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

Percentage Accurate: 84.8% → 92.1%

Time: 10.9s

Precision: binary64

Cost: 2248

Math

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

↓

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot a - t\\ t_2 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{+289}:\\ \;\;\;\;z \cdot \frac{y}{t_1} - \frac{x}{t_1}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* z a) t)) (t_2 (/ (- x (* y z)) (- t (* z a)))))
   (if (<= t_2 -1e+289)
     (- (* z (/ y t_1)) (/ x t_1))
     (if (<= t_2 5e+293) t_2 (/ (- y (/ x z)) a)))))

C

double code(double x, double y, double z, double t, double a) {
	return (x - (y * z)) / (t - (a * z));
}

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * a) - t;
	double t_2 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if (t_2 <= -1e+289) {
		tmp = (z * (y / t_1)) - (x / t_1);
	} else if (t_2 <= 5e+293) {
		tmp = t_2;
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = (x - (y * z)) / (t - (a * z))
end function

↓

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z * a) - t
    t_2 = (x - (y * z)) / (t - (z * a))
    if (t_2 <= (-1d+289)) then
        tmp = (z * (y / t_1)) - (x / t_1)
    else if (t_2 <= 5d+293) then
        tmp = t_2
    else
        tmp = (y - (x / z)) / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return (x - (y * z)) / (t - (a * z));
}

↓

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * a) - t;
	double t_2 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if (t_2 <= -1e+289) {
		tmp = (z * (y / t_1)) - (x / t_1);
	} else if (t_2 <= 5e+293) {
		tmp = t_2;
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a):
	t_1 = (z * a) - t
	t_2 = (x - (y * z)) / (t - (z * a))
	tmp = 0
	if t_2 <= -1e+289:
		tmp = (z * (y / t_1)) - (x / t_1)
	elif t_2 <= 5e+293:
		tmp = t_2
	else:
		tmp = (y - (x / z)) / a
	return tmp

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * a) - t)
	t_2 = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (t_2 <= -1e+289)
		tmp = Float64(Float64(z * Float64(y / t_1)) - Float64(x / t_1));
	elseif (t_2 <= 5e+293)
		tmp = t_2;
	else
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = (x - (y * z)) / (t - (a * z));
end

↓

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * a) - t;
	t_2 = (x - (y * z)) / (t - (z * a));
	tmp = 0.0;
	if (t_2 <= -1e+289)
		tmp = (z * (y / t_1)) - (x / t_1);
	elseif (t_2 <= 5e+293)
		tmp = t_2;
	else
		tmp = (y - (x / z)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+289], N[(N[(z * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+293], t$95$2, N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]]]

Target

Original	84.8%
Target	97.4%
Herbie	92.1%

\[\begin{array}{l} \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z} - \frac{y}{\frac{t}{z} - a}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.0000000000000001e289

   1. Initial program 59.3%

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

   2. Simplified 59.3%

      \[\leadsto \color{blue}{\frac{y \cdot z - x}{z \cdot a - t}} \]
      Step-by-step derivation

      [Start] 59.3%	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      sub-neg [=>] 59.3%	\[ \frac{x - y \cdot z}{\color{blue}{t + \left(-a \cdot z\right)}} \]
      +-commutative [=>] 59.3%	\[ \frac{x - y \cdot z}{\color{blue}{\left(-a \cdot z\right) + t}} \]
      neg-sub0 [=>] 59.3%	\[ \frac{x - y \cdot z}{\color{blue}{\left(0 - a \cdot z\right)} + t} \]
      associate-+l- [=>] 59.3%	\[ \frac{x - y \cdot z}{\color{blue}{0 - \left(a \cdot z - t\right)}} \]
      sub0-neg [=>] 59.3%	\[ \frac{x - y \cdot z}{\color{blue}{-\left(a \cdot z - t\right)}} \]
      neg-mul-1 [=>] 59.3%	\[ \frac{x - y \cdot z}{\color{blue}{-1 \cdot \left(a \cdot z - t\right)}} \]
      sub-neg [=>] 59.3%	\[ \frac{\color{blue}{x + \left(-y \cdot z\right)}}{-1 \cdot \left(a \cdot z - t\right)} \]
      +-commutative [=>] 59.3%	\[ \frac{\color{blue}{\left(-y \cdot z\right) + x}}{-1 \cdot \left(a \cdot z - t\right)} \]
      neg-sub0 [=>] 59.3%	\[ \frac{\color{blue}{\left(0 - y \cdot z\right)} + x}{-1 \cdot \left(a \cdot z - t\right)} \]
      associate-+l- [=>] 59.3%	\[ \frac{\color{blue}{0 - \left(y \cdot z - x\right)}}{-1 \cdot \left(a \cdot z - t\right)} \]
      sub0-neg [=>] 59.3%	\[ \frac{\color{blue}{-\left(y \cdot z - x\right)}}{-1 \cdot \left(a \cdot z - t\right)} \]
      neg-mul-1 [=>] 59.3%	\[ \frac{\color{blue}{-1 \cdot \left(y \cdot z - x\right)}}{-1 \cdot \left(a \cdot z - t\right)} \]
      times-frac [=>] 59.3%	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{y \cdot z - x}{a \cdot z - t}} \]
      metadata-eval [=>] 59.3%	\[ \color{blue}{1} \cdot \frac{y \cdot z - x}{a \cdot z - t} \]
      *-lft-identity [=>] 59.3%	\[ \color{blue}{\frac{y \cdot z - x}{a \cdot z - t}} \]
      *-commutative [=>] 59.3%	\[ \frac{y \cdot z - x}{\color{blue}{z \cdot a} - t} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{y}{\frac{z \cdot a - t}{z}} - \frac{x}{z \cdot a - t}} \]
      Step-by-step derivation

      [Start] 59.3%	\[ \frac{y \cdot z - x}{z \cdot a - t} \]
      div-sub [=>] 59.3%	\[ \color{blue}{\frac{y \cdot z}{z \cdot a - t} - \frac{x}{z \cdot a - t}} \]
      associate-/l* [=>] 100.0%	\[ \color{blue}{\frac{y}{\frac{z \cdot a - t}{z}}} - \frac{x}{z \cdot a - t} \]

   4. Applied egg-rr 92.6%

      \[\leadsto \color{blue}{\frac{y}{z \cdot a - t} \cdot z} - \frac{x}{z \cdot a - t} \]
      Step-by-step derivation

      [Start] 100.0%	\[ \frac{y}{\frac{z \cdot a - t}{z}} - \frac{x}{z \cdot a - t} \]
      associate-/r/ [=>] 92.6%	\[ \color{blue}{\frac{y}{z \cdot a - t} \cdot z} - \frac{x}{z \cdot a - t} \]

   if -1.0000000000000001e289 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 5.00000000000000033e293

   1. Initial program 96.2%

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

   if 5.00000000000000033e293 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 42.9%

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

   2. Simplified 42.9%

      \[\leadsto \color{blue}{\frac{y \cdot z - x}{z \cdot a - t}} \]
      Step-by-step derivation

      [Start] 42.9%	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      sub-neg [=>] 42.9%	\[ \frac{x - y \cdot z}{\color{blue}{t + \left(-a \cdot z\right)}} \]
      +-commutative [=>] 42.9%	\[ \frac{x - y \cdot z}{\color{blue}{\left(-a \cdot z\right) + t}} \]
      neg-sub0 [=>] 42.9%	\[ \frac{x - y \cdot z}{\color{blue}{\left(0 - a \cdot z\right)} + t} \]
      associate-+l- [=>] 42.9%	\[ \frac{x - y \cdot z}{\color{blue}{0 - \left(a \cdot z - t\right)}} \]
      sub0-neg [=>] 42.9%	\[ \frac{x - y \cdot z}{\color{blue}{-\left(a \cdot z - t\right)}} \]
      neg-mul-1 [=>] 42.9%	\[ \frac{x - y \cdot z}{\color{blue}{-1 \cdot \left(a \cdot z - t\right)}} \]
      sub-neg [=>] 42.9%	\[ \frac{\color{blue}{x + \left(-y \cdot z\right)}}{-1 \cdot \left(a \cdot z - t\right)} \]
      +-commutative [=>] 42.9%	\[ \frac{\color{blue}{\left(-y \cdot z\right) + x}}{-1 \cdot \left(a \cdot z - t\right)} \]
      neg-sub0 [=>] 42.9%	\[ \frac{\color{blue}{\left(0 - y \cdot z\right)} + x}{-1 \cdot \left(a \cdot z - t\right)} \]
      associate-+l- [=>] 42.9%	\[ \frac{\color{blue}{0 - \left(y \cdot z - x\right)}}{-1 \cdot \left(a \cdot z - t\right)} \]
      sub0-neg [=>] 42.9%	\[ \frac{\color{blue}{-\left(y \cdot z - x\right)}}{-1 \cdot \left(a \cdot z - t\right)} \]
      neg-mul-1 [=>] 42.9%	\[ \frac{\color{blue}{-1 \cdot \left(y \cdot z - x\right)}}{-1 \cdot \left(a \cdot z - t\right)} \]
      times-frac [=>] 42.9%	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{y \cdot z - x}{a \cdot z - t}} \]
      metadata-eval [=>] 42.9%	\[ \color{blue}{1} \cdot \frac{y \cdot z - x}{a \cdot z - t} \]
      *-lft-identity [=>] 42.9%	\[ \color{blue}{\frac{y \cdot z - x}{a \cdot z - t}} \]
      *-commutative [=>] 42.9%	\[ \frac{y \cdot z - x}{\color{blue}{z \cdot a} - t} \]

   3. Applied egg-rr 80.6%

      \[\leadsto \color{blue}{\frac{y}{\frac{z \cdot a - t}{z}} - \frac{x}{z \cdot a - t}} \]
      Step-by-step derivation

      [Start] 42.9%	\[ \frac{y \cdot z - x}{z \cdot a - t} \]
      div-sub [=>] 42.9%	\[ \color{blue}{\frac{y \cdot z}{z \cdot a - t} - \frac{x}{z \cdot a - t}} \]
      associate-/l* [=>] 80.6%	\[ \color{blue}{\frac{y}{\frac{z \cdot a - t}{z}}} - \frac{x}{z \cdot a - t} \]

   4. Taylor expanded in a around inf 88.4%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1 \cdot 10^{+289}:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t} - \frac{x}{z \cdot a - t}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	92.1%
Cost	2248

\[\begin{array}{l} t_1 := z \cdot a - t\\ t_2 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{+289}:\\ \;\;\;\;z \cdot \frac{y}{t_1} - \frac{x}{t_1}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]

Alternative 2
Accuracy	89.7%
Cost	1476

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]

Alternative 3
Accuracy	69.9%
Cost	1104

\[\begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+169}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{+92}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot a - t}\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-159}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+40}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} - \frac{x}{z \cdot a}\\ \end{array} \]

Alternative 4
Accuracy	88.1%
Cost	1088

\[\begin{array}{l} t_1 := z \cdot a - t\\ \frac{y}{\frac{t_1}{z}} - \frac{x}{t_1} \end{array} \]

Alternative 5
Accuracy	60.4%
Cost	976

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t}\\ \mathbf{if}\;y \leq -3 \cdot 10^{+85}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+47}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+232}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	71.0%
Cost	976

\[\begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+169}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{+91}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot a - t}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-161}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	53.1%
Cost	780

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{-214}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-188}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 8
Accuracy	71.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -105 \lor \neg \left(z \leq 8 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \end{array} \]

Alternative 9
Accuracy	65.9%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+103}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+57}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 10
Accuracy	54.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 11
Accuracy	35.0%
Cost	192

\[\frac{x}{t} \]

Reproduce

herbie shell --seed 2023167 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

  (/ (- x (* y z)) (- t (* a z))))