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

Percentage Accurate: 95.7% → 97.8%

Time: 9.6s

Alternatives: 6

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: 95.7% 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: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+297}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) 1e+297) (/ x (- y (* z t))) (/ (/ x t) (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 1e+297) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / t) / -z;
	}
	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 * t) <= 1d+297) then
        tmp = x / (y - (z * t))
    else
        tmp = (x / t) / -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 1e+297) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= 1e+297:
		tmp = x / (y - (z * t))
	else:
		tmp = (x / t) / -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= 1e+297)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(x / t) / Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= 1e+297)
		tmp = x / (y - (z * t));
	else
		tmp = (x / t) / -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], 1e+297], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 1e297

   1. Initial program 98.2%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   if 1e297 < (*.f64 z t)

   1. Initial program 64.4%

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

      1. clear-num 64.4%

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

      2. associate-/r/ 64.4%

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

   4. Applied egg-rr 64.4%

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

   5. Taylor expanded in y around 0 64.4%

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

      1. mul-1-neg 64.4%

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

      2. associate-/r* 99.9%

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

      3. distribute-neg-frac2 99.9%

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

   7. Simplified 99.9%

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

Alternative 2: 78.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t \cdot \left(-z\right)}\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{-56}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z \cdot \frac{t}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* t (- z)))))
   (if (<= (* z t) -5e-10)
     t_1
     (if (<= (* z t) 1e-56)
       (/ x y)
       (if (<= (* z t) 1e-11)
         t_1
         (if (<= (* z t) 2e+55) (/ x y) (/ -1.0 (* z (/ t x)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / (t * -z);
	double tmp;
	if ((z * t) <= -5e-10) {
		tmp = t_1;
	} else if ((z * t) <= 1e-56) {
		tmp = x / y;
	} else if ((z * t) <= 1e-11) {
		tmp = t_1;
	} else if ((z * t) <= 2e+55) {
		tmp = x / y;
	} else {
		tmp = -1.0 / (z * (t / x));
	}
	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 = x / (t * -z)
    if ((z * t) <= (-5d-10)) then
        tmp = t_1
    else if ((z * t) <= 1d-56) then
        tmp = x / y
    else if ((z * t) <= 1d-11) then
        tmp = t_1
    else if ((z * t) <= 2d+55) then
        tmp = x / y
    else
        tmp = (-1.0d0) / (z * (t / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / (t * -z);
	double tmp;
	if ((z * t) <= -5e-10) {
		tmp = t_1;
	} else if ((z * t) <= 1e-56) {
		tmp = x / y;
	} else if ((z * t) <= 1e-11) {
		tmp = t_1;
	} else if ((z * t) <= 2e+55) {
		tmp = x / y;
	} else {
		tmp = -1.0 / (z * (t / x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / (t * -z)
	tmp = 0
	if (z * t) <= -5e-10:
		tmp = t_1
	elif (z * t) <= 1e-56:
		tmp = x / y
	elif (z * t) <= 1e-11:
		tmp = t_1
	elif (z * t) <= 2e+55:
		tmp = x / y
	else:
		tmp = -1.0 / (z * (t / x))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(t * Float64(-z)))
	tmp = 0.0
	if (Float64(z * t) <= -5e-10)
		tmp = t_1;
	elseif (Float64(z * t) <= 1e-56)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 1e-11)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e+55)
		tmp = Float64(x / y);
	else
		tmp = Float64(-1.0 / Float64(z * Float64(t / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / (t * -z);
	tmp = 0.0;
	if ((z * t) <= -5e-10)
		tmp = t_1;
	elseif ((z * t) <= 1e-56)
		tmp = x / y;
	elseif ((z * t) <= 1e-11)
		tmp = t_1;
	elseif ((z * t) <= 2e+55)
		tmp = x / y;
	else
		tmp = -1.0 / (z * (t / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(t * (-z)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -5e-10], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e-56], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e-11], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e+55], N[(x / y), $MachinePrecision], N[(-1.0 / N[(z * N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.00000000000000031e-10 or 1e-56 < (*.f64 z t) < 9.99999999999999939e-12

   1. Initial program 95.2%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 75.5%

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

      1. associate-*r/ 75.5%

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

      2. neg-mul-1 75.5%

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

   5. Simplified 75.5%

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

   if -5.00000000000000031e-10 < (*.f64 z t) < 1e-56 or 9.99999999999999939e-12 < (*.f64 z t) < 2.00000000000000002e55

   1. Initial program 100.0%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 82.6%

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

   if 2.00000000000000002e55 < (*.f64 z t)

   1. Initial program 87.6%

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

      1. clear-num 87.7%

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

      2. associate-/r/ 87.6%

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

   4. Applied egg-rr 87.6%

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

   5. Taylor expanded in y around 0 76.0%

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

      1. mul-1-neg 76.0%

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

      2. associate-/r* 84.1%

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

      3. distribute-neg-frac2 84.1%

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

   7. Simplified 84.1%

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

      1. div-inv 84.1%

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

      2. neg-mul-1 84.1%

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

      3. times-frac 76.1%

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

   9. Applied egg-rr 76.1%

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

       1. frac-2neg 76.1%

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

       2. metadata-eval 76.1%

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

       3. frac-times 84.1%

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

       4. *-un-lft-identity 84.1%

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

       5. div-inv 84.1%

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

       6. associate-/r* 76.0%

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

       7. clear-num 76.0%

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

       8. frac-2neg 76.0%

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

       9. metadata-eval 76.0%

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

       10. add-sqr-sqrt 32.3%

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

       11. sqrt-unprod 50.0%

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

       12. sqr-neg 50.0%

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

       13. sqrt-unprod 20.4%

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

       14. add-sqr-sqrt 44.5%

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

       15. distribute-frac-neg2 44.5%

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

       16. *-commutative 44.5%

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

       17. *-un-lft-identity 44.5%

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

       18. times-frac 44.3%

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

       19. /-rgt-identity 44.3%

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

       20. add-sqr-sqrt 24.0%

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

       21. sqrt-unprod 58.8%

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

       22. sqr-neg 58.8%

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

       23. sqrt-unprod 45.5%

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

       24. add-sqr-sqrt 84.1%

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

   11. Applied egg-rr 84.1%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{elif}\;z \cdot t \leq 10^{-56}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 10^{-11}:\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z \cdot \frac{t}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t \cdot \left(-z\right)}\\ \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 10^{-56}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* t (- z)))))
   (if (<= (* z t) -5e-10)
     t_1
     (if (<= (* z t) 1e-56)
       (/ x y)
       (if (<= (* z t) 1e-11)
         t_1
         (if (<= (* z t) 2e+55) (/ x y) (/ (/ x t) (- z))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / (t * -z);
	double tmp;
	if ((z * t) <= -5e-10) {
		tmp = t_1;
	} else if ((z * t) <= 1e-56) {
		tmp = x / y;
	} else if ((z * t) <= 1e-11) {
		tmp = t_1;
	} else if ((z * t) <= 2e+55) {
		tmp = x / y;
	} else {
		tmp = (x / t) / -z;
	}
	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 = x / (t * -z)
    if ((z * t) <= (-5d-10)) then
        tmp = t_1
    else if ((z * t) <= 1d-56) then
        tmp = x / y
    else if ((z * t) <= 1d-11) then
        tmp = t_1
    else if ((z * t) <= 2d+55) then
        tmp = x / y
    else
        tmp = (x / t) / -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x / (t * -z);
	double tmp;
	if ((z * t) <= -5e-10) {
		tmp = t_1;
	} else if ((z * t) <= 1e-56) {
		tmp = x / y;
	} else if ((z * t) <= 1e-11) {
		tmp = t_1;
	} else if ((z * t) <= 2e+55) {
		tmp = x / y;
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / (t * -z)
	tmp = 0
	if (z * t) <= -5e-10:
		tmp = t_1
	elif (z * t) <= 1e-56:
		tmp = x / y
	elif (z * t) <= 1e-11:
		tmp = t_1
	elif (z * t) <= 2e+55:
		tmp = x / y
	else:
		tmp = (x / t) / -z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(t * Float64(-z)))
	tmp = 0.0
	if (Float64(z * t) <= -5e-10)
		tmp = t_1;
	elseif (Float64(z * t) <= 1e-56)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 1e-11)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e+55)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / t) / Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / (t * -z);
	tmp = 0.0;
	if ((z * t) <= -5e-10)
		tmp = t_1;
	elseif ((z * t) <= 1e-56)
		tmp = x / y;
	elseif ((z * t) <= 1e-11)
		tmp = t_1;
	elseif ((z * t) <= 2e+55)
		tmp = x / y;
	else
		tmp = (x / t) / -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(t * (-z)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -5e-10], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e-56], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e-11], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e+55], N[(x / y), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.00000000000000031e-10 or 1e-56 < (*.f64 z t) < 9.99999999999999939e-12

   1. Initial program 95.2%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 75.5%

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

      1. associate-*r/ 75.5%

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

      2. neg-mul-1 75.5%

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

   5. Simplified 75.5%

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

   if -5.00000000000000031e-10 < (*.f64 z t) < 1e-56 or 9.99999999999999939e-12 < (*.f64 z t) < 2.00000000000000002e55

   1. Initial program 100.0%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 82.6%

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

   if 2.00000000000000002e55 < (*.f64 z t)

   1. Initial program 87.6%

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

      1. clear-num 87.7%

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

      2. associate-/r/ 87.6%

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

   4. Applied egg-rr 87.6%

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

   5. Taylor expanded in y around 0 76.0%

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

      1. mul-1-neg 76.0%

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

      2. associate-/r* 84.1%

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

      3. distribute-neg-frac2 84.1%

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

   7. Simplified 84.1%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{elif}\;z \cdot t \leq 10^{-56}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 10^{-11}:\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{-10} \lor \neg \left(z \cdot t \leq 10^{-56}\right):\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -5e-10) (not (<= (* z t) 1e-56)))
   (/ x (* t (- z)))
   (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e-10) || !((z * t) <= 1e-56)) {
		tmp = x / (t * -z);
	} 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 * t) <= (-5d-10)) .or. (.not. ((z * t) <= 1d-56))) then
        tmp = x / (t * -z)
    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 * t) <= -5e-10) || !((z * t) <= 1e-56)) {
		tmp = x / (t * -z);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -5e-10) or not ((z * t) <= 1e-56):
		tmp = x / (t * -z)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -5e-10) || !(Float64(z * t) <= 1e-56))
		tmp = Float64(x / Float64(t * Float64(-z)));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -5e-10) || ~(((z * t) <= 1e-56)))
		tmp = x / (t * -z);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -5e-10], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e-56]], $MachinePrecision]], N[(x / N[(t * (-z)), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5.00000000000000031e-10 or 1e-56 < (*.f64 z t)

   1. Initial program 93.3%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 71.6%

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

      1. associate-*r/ 71.6%

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

      2. neg-mul-1 71.6%

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

   5. Simplified 71.6%

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

   if -5.00000000000000031e-10 < (*.f64 z t) < 1e-56

   1. Initial program 100.0%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 85.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{-10} \lor \neg \left(z \cdot t \leq 10^{-56}\right):\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+194} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -5e+194) (not (<= (* z t) 2e+161)))
   (/ x (* z t))
   (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e+194) || !((z * t) <= 2e+161)) {
		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 * t) <= (-5d+194)) .or. (.not. ((z * t) <= 2d+161))) 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 * t) <= -5e+194) || !((z * t) <= 2e+161)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -5e+194) or not ((z * t) <= 2e+161):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -5e+194) || !(Float64(z * t) <= 2e+161))
		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 * t) <= -5e+194) || ~(((z * t) <= 2e+161)))
		tmp = x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -5e+194], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+161]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.99999999999999989e194 or 2.0000000000000001e161 < (*.f64 z t)

   1. Initial program 84.8%

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

      1. clear-num 83.9%

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

      2. associate-/r/ 84.8%

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

   4. Applied egg-rr 84.8%

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

   5. Taylor expanded in y around 0 80.7%

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

      1. mul-1-neg 80.7%

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

      2. associate-/r* 95.8%

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

      3. distribute-neg-frac2 95.8%

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

   7. Simplified 95.8%

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

      1. associate-/l/ 80.7%

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

      2. *-un-lft-identity 80.7%

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

      3. add-sqr-sqrt 45.9%

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

      4. sqrt-unprod 66.3%

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

      5. sqr-neg 66.3%

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

      6. sqrt-unprod 27.7%

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

      7. add-sqr-sqrt 60.0%

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

   9. Applied egg-rr 60.0%

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

       1. *-lft-identity 60.0%

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

       2. *-commutative 60.0%

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

   11. Simplified 60.0%

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

   if -4.99999999999999989e194 < (*.f64 z t) < 2.0000000000000001e161

   1. Initial program 99.8%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 64.0%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+194} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.2% 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 96.1%

   \[\frac{x}{y - z \cdot t} \]
2. Add Preprocessing

3. Taylor expanded in y around inf 52.6%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
4. Add Preprocessing

Developer target: 96.6% accurate, 0.3× 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 2024092 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (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))))