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

Average Error: 2.8 → 0.2

Time: 6.4s

Precision: binary64

Cost: 968

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+277}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+264}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{-1}{t}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -5e+277)
   (/ (/ (- x) z) t)
   (if (<= (* z t) 1e+264) (/ x (- y (* z t))) (/ (* x (/ -1.0 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) <= -5e+277) {
		tmp = (-x / z) / t;
	} else if ((z * t) <= 1e+264) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x * (-1.0 / t)) / 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) <= (-5d+277)) then
        tmp = (-x / z) / t
    else if ((z * t) <= 1d+264) then
        tmp = x / (y - (z * t))
    else
        tmp = (x * ((-1.0d0) / t)) / 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) <= -5e+277) {
		tmp = (-x / z) / t;
	} else if ((z * t) <= 1e+264) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x * (-1.0 / 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) <= -5e+277:
		tmp = (-x / z) / t
	elif (z * t) <= 1e+264:
		tmp = x / (y - (z * t))
	else:
		tmp = (x * (-1.0 / 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) <= -5e+277)
		tmp = Float64(Float64(Float64(-x) / z) / t);
	elseif (Float64(z * t) <= 1e+264)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / 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) <= -5e+277)
		tmp = (-x / z) / t;
	elseif ((z * t) <= 1e+264)
		tmp = x / (y - (z * t));
	else
		tmp = (x * (-1.0 / 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], -5e+277], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+264], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]

Target

Original	2.8
Target	1.7
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -4.99999999999999982e277

   1. Initial program 16.5

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

   2. Taylor expanded in y around 0 17.1

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

   3. Simplified 17.1

      \[\leadsto \color{blue}{\frac{-x}{t \cdot z}} \]
      Proof
      (/.f64 (neg.f64 x) (*.f64 t z)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> neg-mul-1_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. Taylor expanded in x around 0 17.1

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

   5. Simplified 0.7

      \[\leadsto \color{blue}{\frac{\frac{-x}{z}}{t}} \]
      Proof
      (/.f64 (/.f64 (neg.f64 x) z) t): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/l/_binary64 (/.f64 (neg.f64 x) (*.f64 t z))): 51 points increase in error, 42 points decrease in error
      (/.f64 (Rewrite=> neg-mul-1_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

   if -4.99999999999999982e277 < (*.f64 z t) < 1.00000000000000004e264

   1. Initial program 0.1

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

   if 1.00000000000000004e264 < (*.f64 z t)

   1. Initial program 18.2

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

   2. Applied egg-rr 18.2

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

   3. Taylor expanded in y around 0 18.4

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

   4. Applied egg-rr 0.3

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+277}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+264}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{-1}{t}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+277}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+267}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternative 2
Error	19.8
Cost	648

\[\begin{array}{l} t_1 := \frac{-x}{z \cdot t}\\ \mathbf{if}\;z \leq -39000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	18.7
Cost	648

\[\begin{array}{l} t_1 := \frac{\frac{-x}{t}}{z}\\ \mathbf{if}\;z \leq -1.85 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	17.8
Cost	648

\[\begin{array}{l} t_1 := \frac{\frac{-x}{z}}{t}\\ \mathbf{if}\;z \leq -42000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	27.7
Cost	584

\[\begin{array}{l} t_1 := \frac{x}{z \cdot t}\\ \mathbf{if}\;z \leq -3.1 \cdot 10^{+177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	27.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+177}:\\ \;\;\;\;\frac{\frac{x}{z}}{t}\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \]

Alternative 7
Error	30.1
Cost	320

\[\frac{1}{\frac{y}{x}} \]

Alternative 8
Error	29.8
Cost	192

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

Reproduce

herbie shell --seed 2022332 
(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))))