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

Average Error: 2.7 → 0.3

Time: 8.6s

Precision: binary64

Cost: 968

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+222}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+286}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -1e+222)
   (/ (/ (- x) z) t)
   (if (<= (* z t) 4e+286) (/ x (- y (* z t))) (* (/ -1.0 t) (/ x 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) <= -1e+222) {
		tmp = (-x / z) / t;
	} else if ((z * t) <= 4e+286) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-1.0 / t) * (x / z);
	}
	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+222)) then
        tmp = (-x / z) / t
    else if ((z * t) <= 4d+286) then
        tmp = x / (y - (z * t))
    else
        tmp = ((-1.0d0) / t) * (x / z)
    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+222) {
		tmp = (-x / z) / t;
	} else if ((z * t) <= 4e+286) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-1.0 / t) * (x / 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) <= -1e+222:
		tmp = (-x / z) / t
	elif (z * t) <= 4e+286:
		tmp = x / (y - (z * t))
	else:
		tmp = (-1.0 / t) * (x / 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) <= -1e+222)
		tmp = Float64(Float64(Float64(-x) / z) / t);
	elseif (Float64(z * t) <= 4e+286)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(-1.0 / t) * Float64(x / 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) <= -1e+222)
		tmp = (-x / z) / t;
	elseif ((z * t) <= 4e+286)
		tmp = x / (y - (z * t));
	else
		tmp = (-1.0 / t) * (x / 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], -1e+222], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 4e+286], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / t), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Target

Original	2.7
Target	1.8
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1e222

   1. Initial program 13.2

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

   2. Taylor expanded in y around 0 14.6

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

   3. Simplified 14.6

      \[\leadsto \color{blue}{\frac{-x}{t \cdot z}} \]
      Proof
      (/.f64 (neg.f64 x) (*.f64 t z)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x)) (*.f64 t z)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 x (*.f64 t z)))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 1.8

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

   5. Applied egg-rr 1.7

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

   if -1e222 < (*.f64 z t) < 4.00000000000000013e286

   1. Initial program 0.1

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

   if 4.00000000000000013e286 < (*.f64 z t)

   1. Initial program 18.1

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

   2. Taylor expanded in y around 0 18.3

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

   3. Simplified 18.3

      \[\leadsto \color{blue}{\frac{-x}{t \cdot z}} \]
      Proof
      (/.f64 (neg.f64 x) (*.f64 t z)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x)) (*.f64 t z)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 x (*.f64 t z)))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 0.3

      \[\leadsto \color{blue}{\frac{-1}{t} \cdot \frac{x}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+222}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{+286}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.0
Cost	912

\[\begin{array}{l} t_1 := \frac{\frac{-x}{z}}{t}\\ \mathbf{if}\;z \leq -4.306267869110735 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.796534098639511 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -3.5216086590441284 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.3954163648733133 \cdot 10^{-114}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	18.0
Cost	912

\[\begin{array}{l} t_1 := \frac{\frac{-x}{z}}{t}\\ \mathbf{if}\;t \leq -1.888809033072989 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.155048827064724 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t \leq 24939248139413.7:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.3130018544125346 \cdot 10^{+79}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternative 3
Error	19.2
Cost	912

\[\begin{array}{l} t_1 := \frac{\frac{-x}{z}}{t}\\ \mathbf{if}\;t \leq -1.888809033072989 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.155048827064724 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t \leq 24939248139413.7:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.3130018544125346 \cdot 10^{+79}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \]

Alternative 4
Error	19.2
Cost	912

\[\begin{array}{l} \mathbf{if}\;t \leq -1.888809033072989 \cdot 10^{-130}:\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 1.155048827064724 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t \leq 24939248139413.7:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;t \leq 2.3130018544125346 \cdot 10^{+79}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \]

Alternative 5
Error	27.1
Cost	584

\[\begin{array}{l} t_1 := \frac{\frac{x}{t}}{z}\\ \mathbf{if}\;t \leq -3.0091781917561237 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.825397739342984 \cdot 10^{+130}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	29.5
Cost	192

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

Reproduce

herbie shell --seed 2022300 
(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))))