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

Percentage Accurate: 96.2% → 99.5%

Time: 5.9s

Alternatives: 5

Speedup: 0.5×

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: 96.2% 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: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+206}\right):\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) (- INFINITY)) (not (<= (* z t) 5e+206)))
   (/ (/ x t) (- z))
   (/ x (- y (* z t)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -((double) INFINITY)) || !((z * t) <= 5e+206)) {
		tmp = (x / t) / -z;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -Double.POSITIVE_INFINITY) || !((z * t) <= 5e+206)) {
		tmp = (x / t) / -z;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -math.inf) or not ((z * t) <= 5e+206):
		tmp = (x / t) / -z
	else:
		tmp = x / (y - (z * t))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= Float64(-Inf)) || !(Float64(z * t) <= 5e+206))
		tmp = Float64(Float64(x / t) / Float64(-z));
	else
		tmp = Float64(x / Float64(y - Float64(z * t)));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -Inf) || ~(((z * t) <= 5e+206)))
		tmp = (x / t) / -z;
	else
		tmp = x / (y - (z * t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[(z * t), $MachinePrecision], 5e+206]], $MachinePrecision]], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 5.0000000000000002e206 < (*.f64 z t)

   1. Initial program 70.1%

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

      1. clear-num 70.1%

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

      2. associate-/r/ 70.1%

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

   3. Applied egg-rr 70.1%

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

   4. Taylor expanded in y around 0 70.1%

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

      1. associate-/r* 74.4%

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

   6. Simplified 74.4%

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

      1. *-commutative 74.4%

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

      2. frac-2neg 74.4%

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

      3. distribute-neg-frac 74.4%

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

      4. metadata-eval 74.4%

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

      5. associate-*r/ 99.7%

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

      6. add-sqr-sqrt 55.5%

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

      7. sqrt-unprod 70.8%

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

      8. sqr-neg 70.8%

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

      9. sqrt-unprod 22.3%

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

      10. add-sqr-sqrt 58.5%

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

      11. div-inv 58.5%

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

      12. add-sqr-sqrt 22.3%

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

      13. sqrt-unprod 70.9%

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

      14. sqr-neg 70.9%

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

      15. sqrt-unprod 55.5%

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

      16. add-sqr-sqrt 99.9%

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

   8. Applied egg-rr 99.9%

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

   if -inf.0 < (*.f64 z t) < 5.0000000000000002e206

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+206}\right):\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

Alternative 2: 77.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x}{t}}{-z}\\ \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq -2 \cdot 10^{-112}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+124}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (/ x t) (- z))))
   (if (<= (* z t) (- INFINITY))
     t_1
     (if (<= (* z t) -2e-112)
       (/ (- x) (* z t))
       (if (<= (* z t) 1e+124) (/ x y) t_1)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (x / t) / -z;
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = t_1;
	} else if ((z * t) <= -2e-112) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 1e+124) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = (x / t) / -z;
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if ((z * t) <= -2e-112) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 1e+124) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	t_1 = (x / t) / -z
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = t_1
	elif (z * t) <= -2e-112:
		tmp = -x / (z * t)
	elif (z * t) <= 1e+124:
		tmp = x / y
	else:
		tmp = t_1
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(x / t) / Float64(-z))
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = t_1;
	elseif (Float64(z * t) <= -2e-112)
		tmp = Float64(Float64(-x) / Float64(z * t));
	elseif (Float64(z * t) <= 1e+124)
		tmp = Float64(x / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = (x / t) / -z;
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = t_1;
	elseif ((z * t) <= -2e-112)
		tmp = -x / (z * t);
	elseif ((z * t) <= 1e+124)
		tmp = x / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], -2e-112], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+124], N[(x / y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -inf.0 or 9.99999999999999948e123 < (*.f64 z t)

   1. Initial program 78.5%

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

      1. clear-num 78.0%

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

      2. associate-/r/ 78.4%

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

   3. Applied egg-rr 78.4%

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

   4. Taylor expanded in y around 0 70.6%

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

      1. associate-/r* 73.6%

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

   6. Simplified 73.6%

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

      1. *-commutative 73.6%

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

      2. frac-2neg 73.6%

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

      3. distribute-neg-frac 73.6%

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

      4. metadata-eval 73.6%

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

      5. associate-*r/ 91.5%

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

      6. add-sqr-sqrt 48.3%

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

      7. sqrt-unprod 61.1%

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

      8. sqr-neg 61.1%

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

      9. sqrt-unprod 22.6%

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

      10. add-sqr-sqrt 49.0%

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

      11. div-inv 49.0%

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

      12. add-sqr-sqrt 22.6%

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

      13. sqrt-unprod 61.2%

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

      14. sqr-neg 61.2%

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

      15. sqrt-unprod 48.4%

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

      16. add-sqr-sqrt 91.6%

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

   8. Applied egg-rr 91.6%

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

   if -inf.0 < (*.f64 z t) < -1.9999999999999999e-112

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 73.0%

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

      1. associate-*r/ 73.0%

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

      2. neg-mul-1 73.0%

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

   4. Simplified 73.0%

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

   if -1.9999999999999999e-112 < (*.f64 z t) < 9.99999999999999948e123

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 81.4%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{elif}\;z \cdot t \leq -2 \cdot 10^{-112}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+124}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \]

Alternative 3: 63.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+193} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+195}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -4e+193) (not (<= (* z t) 5e+195)))
   (/ x (* z t))
   (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -4e+193) || !((z * t) <= 5e+195)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this 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) <= (-4d+193)) .or. (.not. ((z * t) <= 5d+195))) then
        tmp = x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -4e+193) || !((z * t) <= 5e+195)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -4e+193) or not ((z * t) <= 5e+195):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -4e+193) || !(Float64(z * t) <= 5e+195))
		tmp = Float64(x / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -4e+193) || ~(((z * t) <= 5e+195)))
		tmp = x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -4e+193], N[Not[LessEqual[N[(z * t), $MachinePrecision], 5e+195]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.00000000000000026e193 or 4.9999999999999998e195 < (*.f64 z t)

   1. Initial program 79.3%

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

      1. clear-num 79.1%

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

      2. associate-/r/ 79.2%

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

   3. Applied egg-rr 79.2%

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

   4. Taylor expanded in y around 0 79.2%

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

      1. associate-/r* 82.1%

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

   6. Simplified 82.1%

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

      1. associate-/l/ 79.2%

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

      2. associate-*l/ 79.3%

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

      3. neg-mul-1 79.3%

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

      4. add-sqr-sqrt 37.9%

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

      5. sqrt-unprod 64.5%

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

      6. sqr-neg 64.5%

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

      7. sqrt-unprod 30.4%

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

      8. add-sqr-sqrt 57.5%

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

   8. Applied egg-rr 57.5%

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

   if -4.00000000000000026e193 < (*.f64 z t) < 4.9999999999999998e195

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 67.5%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+193} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+195}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 73.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-37}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -4.3e-37) (/ x y) (if (<= y 5.6e-26) (/ (- x) (* z t)) (/ x y))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.3e-37) {
		tmp = x / y;
	} else if (y <= 5.6e-26) {
		tmp = -x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this 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 (y <= (-4.3d-37)) then
        tmp = x / y
    else if (y <= 5.6d-26) then
        tmp = -x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.3e-37) {
		tmp = x / y;
	} else if (y <= 5.6e-26) {
		tmp = -x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -4.3e-37:
		tmp = x / y
	elif y <= 5.6e-26:
		tmp = -x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -4.3e-37)
		tmp = Float64(x / y);
	elseif (y <= 5.6e-26)
		tmp = Float64(Float64(-x) / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -4.3e-37)
		tmp = x / y;
	elseif (y <= 5.6e-26)
		tmp = -x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[y, -4.3e-37], N[(x / y), $MachinePrecision], If[LessEqual[y, 5.6e-26], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.29999999999999968e-37 or 5.6000000000000002e-26 < y

   1. Initial program 94.9%

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

   2. Taylor expanded in y around inf 76.4%

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

   if -4.29999999999999968e-37 < y < 5.6000000000000002e-26

   1. Initial program 94.8%

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

   2. Taylor expanded in y around 0 76.6%

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

      1. associate-*r/ 76.6%

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

      2. neg-mul-1 76.6%

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

   4. Simplified 76.6%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-37}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 54.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (/ x y))

C

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

Fortran

NOTE: z and t should be sorted in increasing order before calling this 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
    code = x / y
end function

Java

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

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	return x / y

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	return Float64(x / y)
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp = code(x, y, z, t)
	tmp = x / y;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 94.9%

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

2. Taylor expanded in y around inf 54.1%

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

3. Final simplification 54.1%

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

Developer target: 96.7% 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 2023214 
(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))))