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

Average Error: 2.8 → 1.8

Time: 13.5s

Precision: binary64

Cost: 708

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+193}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) 1e+193) (/ 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+193) {
		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+193) 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+193) {
		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+193:
		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+193)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(x / Float64(-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+193)
		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+193], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / (-z)), $MachinePrecision] / t), $MachinePrecision]]

Target

Original	2.8
Target	1.9
Herbie	1.8

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 1.00000000000000007e193

   1. Initial program 1.7

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

   if 1.00000000000000007e193 < (*.f64 z t)

   1. Initial program 11.0

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

   2. Taylor expanded in y around 0 13.1

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

   3. Simplified 13.1

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

      [Start] 13.1	\[ -1 \cdot \frac{x}{t \cdot z} \]
      rational_best-simplify-1 [<=] 13.1	\[ -1 \cdot \frac{x}{\color{blue}{z \cdot t}} \]
      rational_best-simplify-62 [=>] 13.1	\[ \color{blue}{\frac{x \cdot -1}{z \cdot t}} \]
      rational_best-simplify-11 [<=] 13.1	\[ \frac{\color{blue}{-x}}{z \cdot t} \]
      rational_best-simplify-1 [=>] 13.1	\[ \frac{-x}{\color{blue}{t \cdot z}} \]

   4. Applied egg-rr 2.5

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

   5. Applied egg-rr 2.5

      \[\leadsto \color{blue}{\frac{\frac{x}{-z}}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+193}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \end{array} \]

Alternatives

Alternative 1
Error	19.1
Cost	1044

\[\begin{array}{l} t_1 := \frac{\frac{x}{-t}}{z}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -6.4 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-136}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-228}:\\ \;\;\;\;\frac{-x}{t \cdot z}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 2
Error	18.1
Cost	912

\[\begin{array}{l} t_1 := \frac{-x}{t \cdot z}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-136}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-42}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3
Error	19.1
Cost	912

\[\begin{array}{l} t_1 := \frac{\frac{x}{-z}}{t}\\ \mathbf{if}\;y \leq -5.1 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-136}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-42}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4
Error	30.1
Cost	192

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

Reproduce

herbie shell --seed 2023101 
(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))))