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

Average Error: 2.9 → 1.5

Time: 7.9s

Precision: binary64

Cost: 708

\[ \begin{array}{c}[z, t] = \mathsf{sort}([z, t])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+253}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) 1e+253) (/ x (- y (* z t))) (/ (/ (- x) z) t)))

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) <= 1e+253) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-x / z) / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x / (y - (z * t))
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * t) <= 1d+253) then
        tmp = x / (y - (z * t))
    else
        tmp = (-x / z) / t
    end if
    code = tmp
end function

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) <= 1e+253) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-x / z) / t;
	}
	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) <= 1e+253:
		tmp = x / (y - (z * t))
	else:
		tmp = (-x / z) / t
	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) <= 1e+253)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(Float64(-x) / z) / t);
	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) <= 1e+253)
		tmp = x / (y - (z * t));
	else
		tmp = (-x / z) / t;
	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], 1e+253], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision]]

Target

Original	2.9
Target	1.7
Herbie	1.5

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 9.9999999999999994e252

   1. Initial program 1.6

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

   if 9.9999999999999994e252 < (*.f64 z t)

   1. Initial program 15.8

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

   2. Applied egg-rr 46.7

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

   3. Simplified 46.7

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

      [Start] 46.7	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\mathsf{fma}\left(-t, z, z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right)} \]
      associate-+r+ [=>] 46.6	\[ \frac{x}{\color{blue}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)}} \]
      fma-udef [=>] 46.6	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \color{blue}{\left(\left(-t\right) \cdot z + z \cdot t\right)}} \]
      neg-mul-1 [=>] 46.6	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{\left(-1 \cdot t\right)} \cdot z + z \cdot t\right)} \]
      associate-*r* [<=] 46.6	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{-1 \cdot \left(t \cdot z\right)} + z \cdot t\right)} \]
      *-commutative [<=] 46.6	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(-1 \cdot \color{blue}{\left(z \cdot t\right)} + z \cdot t\right)} \]
      mul-1-neg [=>] 46.6	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{\left(-z \cdot t\right)} + z \cdot t\right)} \]
      *-rgt-identity [<=] 46.6	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \left(\color{blue}{\left(-z \cdot t\right) \cdot 1} + z \cdot t\right)} \]
      fma-udef [<=] 46.6	\[ \frac{x}{\left(\left(y - z \cdot t\right) + \mathsf{fma}\left(-t, z, z \cdot t\right)\right) + \color{blue}{\mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)}} \]
      associate-+r+ [<=] 46.6	\[ \frac{x}{\color{blue}{\left(y - z \cdot t\right) + \left(\mathsf{fma}\left(-t, z, z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)}} \]
      fma-udef [=>] 46.6	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\color{blue}{\left(\left(-t\right) \cdot z + z \cdot t\right)} + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)} \]
      distribute-lft-neg-in [<=] 46.6	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\left(\color{blue}{\left(-t \cdot z\right)} + z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)} \]
      *-commutative [<=] 46.6	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\left(\left(-\color{blue}{z \cdot t}\right) + z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)} \]
      associate-+l+ [=>] 46.6	\[ \frac{x}{\left(y - z \cdot t\right) + \color{blue}{\left(\left(-z \cdot t\right) + \left(z \cdot t + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)\right)}} \]
      *-rgt-identity [<=] 46.6	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\color{blue}{\left(-z \cdot t\right) \cdot 1} + \left(z \cdot t + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)\right)} \]
      associate-+l+ [<=] 46.6	\[ \frac{x}{\left(y - z \cdot t\right) + \color{blue}{\left(\left(\left(-z \cdot t\right) \cdot 1 + z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)}} \]
      fma-udef [<=] 46.6	\[ \frac{x}{\left(y - z \cdot t\right) + \left(\color{blue}{\mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)} + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)} \]
      *-commutative [=>] 46.6	\[ \frac{x}{\left(y - \color{blue}{t \cdot z}\right) + \left(\mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right) + \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)\right)} \]
      count-2 [=>] 46.6	\[ \frac{x}{\left(y - t \cdot z\right) + \color{blue}{2 \cdot \mathsf{fma}\left(-z \cdot t, 1, z \cdot t\right)}} \]
      fma-udef [=>] 46.6	\[ \frac{x}{\left(y - t \cdot z\right) + 2 \cdot \color{blue}{\left(\left(-z \cdot t\right) \cdot 1 + z \cdot t\right)}} \]
      *-rgt-identity [=>] 46.6	\[ \frac{x}{\left(y - t \cdot z\right) + 2 \cdot \left(\color{blue}{\left(-z \cdot t\right)} + z \cdot t\right)} \]
      distribute-rgt-neg-in [=>] 46.6	\[ \frac{x}{\left(y - t \cdot z\right) + 2 \cdot \left(\color{blue}{z \cdot \left(-t\right)} + z \cdot t\right)} \]
      *-commutative [=>] 46.6	\[ \frac{x}{\left(y - t \cdot z\right) + 2 \cdot \left(\color{blue}{\left(-t\right) \cdot z} + z \cdot t\right)} \]
      fma-udef [<=] 46.7	\[ \frac{x}{\left(y - t \cdot z\right) + 2 \cdot \color{blue}{\mathsf{fma}\left(-t, z, z \cdot t\right)}} \]
      *-commutative [=>] 46.7	\[ \frac{x}{\left(y - t \cdot z\right) + 2 \cdot \mathsf{fma}\left(-t, z, \color{blue}{t \cdot z}\right)} \]

   4. Taylor expanded in t around -inf 16.3

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

   5. Simplified 0.6

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

      [Start] 16.3	\[ -1 \cdot \frac{x}{t \cdot \left(2 \cdot \left(-1 \cdot z + z\right) - -1 \cdot z\right)} \]
      associate-*r/ [=>] 16.3	\[ \color{blue}{\frac{-1 \cdot x}{t \cdot \left(2 \cdot \left(-1 \cdot z + z\right) - -1 \cdot z\right)}} \]
      neg-mul-1 [<=] 16.3	\[ \frac{\color{blue}{-x}}{t \cdot \left(2 \cdot \left(-1 \cdot z + z\right) - -1 \cdot z\right)} \]
      *-commutative [=>] 16.3	\[ \frac{-x}{\color{blue}{\left(2 \cdot \left(-1 \cdot z + z\right) - -1 \cdot z\right) \cdot t}} \]
      associate-/r* [=>] 0.6	\[ \color{blue}{\frac{\frac{-x}{2 \cdot \left(-1 \cdot z + z\right) - -1 \cdot z}}{t}} \]
      distribute-lft1-in [=>] 0.6	\[ \frac{\frac{-x}{2 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot z\right)} - -1 \cdot z}}{t} \]
      metadata-eval [=>] 0.6	\[ \frac{\frac{-x}{2 \cdot \left(\color{blue}{0} \cdot z\right) - -1 \cdot z}}{t} \]
      mul0-lft [=>] 0.6	\[ \frac{\frac{-x}{2 \cdot \color{blue}{0} - -1 \cdot z}}{t} \]
      metadata-eval [=>] 0.6	\[ \frac{\frac{-x}{\color{blue}{0} - -1 \cdot z}}{t} \]
      neg-sub0 [<=] 0.6	\[ \frac{\frac{-x}{\color{blue}{--1 \cdot z}}}{t} \]
      mul-1-neg [=>] 0.6	\[ \frac{\frac{-x}{-\color{blue}{\left(-z\right)}}}{t} \]
      remove-double-neg [=>] 0.6	\[ \frac{\frac{-x}{\color{blue}{z}}}{t} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+253}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternatives

Alternative 1
Error	20.1
Cost	1179

\[\begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{+137} \lor \neg \left(z \leq -9.5 \cdot 10^{+110} \lor \neg \left(z \leq -7.2 \cdot 10^{+74}\right) \land \left(z \leq 4.6 \cdot 10^{-185} \lor \neg \left(z \leq 4.8 \cdot 10^{-172}\right) \land z \leq 4.2 \cdot 10^{-112}\right)\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 2
Error	19.3
Cost	1176

\[\begin{array}{l} t_1 := \frac{\frac{-x}{t}}{z}\\ t_2 := \frac{-x}{z \cdot t}\\ \mathbf{if}\;z \leq -1.82 \cdot 10^{+137}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{+111}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-185}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-172}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-112}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	19.2
Cost	1176

\[\begin{array}{l} t_1 := \frac{-x}{z \cdot t}\\ \mathbf{if}\;z \leq -1.82 \cdot 10^{+137}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{+111}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -8.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-185}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternative 4
Error	27.0
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+197} \lor \neg \left(z \leq 1900000000000\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5
Error	30.1
Cost	192

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

Reproduce

herbie shell --seed 2022356 
(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))))