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

Average Error: 10.2 → 1.5

Time: 15.7s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y}{\frac{t}{z} - a}\\ t_2 := t - z \cdot a\\ \mathbf{if}\;z \leq -3.309001054250559 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{t_2} - t_1\\ \mathbf{elif}\;z \leq 2.122287145817166 \cdot 10^{+25}:\\ \;\;\;\;\frac{x - z \cdot y}{t_2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{t_2} - t_1\\ \end{array} \]

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 (/ y (- (/ t z) a))) (t_2 (- t (* z a))))
   (if (<= z -3.309001054250559e+22)
     (- (/ x t_2) t_1)
     (if (<= z 2.122287145817166e+25)
       (/ (- x (* z y)) t_2)
       (- (* x (/ 1.0 t_2)) t_1)))))

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 = y / ((t / z) - a);
	double t_2 = t - (z * a);
	double tmp;
	if (z <= -3.309001054250559e+22) {
		tmp = (x / t_2) - t_1;
	} else if (z <= 2.122287145817166e+25) {
		tmp = (x - (z * y)) / t_2;
	} else {
		tmp = (x * (1.0 / t_2)) - t_1;
	}
	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 = y / ((t / z) - a)
    t_2 = t - (z * a)
    if (z <= (-3.309001054250559d+22)) then
        tmp = (x / t_2) - t_1
    else if (z <= 2.122287145817166d+25) then
        tmp = (x - (z * y)) / t_2
    else
        tmp = (x * (1.0d0 / t_2)) - t_1
    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 = y / ((t / z) - a);
	double t_2 = t - (z * a);
	double tmp;
	if (z <= -3.309001054250559e+22) {
		tmp = (x / t_2) - t_1;
	} else if (z <= 2.122287145817166e+25) {
		tmp = (x - (z * y)) / t_2;
	} else {
		tmp = (x * (1.0 / t_2)) - t_1;
	}
	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 = y / ((t / z) - a)
	t_2 = t - (z * a)
	tmp = 0
	if z <= -3.309001054250559e+22:
		tmp = (x / t_2) - t_1
	elif z <= 2.122287145817166e+25:
		tmp = (x - (z * y)) / t_2
	else:
		tmp = (x * (1.0 / t_2)) - t_1
	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(y / Float64(Float64(t / z) - a))
	t_2 = Float64(t - Float64(z * a))
	tmp = 0.0
	if (z <= -3.309001054250559e+22)
		tmp = Float64(Float64(x / t_2) - t_1);
	elseif (z <= 2.122287145817166e+25)
		tmp = Float64(Float64(x - Float64(z * y)) / t_2);
	else
		tmp = Float64(Float64(x * Float64(1.0 / t_2)) - t_1);
	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 = y / ((t / z) - a);
	t_2 = t - (z * a);
	tmp = 0.0;
	if (z <= -3.309001054250559e+22)
		tmp = (x / t_2) - t_1;
	elseif (z <= 2.122287145817166e+25)
		tmp = (x - (z * y)) / t_2;
	else
		tmp = (x * (1.0 / t_2)) - t_1;
	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[(y / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.309001054250559e+22], N[(N[(x / t$95$2), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[z, 2.122287145817166e+25], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(x * N[(1.0 / t$95$2), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]

Target

Original	10.2
Target	1.6
Herbie	1.5

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

Derivation

1. Split input into 3 regimes

2. if z < -3.309001054250559e22

   1. Initial program 21.5

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

   2. Applied egg-rr 13.5

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

   3. Taylor expanded in t around 0 2.4

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

   4. Simplified 2.4

      \[\leadsto \frac{x}{t - z \cdot a} - \frac{y}{\color{blue}{\frac{t}{z} - a}} \]
      Proof
      (-.f64 (/.f64 t z) a): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 t z) (neg.f64 a))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 t z) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 a))): 0 points increase in error, 0 points decrease in error

   if -3.309001054250559e22 < z < 2.1222871458171659e25

   1. Initial program 0.3

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

   if 2.1222871458171659e25 < z

   1. Initial program 22.2

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

   2. Applied egg-rr 13.6

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

   3. Taylor expanded in t around 0 3.1

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

   4. Simplified 3.1

      \[\leadsto \frac{x}{t - z \cdot a} - \frac{y}{\color{blue}{\frac{t}{z} - a}} \]
      Proof
      (-.f64 (/.f64 t z) a): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 t z) (neg.f64 a))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 t z) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 a))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 3.2

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.309001054250559 \cdot 10^{+22}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \leq 2.122287145817166 \cdot 10^{+25}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{t - z \cdot a} - \frac{y}{\frac{t}{z} - a}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.6
Cost	3020

\[\begin{array}{l} t_1 := \frac{-y}{\frac{t}{z} - a}\\ t_2 := \frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-313}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+279}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	27.7
Cost	1372

\[\begin{array}{l} t_1 := \frac{z \cdot y}{-t}\\ t_2 := \frac{x}{t - z \cdot a}\\ \mathbf{if}\;x \leq -1.0676218403042932 \cdot 10^{-96}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.029279660113034 \cdot 10^{-171}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.0987278852861932 \cdot 10^{-237}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;x \leq 8.29903875057332 \cdot 10^{-278}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.950824803351394 \cdot 10^{-230}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;x \leq 3.5925160872199716 \cdot 10^{-181}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.186453974266548 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	30.9
Cost	1308

\[\begin{array}{l} t_1 := \frac{z \cdot y}{-t}\\ \mathbf{if}\;z \leq -1.740572565175928 \cdot 10^{+178}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -4.1918545473539674 \cdot 10^{+138}:\\ \;\;\;\;y \cdot \frac{-z}{t}\\ \mathbf{elif}\;z \leq -3.309001054250559 \cdot 10^{+22}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.3225137737902944 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.42110299656921 \cdot 10^{-25}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3.472127675153732 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq -1.674365284558833 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.657591809318698 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 4
Error	22.0
Cost	1240

\[\begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ t_2 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;a \leq -1.9580498169577897 \cdot 10^{+125}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.2832277269221216 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.6796066547833117 \cdot 10^{-86}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -2.2059666655146636 \cdot 10^{-92}:\\ \;\;\;\;y \cdot \frac{-z}{t}\\ \mathbf{elif}\;a \leq 5.546298441910733 \cdot 10^{-10}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;a \leq 1.0776072547129701 \cdot 10^{+183}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	1.4
Cost	1224

\[\begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x}{t_1} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{if}\;z \leq -3.309001054250559 \cdot 10^{+22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 8.657591809318698 \cdot 10^{-13}:\\ \;\;\;\;\frac{x - z \cdot y}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	30.1
Cost	1044

\[\begin{array}{l} t_1 := \frac{z \cdot y}{-t}\\ \mathbf{if}\;z \leq -3.309001054250559 \cdot 10^{+22}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.3225137737902944 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.42110299656921 \cdot 10^{-25}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3.472127675153732 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq -1.674365284558833 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.657591809318698 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 7
Error	17.9
Cost	1040

\[\begin{array}{l} t_1 := \frac{-y}{\frac{t}{z} - a}\\ \mathbf{if}\;z \leq -9.42110299656921 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.3577581840398864 \cdot 10^{-166}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.7964384589974274 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 10^{+210}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]

Alternative 8
Error	17.9
Cost	1040

\[\begin{array}{l} t_1 := \frac{-y}{\frac{t}{z} - a}\\ \mathbf{if}\;z \leq -9.42110299656921 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.3577581840398864 \cdot 10^{-166}:\\ \;\;\;\;\frac{x}{t} - \frac{z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.7964384589974274 \cdot 10^{-20}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 10^{+210}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]

Alternative 9
Error	18.3
Cost	712

\[\begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -9.42110299656921 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.936842789862069 \cdot 10^{+65}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	29.9
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -9.42110299656921 \cdot 10^{-25}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 8.657591809318698 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 11
Error	42.7
Cost	192

\[\frac{y}{a} \]

Reproduce

herbie shell --seed 2022302 
(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))))