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

Percentage Accurate: 95.3% → 95.3%

Time: 6.8s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

2. Final simplification 97.3%

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

Alternative 2: 72.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-9} \lor \neg \left(y \leq -2.8 \cdot 10^{-80}\right) \land y \leq 6.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.3e+41)
   (/ 1.0 (/ y x))
   (if (or (<= y -2.9e-9) (and (not (<= y -2.8e-80)) (<= y 6.2e-23)))
     (/ (- x) (* z t))
     (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.3e+41) {
		tmp = 1.0 / (y / x);
	} else if ((y <= -2.9e-9) || (!(y <= -2.8e-80) && (y <= 6.2e-23))) {
		tmp = -x / (z * t);
	} else {
		tmp = x / y;
	}
	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
    real(8) :: tmp
    if (y <= (-2.3d+41)) then
        tmp = 1.0d0 / (y / x)
    else if ((y <= (-2.9d-9)) .or. (.not. (y <= (-2.8d-80))) .and. (y <= 6.2d-23)) then
        tmp = -x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.3e+41) {
		tmp = 1.0 / (y / x);
	} else if ((y <= -2.9e-9) || (!(y <= -2.8e-80) && (y <= 6.2e-23))) {
		tmp = -x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -2.3e+41:
		tmp = 1.0 / (y / x)
	elif (y <= -2.9e-9) or (not (y <= -2.8e-80) and (y <= 6.2e-23)):
		tmp = -x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.3e+41)
		tmp = Float64(1.0 / Float64(y / x));
	elseif ((y <= -2.9e-9) || (!(y <= -2.8e-80) && (y <= 6.2e-23)))
		tmp = Float64(Float64(-x) / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -2.3e+41)
		tmp = 1.0 / (y / x);
	elseif ((y <= -2.9e-9) || (~((y <= -2.8e-80)) && (y <= 6.2e-23)))
		tmp = -x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.3e+41], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -2.9e-9], And[N[Not[LessEqual[y, -2.8e-80]], $MachinePrecision], LessEqual[y, 6.2e-23]]], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2999999999999998e41

   1. Initial program 97.4%

      \[\frac{x}{y - z \cdot t} \]
   2. Step-by-step derivation

      1. clear-num 97.0%

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

      2. associate-/r/ 97.2%

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

   3. Applied egg-rr 97.2%

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

   4. Taylor expanded in y around inf 86.0%

      \[\leadsto \color{blue}{\frac{1}{y}} \cdot x \]
   5. Step-by-step derivation

      1. associate-*l/ 86.2%

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

      2. *-un-lft-identity 86.2%

         \[\leadsto \frac{\color{blue}{x}}{y} \]

      3. clear-num 86.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

   6. Applied egg-rr 86.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

   if -2.2999999999999998e41 < y < -2.89999999999999991e-9 or -2.79999999999999989e-80 < y < 6.1999999999999998e-23

   1. Initial program 97.3%

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

   2. Taylor expanded in y around 0 71.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{t \cdot z}} \]
   3. Step-by-step derivation

      1. associate-*r/ 71.7%

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

      2. neg-mul-1 71.7%

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

   4. Simplified 71.7%

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

   if -2.89999999999999991e-9 < y < -2.79999999999999989e-80 or 6.1999999999999998e-23 < y

   1. Initial program 97.2%

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

   2. Taylor expanded in y around inf 75.0%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-9} \lor \neg \left(y \leq -2.8 \cdot 10^{-80}\right) \land y \leq 6.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3: 55.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+272} \lor \neg \left(z \leq 1.1 \cdot 10^{+60}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.2e+272) (not (<= z 1.1e+60))) (/ x (* z t)) (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.2e+272) || !(z <= 1.1e+60)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	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
    real(8) :: tmp
    if ((z <= (-2.2d+272)) .or. (.not. (z <= 1.1d+60))) then
        tmp = x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.2e+272) || !(z <= 1.1e+60)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.2e+272) or not (z <= 1.1e+60):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.2e+272) || !(z <= 1.1e+60))
		tmp = Float64(x / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2.2e+272) || ~((z <= 1.1e+60)))
		tmp = x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.2e+272], N[Not[LessEqual[z, 1.1e+60]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.20000000000000008e272 or 1.09999999999999998e60 < z

   1. Initial program 93.5%

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

   2. Taylor expanded in y around 0 75.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{t \cdot z}} \]
   3. Step-by-step derivation

      1. associate-*r/ 75.1%

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

      2. neg-mul-1 75.1%

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

   4. Simplified 75.1%

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

      1. expm1-log1p-u 67.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-x}{t \cdot z}\right)\right)} \]

      2. expm1-udef 51.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-x}{t \cdot z}\right)} - 1} \]

      3. add-sqr-sqrt 26.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{t \cdot z}\right)} - 1 \]

      4. sqrt-unprod 41.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{t \cdot z}\right)} - 1 \]

      5. sqr-neg 41.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{x \cdot x}}}{t \cdot z}\right)} - 1 \]

      6. sqrt-unprod 18.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{t \cdot z}\right)} - 1 \]

      7. add-sqr-sqrt 41.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{t \cdot z}\right)} - 1 \]

   6. Applied egg-rr 41.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{t \cdot z}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 37.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{t \cdot z}\right)\right)} \]

      2. expm1-log1p 38.0%

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

   8. Simplified 38.0%

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

   if -2.20000000000000008e272 < z < 1.09999999999999998e60

   1. Initial program 98.4%

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

   2. Taylor expanded in y around inf 66.4%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+272} \lor \neg \left(z \leq 1.1 \cdot 10^{+60}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 54.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 97.3%

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

2. Taylor expanded in y around inf 58.5%

   \[\leadsto \color{blue}{\frac{x}{y}} \]

3. Final simplification 58.5%

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

Developer target: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{if}\;x < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x < 2.1378306434876444 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (- (/ y x) (* (/ z x) t)))))
   (if (< x -1.618195973607049e+50)
     t_1
     (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / ((y / x) - ((z / x) * t))
    if (x < (-1.618195973607049d+50)) then
        tmp = t_1
    else if (x < 2.1378306434876444d+131) 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) {
	double t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 / ((y / x) - ((z / x) * t))
	tmp = 0
	if x < -1.618195973607049e+50:
		tmp = t_1
	elif x < 2.1378306434876444e+131:
		tmp = x / (y - (z * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(Float64(y / x) - Float64(Float64(z / x) * t)))
	tmp = 0.0
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 / ((y / x) - ((z / x) * t));
	tmp = 0.0;
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = x / (y - (z * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 / N[(N[(y / x), $MachinePrecision] - N[(N[(z / x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[x, -1.618195973607049e+50], t$95$1, If[Less[x, 2.1378306434876444e+131], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2023238 
(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))))