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

Average Error: 2.8 → 0.1

Time: 9.8s

Precision: binary64

Cost: 968

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+287}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{t}}{\frac{z}{x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -2e+287)
   (/ (/ (- x) z) t)
   (if (<= (* z t) 5e+307) (/ x (- y (* z t))) (/ (/ -1.0 t) (/ z x)))))

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) <= -2e+287) {
		tmp = (-x / z) / t;
	} else if ((z * t) <= 5e+307) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-1.0 / t) / (z / x);
	}
	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) <= (-2d+287)) then
        tmp = (-x / z) / t
    else if ((z * t) <= 5d+307) then
        tmp = x / (y - (z * t))
    else
        tmp = ((-1.0d0) / t) / (z / x)
    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) <= -2e+287) {
		tmp = (-x / z) / t;
	} else if ((z * t) <= 5e+307) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-1.0 / t) / (z / x);
	}
	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) <= -2e+287:
		tmp = (-x / z) / t
	elif (z * t) <= 5e+307:
		tmp = x / (y - (z * t))
	else:
		tmp = (-1.0 / t) / (z / x)
	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) <= -2e+287)
		tmp = Float64(Float64(Float64(-x) / z) / t);
	elseif (Float64(z * t) <= 5e+307)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(-1.0 / t) / Float64(z / x));
	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) <= -2e+287)
		tmp = (-x / z) / t;
	elseif ((z * t) <= 5e+307)
		tmp = x / (y - (z * t));
	else
		tmp = (-1.0 / t) / (z / x);
	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], -2e+287], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+307], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / t), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision]]]

Target

Original	2.8
Target	1.7
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.0000000000000002e287

   1. Initial program 14.9

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

   2. Applied egg-rr 14.9

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

   3. Taylor expanded in y around 0 15.1

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

   4. Simplified 0.3

      \[\leadsto \color{blue}{\frac{\frac{-x}{t}}{z}} \]
      Proof
      (/.f64 (/.f64 (neg.f64 x) t) z): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.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)): 6 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 -1 x) (Rewrite<= *-commutative_binary64 (*.f64 z t))): 6 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 x (*.f64 z t)))): 0 points increase in error, 6 points decrease in error
      (*.f64 -1 (/.f64 x (Rewrite=> *-commutative_binary64 (*.f64 t z)))): 0 points increase in error, 1 points decrease in error

   5. Taylor expanded in x around 0 15.1

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

   6. Simplified 0.3

      \[\leadsto \color{blue}{\frac{\frac{-x}{z}}{t}} \]
      Proof
      (/.f64 (/.f64 (neg.f64 x) z) t): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x)) z) t): 0 points increase in error, 2 points decrease in error
      (Rewrite=> associate-/l/_binary64 (/.f64 (*.f64 -1 x) (*.f64 t z))): 4 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 x (*.f64 t z)))): 4 points increase in error, 0 points decrease in error

   if -2.0000000000000002e287 < (*.f64 z t) < 5e307

   1. Initial program 0.1

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

   if 5e307 < (*.f64 z t)

   1. Initial program 21.7

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

   2. Applied egg-rr 21.7

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

   3. Taylor expanded in y around 0 21.7

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

   4. Simplified 19.7

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

   5. Applied egg-rr 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+287}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{t}}{\frac{z}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	19.7
Cost	1176

\[\begin{array}{l} t_1 := \frac{1}{\frac{y}{x}}\\ t_2 := \frac{\frac{-x}{t}}{z}\\ \mathbf{if}\;y \leq -5.2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 10^{+89}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+136}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+157}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.2
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+287}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+221}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z} \cdot \frac{x}{t}\\ \end{array} \]

Alternative 3
Error	20.0
Cost	912

\[\begin{array}{l} t_1 := \frac{1}{\frac{y}{x}}\\ t_2 := \frac{-x}{z \cdot t}\\ \mathbf{if}\;y \leq -0.215:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-90}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+157}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	17.6
Cost	648

\[\begin{array}{l} \mathbf{if}\;z \leq -7800000000:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternative 5
Error	27.4
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+157} \lor \neg \left(z \leq 1.75\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6
Error	30.5
Cost	192

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

Reproduce

herbie shell --seed 2022340 
(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))))