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

Average Error: 2.8 → 1.0

Time: 7.6s

Precision: binary64

Cost: 968

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\frac{-x}{z}}{t}\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+304}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (/ (- x) z) t)))
   (if (<= (* z t) -2e+304)
     t_1
     (if (<= (* z t) 2e+129) (/ x (- y (* z t))) t_1))))

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 t_1 = (-x / z) / t;
	double tmp;
	if ((z * t) <= -2e+304) {
		tmp = t_1;
	} else if ((z * t) <= 2e+129) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (-x / z) / t
    if ((z * t) <= (-2d+304)) then
        tmp = t_1
    else if ((z * t) <= 2d+129) then
        tmp = x / (y - (z * t))
    else
        tmp = t_1
    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 t_1 = (-x / z) / t;
	double tmp;
	if ((z * t) <= -2e+304) {
		tmp = t_1;
	} else if ((z * t) <= 2e+129) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return x / (y - (z * t))

↓

def code(x, y, z, t):
	t_1 = (-x / z) / t
	tmp = 0
	if (z * t) <= -2e+304:
		tmp = t_1
	elif (z * t) <= 2e+129:
		tmp = x / (y - (z * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	return Float64(x / Float64(y - Float64(z * t)))
end

↓

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(-x) / z) / t)
	tmp = 0.0
	if (Float64(z * t) <= -2e+304)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e+129)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = t_1;
	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)
	t_1 = (-x / z) / t;
	tmp = 0.0;
	if ((z * t) <= -2e+304)
		tmp = t_1;
	elseif ((z * t) <= 2e+129)
		tmp = x / (y - (z * t));
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e+304], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e+129], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Target

Original	2.8
Target	1.7
Herbie	1.0

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.9999999999999999e304 or 2e129 < (*.f64 z t)

   1. Initial program 11.4

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

   2. Applied egg-rr 11.4

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

   3. Applied egg-rr 11.7

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

   4. Taylor expanded in y around 0 14.7

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

   5. Simplified 4.1

      \[\leadsto \color{blue}{\frac{\frac{-x}{z}}{t}} \]
      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, 0 points decrease in error
      (Rewrite=> associate-/l/_binary64 (/.f64 (*.f64 -1 x) (*.f64 t z))): 54 points increase in error, 34 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 -1.9999999999999999e304 < (*.f64 z t) < 2e129

   1. Initial program 0.1

      \[\frac{x}{y - z \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+304}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternatives

Alternative 1
Error	21.4
Cost	912

\[\begin{array}{l} t_1 := \frac{\frac{-x}{z}}{t}\\ \mathbf{if}\;t \leq -1.2649896080357107 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.683584477241817 \cdot 10^{+92}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t \leq 9.72687571907555 \cdot 10^{+140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.234516731215059 \cdot 10^{+156}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	19.9
Cost	648

\[\begin{array}{l} t_1 := \frac{\frac{-x}{t}}{z}\\ \mathbf{if}\;t \leq -1.2649896080357107 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.103813028568 \cdot 10^{+54}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	30.0
Cost	192

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

Reproduce

herbie shell --seed 2022308 
(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))))