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

Average Accuracy: 83.3% → 93.2%

Time: 16.5s

Precision: binary64

Cost: 3020

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-316}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{y \cdot z - x}{a} \cdot \frac{1}{z}\\ \mathbf{elif}\;t_1 \leq 10^{+273}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]

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 (/ (- x (* y z)) (- t (* z a)))))
   (if (<= t_1 -5e-316)
     t_1
     (if (<= t_1 0.0)
       (* (/ (- (* y z) x) a) (/ 1.0 z))
       (if (<= t_1 1e+273) t_1 (/ y (- a (/ t z))))))))

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 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if (t_1 <= -5e-316) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (((y * z) - x) / a) * (1.0 / z);
	} else if (t_1 <= 1e+273) {
		tmp = t_1;
	} else {
		tmp = y / (a - (t / z));
	}
	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) :: tmp
    t_1 = (x - (y * z)) / (t - (z * a))
    if (t_1 <= (-5d-316)) then
        tmp = t_1
    else if (t_1 <= 0.0d0) then
        tmp = (((y * z) - x) / a) * (1.0d0 / z)
    else if (t_1 <= 1d+273) then
        tmp = t_1
    else
        tmp = y / (a - (t / z))
    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 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if (t_1 <= -5e-316) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (((y * z) - x) / a) * (1.0 / z);
	} else if (t_1 <= 1e+273) {
		tmp = t_1;
	} else {
		tmp = y / (a - (t / z));
	}
	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 = (x - (y * z)) / (t - (z * a))
	tmp = 0
	if t_1 <= -5e-316:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = (((y * z) - x) / a) * (1.0 / z)
	elif t_1 <= 1e+273:
		tmp = t_1
	else:
		tmp = y / (a - (t / z))
	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(x - Float64(y * z)) / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (t_1 <= -5e-316)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(y * z) - x) / a) * Float64(1.0 / z));
	elseif (t_1 <= 1e+273)
		tmp = t_1;
	else
		tmp = Float64(y / Float64(a - Float64(t / z)));
	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 = (x - (y * z)) / (t - (z * a));
	tmp = 0.0;
	if (t_1 <= -5e-316)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = (((y * z) - x) / a) * (1.0 / z);
	elseif (t_1 <= 1e+273)
		tmp = t_1;
	else
		tmp = y / (a - (t / z));
	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[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-316], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / a), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+273], t$95$1, N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	83.3%
Target	97.0%
Herbie	93.2%

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -5.000000017e-316 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 9.99999999999999945e272

   1. Initial program 96.0%

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

   if -5.000000017e-316 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 59.1%

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

   2. Simplified 59.1%

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

      [Start] 59.1	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      sub-neg [=>] 59.1	\[ \frac{x - y \cdot z}{\color{blue}{t + \left(-a \cdot z\right)}} \]
      +-commutative [=>] 59.1	\[ \frac{x - y \cdot z}{\color{blue}{\left(-a \cdot z\right) + t}} \]
      neg-sub0 [=>] 59.1	\[ \frac{x - y \cdot z}{\color{blue}{\left(0 - a \cdot z\right)} + t} \]
      associate-+l- [=>] 59.1	\[ \frac{x - y \cdot z}{\color{blue}{0 - \left(a \cdot z - t\right)}} \]
      sub0-neg [=>] 59.1	\[ \frac{x - y \cdot z}{\color{blue}{-\left(a \cdot z - t\right)}} \]
      neg-mul-1 [=>] 59.1	\[ \frac{x - y \cdot z}{\color{blue}{-1 \cdot \left(a \cdot z - t\right)}} \]
      sub-neg [=>] 59.1	\[ \frac{\color{blue}{x + \left(-y \cdot z\right)}}{-1 \cdot \left(a \cdot z - t\right)} \]
      +-commutative [=>] 59.1	\[ \frac{\color{blue}{\left(-y \cdot z\right) + x}}{-1 \cdot \left(a \cdot z - t\right)} \]
      neg-sub0 [=>] 59.1	\[ \frac{\color{blue}{\left(0 - y \cdot z\right)} + x}{-1 \cdot \left(a \cdot z - t\right)} \]
      associate-+l- [=>] 59.1	\[ \frac{\color{blue}{0 - \left(y \cdot z - x\right)}}{-1 \cdot \left(a \cdot z - t\right)} \]
      sub0-neg [=>] 59.1	\[ \frac{\color{blue}{-\left(y \cdot z - x\right)}}{-1 \cdot \left(a \cdot z - t\right)} \]
      neg-mul-1 [=>] 59.1	\[ \frac{\color{blue}{-1 \cdot \left(y \cdot z - x\right)}}{-1 \cdot \left(a \cdot z - t\right)} \]
      times-frac [=>] 59.1	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{y \cdot z - x}{a \cdot z - t}} \]
      metadata-eval [=>] 59.1	\[ \color{blue}{1} \cdot \frac{y \cdot z - x}{a \cdot z - t} \]
      *-lft-identity [=>] 59.1	\[ \color{blue}{\frac{y \cdot z - x}{a \cdot z - t}} \]
      *-commutative [=>] 59.1	\[ \frac{y \cdot z - x}{\color{blue}{z \cdot a} - t} \]

   3. Taylor expanded in a around inf 33.1%

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

   4. Simplified 33.1%

      \[\leadsto \color{blue}{\frac{z \cdot y - x}{a \cdot z}} \]
      Proof

      [Start] 33.1	\[ \frac{y \cdot z - x}{a \cdot z} \]
      *-commutative [=>] 33.1	\[ \frac{\color{blue}{z \cdot y} - x}{a \cdot z} \]

   5. Applied egg-rr 78.6%

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

      [Start] 33.1	\[ \frac{z \cdot y - x}{a \cdot z} \]
      associate-/r* [=>] 78.7	\[ \color{blue}{\frac{\frac{z \cdot y - x}{a}}{z}} \]
      div-inv [=>] 78.6	\[ \color{blue}{\frac{z \cdot y - x}{a} \cdot \frac{1}{z}} \]

   if 9.99999999999999945e272 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 10.9%

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

   2. Simplified 10.9%

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

      [Start] 10.9	\[ \frac{x - y \cdot z}{t - a \cdot z} \]
      sub-neg [=>] 10.9	\[ \frac{x - y \cdot z}{\color{blue}{t + \left(-a \cdot z\right)}} \]
      +-commutative [=>] 10.9	\[ \frac{x - y \cdot z}{\color{blue}{\left(-a \cdot z\right) + t}} \]
      neg-sub0 [=>] 10.9	\[ \frac{x - y \cdot z}{\color{blue}{\left(0 - a \cdot z\right)} + t} \]
      associate-+l- [=>] 10.9	\[ \frac{x - y \cdot z}{\color{blue}{0 - \left(a \cdot z - t\right)}} \]
      sub0-neg [=>] 10.9	\[ \frac{x - y \cdot z}{\color{blue}{-\left(a \cdot z - t\right)}} \]
      neg-mul-1 [=>] 10.9	\[ \frac{x - y \cdot z}{\color{blue}{-1 \cdot \left(a \cdot z - t\right)}} \]
      sub-neg [=>] 10.9	\[ \frac{\color{blue}{x + \left(-y \cdot z\right)}}{-1 \cdot \left(a \cdot z - t\right)} \]
      +-commutative [=>] 10.9	\[ \frac{\color{blue}{\left(-y \cdot z\right) + x}}{-1 \cdot \left(a \cdot z - t\right)} \]
      neg-sub0 [=>] 10.9	\[ \frac{\color{blue}{\left(0 - y \cdot z\right)} + x}{-1 \cdot \left(a \cdot z - t\right)} \]
      associate-+l- [=>] 10.9	\[ \frac{\color{blue}{0 - \left(y \cdot z - x\right)}}{-1 \cdot \left(a \cdot z - t\right)} \]
      sub0-neg [=>] 10.9	\[ \frac{\color{blue}{-\left(y \cdot z - x\right)}}{-1 \cdot \left(a \cdot z - t\right)} \]
      neg-mul-1 [=>] 10.9	\[ \frac{\color{blue}{-1 \cdot \left(y \cdot z - x\right)}}{-1 \cdot \left(a \cdot z - t\right)} \]
      times-frac [=>] 10.9	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{y \cdot z - x}{a \cdot z - t}} \]
      metadata-eval [=>] 10.9	\[ \color{blue}{1} \cdot \frac{y \cdot z - x}{a \cdot z - t} \]
      *-lft-identity [=>] 10.9	\[ \color{blue}{\frac{y \cdot z - x}{a \cdot z - t}} \]
      *-commutative [=>] 10.9	\[ \frac{y \cdot z - x}{\color{blue}{z \cdot a} - t} \]

   3. Applied egg-rr 10.9%

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

      [Start] 10.9	\[ \frac{y \cdot z - x}{z \cdot a - t} \]
      clear-num [=>] 10.9	\[ \color{blue}{\frac{1}{\frac{z \cdot a - t}{y \cdot z - x}}} \]
      associate-/r/ [=>] 10.9	\[ \color{blue}{\frac{1}{z \cdot a - t} \cdot \left(y \cdot z - x\right)} \]

   4. Taylor expanded in y around inf 3.1%

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

   5. Simplified 36.8%

      \[\leadsto \color{blue}{\frac{y}{\frac{a \cdot z - t}{z}}} \]
      Proof

      [Start] 3.1	\[ \frac{y \cdot z}{a \cdot z - t} \]
      associate-/l* [=>] 36.8	\[ \color{blue}{\frac{y}{\frac{a \cdot z - t}{z}}} \]

   6. Taylor expanded in a around 0 92.0%

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

   7. Simplified 92.0%

      \[\leadsto \frac{y}{\color{blue}{a - \frac{t}{z}}} \]
      Proof

      [Start] 92.0	\[ \frac{y}{a + -1 \cdot \frac{t}{z}} \]
      neg-mul-1 [<=] 92.0	\[ \frac{y}{a + \color{blue}{\left(-\frac{t}{z}\right)}} \]
      sub-neg [<=] 92.0	\[ \frac{y}{\color{blue}{a - \frac{t}{z}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -5 \cdot 10^{-316}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y \cdot z - x}{a} \cdot \frac{1}{z}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 10^{+273}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	52.9%
Cost	1044

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+64}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{-x}{z}}{a}\\ \mathbf{elif}\;z \leq -6 \cdot 10^{-40}:\\ \;\;\;\;z \cdot \frac{-y}{t}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{+22}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 2
Accuracy	72.3%
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ t_2 := \frac{y}{a - \frac{t}{z}}\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{+25}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -9.6 \cdot 10^{-156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-292}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	71.6%
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ \mathbf{if}\;z \leq -1.02 \cdot 10^{+24}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-293}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 165000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]

Alternative 4
Accuracy	52.9%
Cost	780

\[\begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+17}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 61000000000000:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 5
Accuracy	63.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+26} \lor \neg \left(z \leq 3.5 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]

Alternative 6
Accuracy	72.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -2.95 \cdot 10^{+23} \lor \neg \left(z \leq 2.75 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]

Alternative 7
Accuracy	52.8%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{+17}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 8
Accuracy	34.1%
Cost	192

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

Reproduce

herbie shell --seed 2023146 
(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))))