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

Average Error: 10.7 → 6.0

Time: 6.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x - y \cdot z}{t_1}\\ t_3 := \frac{\mathsf{fma}\left(y, -z, x\right)}{t_1}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;t_2 \leq -1.20727392575 \cdot 10^{-313}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;t_2 \leq 3.0060081326494514 \cdot 10^{+255}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{a} + \frac{t}{a \cdot a} \cdot \left(\frac{y}{z} - \frac{x}{z \cdot z}\right)\right) - \frac{x}{z \cdot a}\\ \end{array} \]

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 (- t (* z a)))
        (t_2 (/ (- x (* y z)) t_1))
        (t_3 (/ (fma y (- z) x) t_1)))
   (if (<= t_2 (- INFINITY))
     (/ y a)
     (if (<= t_2 -1.20727392575e-313)
       t_3
       (if (<= t_2 0.0)
         (/ y a)
         (if (<= t_2 3.0060081326494514e+255)
           t_3
           (-
            (+ (/ y a) (* (/ t (* a a)) (- (/ y z) (/ x (* z z)))))
            (/ x (* z 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 = t - (z * a);
	double t_2 = (x - (y * z)) / t_1;
	double t_3 = fma(y, -z, x) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = y / a;
	} else if (t_2 <= -1.20727392575e-313) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = y / a;
	} else if (t_2 <= 3.0060081326494514e+255) {
		tmp = t_3;
	} else {
		tmp = ((y / a) + ((t / (a * a)) * ((y / z) - (x / (z * z))))) - (x / (z * 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(t - Float64(z * a))
	t_2 = Float64(Float64(x - Float64(y * z)) / t_1)
	t_3 = Float64(fma(y, Float64(-z), x) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(y / a);
	elseif (t_2 <= -1.20727392575e-313)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = Float64(y / a);
	elseif (t_2 <= 3.0060081326494514e+255)
		tmp = t_3;
	else
		tmp = Float64(Float64(Float64(y / a) + Float64(Float64(t / Float64(a * a)) * Float64(Float64(y / z) - Float64(x / Float64(z * z))))) - Float64(x / Float64(z * 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[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * (-z) + x), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(y / a), $MachinePrecision], If[LessEqual[t$95$2, -1.20727392575e-313], t$95$3, If[LessEqual[t$95$2, 0.0], N[(y / a), $MachinePrecision], If[LessEqual[t$95$2, 3.0060081326494514e+255], t$95$3, N[(N[(N[(y / a), $MachinePrecision] + N[(N[(t / N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(y / z), $MachinePrecision] - N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	10.7
Target	1.6
Herbie	6.0

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0 or -1.20727392575e-313 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 31.7

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

   2. Taylor expanded in z around inf 23.3

      \[\leadsto \color{blue}{\frac{y}{a}} \]

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.20727392575e-313 or -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 3.0060081326494514e255

   1. Initial program 0.2

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

   2. Applied egg-rr 0.2

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

   if 3.0060081326494514e255 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 52.5

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

   2. Taylor expanded in t around 0 31.1

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

   3. Simplified 18.3

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

4. Final simplification 6.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1.20727392575 \cdot 10^{-313}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -z, x\right)}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 3.0060081326494514 \cdot 10^{+255}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, -z, x\right)}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{a} + \frac{t}{a \cdot a} \cdot \left(\frac{y}{z} - \frac{x}{z \cdot z}\right)\right) - \frac{x}{z \cdot a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022134 
(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))))