Average Error: 11.0 → 1.7
Time: 5.5s
Precision: binary64
\[\frac{x - y \cdot z}{t - a \cdot z} \]
↓
\[\begin{array}{l} t_1 := \frac{y}{\frac{t}{z} - a}\\ t_2 := t - z \cdot a\\ \mathbf{if}\;z \leq -1.965510614121794 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{t_2} - t_1\\ \mathbf{elif}\;z \leq 4.4228944462314166 \cdot 10^{-44}:\\ \;\;\;\;\frac{x - z \cdot y}{\mathsf{fma}\left(-a, z, t\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t_2}{x}\right)}^{-1} - t_1\\ \end{array} \]
(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 (/ y (- (/ t z) a))) (t_2 (- t (* z a))))
   (if (<= z -1.965510614121794e+42)
     (- (/ x t_2) t_1)
     (if (<= z 4.4228944462314166e-44)
       (/ (- x (* z y)) (fma (- a) z t))
       (- (pow (/ t_2 x) -1.0) t_1)))))
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 = y / ((t / z) - a);
	double t_2 = t - (z * a);
	double tmp;
	if (z <= -1.965510614121794e+42) {
		tmp = (x / t_2) - t_1;
	} else if (z <= 4.4228944462314166e-44) {
		tmp = (x - (z * y)) / fma(-a, z, t);
	} else {
		tmp = pow((t_2 / x), -1.0) - t_1;
	}
	return tmp;
}

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(y / Float64(Float64(t / z) - a))
	t_2 = Float64(t - Float64(z * a))
	tmp = 0.0
	if (z <= -1.965510614121794e+42)
		tmp = Float64(Float64(x / t_2) - t_1);
	elseif (z <= 4.4228944462314166e-44)
		tmp = Float64(Float64(x - Float64(z * y)) / fma(Float64(-a), z, t));
	else
		tmp = Float64((Float64(t_2 / x) ^ -1.0) - t_1);
	end
	return tmp
end

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[(y / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.965510614121794e+42], N[(N[(x / t$95$2), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[z, 4.4228944462314166e-44], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[((-a) * z + t), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(t$95$2 / x), $MachinePrecision], -1.0], $MachinePrecision] - t$95$1), $MachinePrecision]]]]]

\frac{x - y \cdot z}{t - a \cdot z}
↓
\begin{array}{l}
t_1 := \frac{y}{\frac{t}{z} - a}\\
t_2 := t - z \cdot a\\
\mathbf{if}\;z \leq -1.965510614121794 \cdot 10^{+42}:\\
\;\;\;\;\frac{x}{t_2} - t_1\\

\mathbf{elif}\;z \leq 4.4228944462314166 \cdot 10^{-44}:\\
\;\;\;\;\frac{x - z \cdot y}{\mathsf{fma}\left(-a, z, t\right)}\\

\mathbf{else}:\\
\;\;\;\;{\left(\frac{t_2}{x}\right)}^{-1} - t_1\\


\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original11.0
Target1.8
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.965510614121794e42

    1. Initial program 24.7

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

    2. Taylor expanded in x around 0 24.7

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

    3. Simplified15.4

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

    4. Taylor expanded in t around 0 3.5

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

    if -1.965510614121794e42 < z < 4.4228944462314166e-44

    1. Initial program 0.3

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

    2. Applied egg-rr0.3

      \[\leadsto \frac{x - y \cdot z}{\color{blue}{\mathsf{fma}\left(\left(-a\right) \cdot 1, z, t\right)}} \]

    if 4.4228944462314166e-44 < z

    1. Initial program 18.8

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

    2. Taylor expanded in x around 0 18.8

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

    3. Simplified12.1

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

    4. Taylor expanded in t around 0 2.7

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

    5. Applied egg-rr2.8

      \[\leadsto \color{blue}{{\left(\frac{t - z \cdot a}{x}\right)}^{-1}} - \frac{y}{\frac{t}{z} - a} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.965510614121794 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \leq 4.4228944462314166 \cdot 10^{-44}:\\ \;\;\;\;\frac{x - z \cdot y}{\mathsf{fma}\left(-a, z, t\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t - z \cdot a}{x}\right)}^{-1} - \frac{y}{\frac{t}{z} - a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022152 
(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))))