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

Average Accuracy: 95.4% → 97.6%

Time: 10.3s

Precision: binary64

Cost: 708

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+298}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -5e+298) (/ (/ (- x) z) t) (/ x (- y (* z t)))))

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) <= -5e+298) {
		tmp = (-x / z) / t;
	} else {
		tmp = x / (y - (z * t));
	}
	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) <= (-5d+298)) then
        tmp = (-x / z) / t
    else
        tmp = x / (y - (z * t))
    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) <= -5e+298) {
		tmp = (-x / z) / t;
	} else {
		tmp = x / (y - (z * t));
	}
	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) <= -5e+298:
		tmp = (-x / z) / t
	else:
		tmp = x / (y - (z * t))
	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) <= -5e+298)
		tmp = Float64(Float64(Float64(-x) / z) / t);
	else
		tmp = Float64(x / Float64(y - Float64(z * t)));
	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) <= -5e+298)
		tmp = (-x / z) / t;
	else
		tmp = x / (y - (z * t));
	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], -5e+298], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	95.4%
Target	97.3%
Herbie	97.6%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5.0000000000000003e298

   1. Initial program 69.2%

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

   2. Applied egg-rr 69.0%

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

      [Start] 69.2	\[ \frac{x}{y - z \cdot t} \]
      clear-num [=>] 69.0	\[ \color{blue}{\frac{1}{\frac{y - z \cdot t}{x}}} \]
      inv-pow [=>] 69.0	\[ \color{blue}{{\left(\frac{y - z \cdot t}{x}\right)}^{-1}} \]

   3. Taylor expanded in y around 0 69.0%

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

   4. Simplified 98.6%

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

      [Start] 69.0	\[ {\left(-1 \cdot \frac{t \cdot z}{x}\right)}^{-1} \]
      mul-1-neg [=>] 69.0	\[ {\color{blue}{\left(-\frac{t \cdot z}{x}\right)}}^{-1} \]
      associate-/l* [=>] 98.6	\[ {\left(-\color{blue}{\frac{t}{\frac{x}{z}}}\right)}^{-1} \]

   5. Taylor expanded in t around 0 69.2%

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

   6. Simplified 99.9%

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

      [Start] 69.2	\[ -1 \cdot \frac{x}{t \cdot z} \]
      mul-1-neg [=>] 69.2	\[ \color{blue}{-\frac{x}{t \cdot z}} \]
      associate-/l/ [<=] 99.9	\[ -\color{blue}{\frac{\frac{x}{z}}{t}} \]
      distribute-neg-frac [=>] 99.9	\[ \color{blue}{\frac{-\frac{x}{z}}{t}} \]
      distribute-neg-frac [=>] 99.9	\[ \frac{\color{blue}{\frac{-x}{z}}}{t} \]

   if -5.0000000000000003e298 < (*.f64 z t)

   1. Initial program 97.4%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+298}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	56.0%
Cost	717

\[\begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{+36} \lor \neg \left(t \leq 4.3 \cdot 10^{+176}\right) \land t \leq 1.12 \cdot 10^{+264}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 2
Accuracy	73.0%
Cost	649

\[\begin{array}{l} \mathbf{if}\;t \leq -2.35 \cdot 10^{-39} \lor \neg \left(t \leq 3 \cdot 10^{+55}\right):\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3
Accuracy	72.3%
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+72} \lor \neg \left(z \leq 1.32 \cdot 10^{-108}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4
Accuracy	72.1%
Cost	648

\[\begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{-1}{t \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-108}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternative 5
Accuracy	58.1%
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+36}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+173}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 6
Accuracy	53.4%
Cost	192

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

Reproduce

herbie shell --seed 2023151 
(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))))