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

Average Error: 10.7 → 3.2

Time: 9.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := z \cdot a - t\\ t_2 := \frac{y}{\frac{t_1}{z}} - \frac{x}{t_1}\\ t_3 := \frac{x - y \cdot z}{t - z \cdot a}\\ t_4 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{-313}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_3 \leq 10^{-23}:\\ \;\;\;\;\frac{y \cdot z - x}{\mathsf{fma}\left(z, a, -t\right)}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

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 (- (* z a) t))
        (t_2 (- (/ y (/ t_1 z)) (/ x t_1)))
        (t_3 (/ (- x (* y z)) (- t (* z a))))
        (t_4 (/ (- y (/ x z)) a)))
   (if (<= t_3 -1e-313)
     t_2
     (if (<= t_3 0.0)
       t_4
       (if (<= t_3 1e-23)
         (/ (- (* y z) x) (fma z a (- t)))
         (if (<= t_3 INFINITY) t_2 t_4))))))

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 = (z * a) - t;
	double t_2 = (y / (t_1 / z)) - (x / t_1);
	double t_3 = (x - (y * z)) / (t - (z * a));
	double t_4 = (y - (x / z)) / a;
	double tmp;
	if (t_3 <= -1e-313) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = t_4;
	} else if (t_3 <= 1e-23) {
		tmp = ((y * z) - x) / fma(z, a, -t);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = t_4;
	}
	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(Float64(z * a) - t)
	t_2 = Float64(Float64(y / Float64(t_1 / z)) - Float64(x / t_1))
	t_3 = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(z * a)))
	t_4 = Float64(Float64(y - Float64(x / z)) / a)
	tmp = 0.0
	if (t_3 <= -1e-313)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = t_4;
	elseif (t_3 <= 1e-23)
		tmp = Float64(Float64(Float64(y * z) - x) / fma(z, a, Float64(-t)));
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = t_4;
	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 * a), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision] - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-313], t$95$2, If[LessEqual[t$95$3, 0.0], t$95$4, If[LessEqual[t$95$3, 1e-23], N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(z * a + (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$2, t$95$4]]]]]]]]

Target

Original	10.7
Target	1.8
Herbie	3.2

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.00000000001e-313 or 9.9999999999999996e-24 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 6.4

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

   2. Simplified 6.4

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

   3. Applied egg-rr 1.3

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

   if -1.00000000001e-313 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0 or +inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 35.1

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

   2. Simplified 35.1

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

   3. Applied egg-rr 34.6

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

   4. Applied egg-rr 34.6

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

   5. Taylor expanded in a around inf 12.6

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

   6. Simplified 12.6

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

   if 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 9.9999999999999996e-24

   1. Initial program 0.2

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

   2. Simplified 0.2

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

   3. Applied egg-rr 0.2

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

4. Final simplification 3.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1 \cdot 10^{-313}:\\ \;\;\;\;\frac{y}{\frac{z \cdot a - t}{z}} - \frac{x}{z \cdot a - t}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 10^{-23}:\\ \;\;\;\;\frac{y \cdot z - x}{\mathsf{fma}\left(z, a, -t\right)}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\frac{y}{\frac{z \cdot a - t}{z}} - \frac{x}{z \cdot a - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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))))