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

Average Error: 10.9 → 5.2

Time: 5.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x}{t_1}\\ t_3 := \frac{x - y \cdot z}{t - a \cdot z}\\ t_4 := t_2 - \frac{-z \cdot y}{\left(t + z \cdot a\right) - \left(t + t\right)}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;t_2 - \frac{z}{\frac{t_1}{y}}\\ \mathbf{elif}\;t_3 \leq -5 \cdot 10^{-321}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{\frac{x}{a} - \frac{y \cdot t}{a \cdot a}}{z}, \frac{y}{a}\right)\\ \mathbf{elif}\;t_3 \leq 5 \cdot 10^{+112}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_2 - \frac{y}{-a}\\ \end{array} \]

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 (- t (* z a)))
        (t_2 (/ x t_1))
        (t_3 (/ (- x (* y z)) (- t (* a z))))
        (t_4 (- t_2 (/ (- (* z y)) (- (+ t (* z a)) (+ t t))))))
   (if (<= t_3 (- INFINITY))
     (- t_2 (/ z (/ t_1 y)))
     (if (<= t_3 -5e-321)
       t_4
       (if (<= t_3 0.0)
         (fma -1.0 (/ (- (/ x a) (/ (* y t) (* a a))) z) (/ y a))
         (if (<= t_3 5e+112) t_4 (- 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 = t - (z * a);
	double t_2 = x / t_1;
	double t_3 = (x - (y * z)) / (t - (a * z));
	double t_4 = t_2 - (-(z * y) / ((t + (z * a)) - (t + t)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_2 - (z / (t_1 / y));
	} else if (t_3 <= -5e-321) {
		tmp = t_4;
	} else if (t_3 <= 0.0) {
		tmp = fma(-1.0, (((x / a) - ((y * t) / (a * a))) / z), (y / a));
	} else if (t_3 <= 5e+112) {
		tmp = t_4;
	} else {
		tmp = t_2 - (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(t - Float64(z * a))
	t_2 = Float64(x / t_1)
	t_3 = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(a * z)))
	t_4 = Float64(t_2 - Float64(Float64(-Float64(z * y)) / Float64(Float64(t + Float64(z * a)) - Float64(t + t))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(t_2 - Float64(z / Float64(t_1 / y)));
	elseif (t_3 <= -5e-321)
		tmp = t_4;
	elseif (t_3 <= 0.0)
		tmp = fma(-1.0, Float64(Float64(Float64(x / a) - Float64(Float64(y * t) / Float64(a * a))) / z), Float64(y / a));
	elseif (t_3 <= 5e+112)
		tmp = t_4;
	else
		tmp = Float64(t_2 - Float64(y / Float64(-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[(x / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 - N[((-N[(z * y), $MachinePrecision]) / N[(N[(t + N[(z * a), $MachinePrecision]), $MachinePrecision] - N[(t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(t$95$2 - N[(z / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-321], t$95$4, If[LessEqual[t$95$3, 0.0], N[(-1.0 * 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+112], t$95$4, N[(t$95$2 - N[(y / (-a)), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Target

Original	10.9
Target	1.7
Herbie	5.2

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

Derivation

1. Split input into 4 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. Applied egg-rr 0.4

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

   3. Taylor expanded in y around 0 64.0

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

   4. Simplified 0.3

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -4.99994e-321 or -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 5e112

   1. Initial program 0.2

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

   2. Applied egg-rr 4.1

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

   3. Applied egg-rr 0.2

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

   if -4.99994e-321 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 26.3

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

   2. Taylor expanded in z around inf 28.0

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

   3. Simplified 17.1

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

   if 5e112 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 28.2

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

   2. Applied egg-rr 20.9

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

   3. Taylor expanded in y around 0 28.2

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

   4. Simplified 20.4

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

   5. Taylor expanded in z around inf 15.2

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

   6. Simplified 15.2

      \[\leadsto \frac{x}{t - z \cdot a} - \color{blue}{\frac{y}{-a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - a \cdot z} \leq -\infty:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{z}{\frac{t - z \cdot a}{y}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq -5 \cdot 10^{-321}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{-z \cdot y}{\left(t + z \cdot a\right) - \left(t + t\right)}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{\frac{x}{a} - \frac{y \cdot t}{a \cdot a}}{z}, \frac{y}{a}\right)\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - a \cdot z} \leq 5 \cdot 10^{+112}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{-z \cdot y}{\left(t + z \cdot a\right) - \left(t + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{-a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022203 
(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))))