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

Average Error: 2.8 → 0.4

Time: 9.5s

Precision: binary64

Cost: 968

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+201}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (/ (/ (- x) z) t)
   (if (<= (* z t) 2e+201) (/ x (- y (* z t))) (/ (/ x (- t)) z))))

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) <= -((double) INFINITY)) {
		tmp = (-x / z) / t;
	} else if ((z * t) <= 2e+201) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / -t) / z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	return x / (y - (z * t));
}

↓

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = (-x / z) / t;
	} else if ((z * t) <= 2e+201) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / -t) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return x / (y - (z * t))

↓

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = (-x / z) / t
	elif (z * t) <= 2e+201:
		tmp = x / (y - (z * t))
	else:
		tmp = (x / -t) / z
	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) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-x) / z) / t);
	elseif (Float64(z * t) <= 2e+201)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(x / Float64(-t)) / z);
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x / (y - (z * t));
end

↓

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = (-x / z) / t;
	elseif ((z * t) <= 2e+201)
		tmp = x / (y - (z * t));
	else
		tmp = (x / -t) / z;
	end
	tmp_2 = 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], (-Infinity)], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+201], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / (-t)), $MachinePrecision] / z), $MachinePrecision]]]

Target

Original	2.8
Target	1.9
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -inf.0

   1. Initial program 21.8

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

   2. Applied egg-rr 64.0

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

   3. Simplified 64.0

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

      [Start] 64.0	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\mathsf{fma}\left(-t, z, z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right)} \]
      associate-+r+ [=>] 64.0	\[ \frac{x}{\color{blue}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)}} \]
      fma-udef [=>] 64.0	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \color{blue}{\left(\left(-t\right) \cdot z + z \cdot t\right)}} \]
      neg-mul-1 [=>] 64.0	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{\left(-1 \cdot t\right)} \cdot z + z \cdot t\right)} \]
      associate-*r* [<=] 64.0	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{-1 \cdot \left(t \cdot z\right)} + z \cdot t\right)} \]
      *-commutative [<=] 64.0	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(-1 \cdot \color{blue}{\left(z \cdot t\right)} + z \cdot t\right)} \]
      mul-1-neg [=>] 64.0	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{\left(-z \cdot t\right)} + z \cdot t\right)} \]
      *-rgt-identity [<=] 64.0	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{\left(-z \cdot t\right) \cdot 1} + z \cdot t\right)} \]
      fma-udef [<=] 64.0	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \color{blue}{\mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)}} \]
      associate-+r+ [<=] 64.0	\[ \frac{x}{\color{blue}{\left(y - z \cdot t\right) + \left(\mathsf{fma}\left(-t, z, z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)}} \]
      fma-udef [=>] 64.0	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\color{blue}{\left(\left(-t\right) \cdot z + z \cdot t\right)} + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)} \]
      distribute-lft-neg-in [<=] 64.0	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\left(\color{blue}{\left(-t \cdot z\right)} + z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)} \]
      *-commutative [<=] 64.0	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\left(\left(-\color{blue}{z \cdot t}\right) + z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)} \]
      associate-+l+ [=>] 64.0	\[ \frac{x}{\left(y - z \cdot t\right) + \color{blue}{\left(\left(-z \cdot t\right) + \left(z \cdot t + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)\right)}} \]
      *-rgt-identity [<=] 64.0	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\color{blue}{\left(-z \cdot t\right) \cdot 1} + \left(z \cdot t + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)\right)} \]
      associate-+l+ [<=] 64.0	\[ \frac{x}{\left(y - z \cdot t\right) + \color{blue}{\left(\left(\left(-z \cdot t\right) \cdot 1 + z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)}} \]

   4. Taylor expanded in t around -inf 21.8

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

   5. Simplified 0.1

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

      [Start] 21.8	\[ -1 \cdot \frac{x}{t \cdot \left(2 \cdot \left(-1 \cdot z + z\right) - -1 \cdot z\right)} \]
      associate-*r/ [=>] 21.8	\[ \color{blue}{\frac{-1 \cdot x}{t \cdot \left(2 \cdot \left(-1 \cdot z + z\right) - -1 \cdot z\right)}} \]
      neg-mul-1 [<=] 21.8	\[ \frac{\color{blue}{-x}}{t \cdot \left(2 \cdot \left(-1 \cdot z + z\right) - -1 \cdot z\right)} \]
      *-commutative [=>] 21.8	\[ \frac{-x}{\color{blue}{\left(2 \cdot \left(-1 \cdot z + z\right) - -1 \cdot z\right) \cdot t}} \]
      associate-/r* [=>] 0.1	\[ \color{blue}{\frac{\frac{-x}{2 \cdot \left(-1 \cdot z + z\right) - -1 \cdot z}}{t}} \]
      distribute-lft1-in [=>] 0.1	\[ \frac{\frac{-x}{2 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot z\right)} - -1 \cdot z}}{t} \]
      metadata-eval [=>] 0.1	\[ \frac{\frac{-x}{2 \cdot \left(\color{blue}{0} \cdot z\right) - -1 \cdot z}}{t} \]
      mul0-lft [=>] 0.1	\[ \frac{\frac{-x}{2 \cdot \color{blue}{0} - -1 \cdot z}}{t} \]
      metadata-eval [=>] 0.1	\[ \frac{\frac{-x}{\color{blue}{0} - -1 \cdot z}}{t} \]
      neg-sub0 [<=] 0.1	\[ \frac{\frac{-x}{\color{blue}{--1 \cdot z}}}{t} \]
      mul-1-neg [=>] 0.1	\[ \frac{\frac{-x}{-\color{blue}{\left(-z\right)}}}{t} \]
      remove-double-neg [=>] 0.1	\[ \frac{\frac{-x}{\color{blue}{z}}}{t} \]

   if -inf.0 < (*.f64 z t) < 2.00000000000000008e201

   1. Initial program 0.1

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

   if 2.00000000000000008e201 < (*.f64 z t)

   1. Initial program 11.4

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

   2. Applied egg-rr 35.7

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

   3. Simplified 35.7

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

      [Start] 35.7	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\mathsf{fma}\left(-t, z, z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right)} \]
      associate-+r+ [=>] 35.7	\[ \frac{x}{\color{blue}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)}} \]
      fma-udef [=>] 35.7	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \color{blue}{\left(\left(-t\right) \cdot z + z \cdot t\right)}} \]
      neg-mul-1 [=>] 35.7	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{\left(-1 \cdot t\right)} \cdot z + z \cdot t\right)} \]
      associate-*r* [<=] 35.7	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{-1 \cdot \left(t \cdot z\right)} + z \cdot t\right)} \]
      *-commutative [<=] 35.7	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(-1 \cdot \color{blue}{\left(z \cdot t\right)} + z \cdot t\right)} \]
      mul-1-neg [=>] 35.7	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{\left(-z \cdot t\right)} + z \cdot t\right)} \]
      *-rgt-identity [<=] 35.7	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{\left(-z \cdot t\right) \cdot 1} + z \cdot t\right)} \]
      fma-udef [<=] 35.7	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \color{blue}{\mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)}} \]
      associate-+r+ [<=] 35.7	\[ \frac{x}{\color{blue}{\left(y - z \cdot t\right) + \left(\mathsf{fma}\left(-t, z, z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)}} \]
      fma-udef [=>] 35.7	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\color{blue}{\left(\left(-t\right) \cdot z + z \cdot t\right)} + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)} \]
      distribute-lft-neg-in [<=] 35.7	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\left(\color{blue}{\left(-t \cdot z\right)} + z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)} \]
      *-commutative [<=] 35.7	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\left(\left(-\color{blue}{z \cdot t}\right) + z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)} \]
      associate-+l+ [=>] 35.7	\[ \frac{x}{\left(y - z \cdot t\right) + \color{blue}{\left(\left(-z \cdot t\right) + \left(z \cdot t + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)\right)}} \]
      *-rgt-identity [<=] 35.7	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\color{blue}{\left(-z \cdot t\right) \cdot 1} + \left(z \cdot t + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)\right)} \]
      associate-+l+ [<=] 35.7	\[ \frac{x}{\left(y - z \cdot t\right) + \color{blue}{\left(\left(\left(-z \cdot t\right) \cdot 1 + z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)}} \]

   4. Taylor expanded in y around 0 37.2

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

   5. Simplified 2.3

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

      [Start] 37.2	\[ \frac{x}{2 \cdot \left(-1 \cdot \left(t \cdot z\right) + t \cdot z\right) - t \cdot z} \]
      +-commutative [=>] 37.2	\[ \frac{x}{2 \cdot \color{blue}{\left(t \cdot z + -1 \cdot \left(t \cdot z\right)\right)} - t \cdot z} \]
      mul-1-neg [=>] 37.2	\[ \frac{x}{2 \cdot \left(t \cdot z + \color{blue}{\left(-t \cdot z\right)}\right) - t \cdot z} \]
      sub-neg [<=] 37.2	\[ \frac{x}{2 \cdot \color{blue}{\left(t \cdot z - t \cdot z\right)} - t \cdot z} \]
      +-inverses [=>] 12.9	\[ \frac{x}{2 \cdot \color{blue}{0} - t \cdot z} \]
      metadata-eval [=>] 12.9	\[ \frac{x}{\color{blue}{0} - t \cdot z} \]
      sub0-neg [=>] 12.9	\[ \frac{x}{\color{blue}{-t \cdot z}} \]
      *-commutative [=>] 12.9	\[ \frac{x}{-\color{blue}{z \cdot t}} \]
      distribute-rgt-neg-in [=>] 12.9	\[ \frac{x}{\color{blue}{z \cdot \left(-t\right)}} \]
      neg-sub0 [=>] 12.9	\[ \frac{x}{z \cdot \color{blue}{\left(0 - t\right)}} \]
      metadata-eval [<=] 12.9	\[ \frac{x}{z \cdot \left(\color{blue}{2 \cdot 0} - t\right)} \]
      mul0-lft [<=] 12.9	\[ \frac{x}{z \cdot \left(2 \cdot \color{blue}{\left(0 \cdot t\right)} - t\right)} \]
      metadata-eval [<=] 12.9	\[ \frac{x}{z \cdot \left(2 \cdot \left(\color{blue}{\left(-1 + 1\right)} \cdot t\right) - t\right)} \]
      distribute-lft1-in [<=] 12.9	\[ \frac{x}{z \cdot \left(2 \cdot \color{blue}{\left(-1 \cdot t + t\right)} - t\right)} \]
      associate-/l/ [<=] 2.3	\[ \color{blue}{\frac{\frac{x}{2 \cdot \left(-1 \cdot t + t\right) - t}}{z}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+201}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-t}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.3
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-64}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-33}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 2
Error	18.2
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -2.65 \cdot 10^{-64}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3
Error	29.7
Cost	585

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{+29} \lor \neg \left(t \leq 1.55 \cdot 10^{+265}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4
Error	29.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+264}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 5
Error	30.1
Cost	192

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

Reproduce

herbie shell --seed 2023066 
(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))))