq 6.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 2
Accuracy	71.0%
Cost	1177

\[\begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-77}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+78} \lor \neg \left(y \leq 6.6 \cdot 10^{+131}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternative 3
Accuracy	71.1%
Cost	1177

\[\begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-77}:\\ \;\;\;\;x \cdot \frac{\frac{-1}{t}}{z}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{+78} \lor \neg \left(y \leq 6.5 \cdot 10^{+131}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternative 4
Accuracy	70.9%
Cost	1176

\[\begin{array}{l} t_1 := \frac{-x}{z \cdot t}\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 50000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+131}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5
Accuracy	99.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+265}:\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+288}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternative 6
Accuracy	52.7%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{+203} \lor \neg \left(z \leq 4.5 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \]

Alternative 7
Accuracy	52.8%
Cost	320

\[\frac{1}{\frac{y}{x}} \]

Alternative 8
Accuracy	53.2%
Cost	192

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

Reproduce

herbie shell --seed 2023137 
(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))))