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

Percentage Accurate: 95.7% → 97.7%

Time: 6.3s

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: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+265}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) 2e+265) (/ x (fma z (- t) y)) (* (/ -1.0 t) (/ x z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 2e+265) {
		tmp = x / fma(z, -t, y);
	} else {
		tmp = (-1.0 / t) * (x / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= 2e+265)
		tmp = Float64(x / fma(z, Float64(-t), y));
	else
		tmp = Float64(Float64(-1.0 / t) * Float64(x / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], 2e+265], N[(x / N[(z * (-t) + y), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / t), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 2.00000000000000013e265

   1. Initial program 98.2%

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

      1. cancel-sign-sub-inv 98.2%

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

      2. +-commutative 98.2%

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

      3. distribute-lft-neg-out 98.2%

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

      4. distribute-rgt-neg-out 98.2%

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

      5. fma-def 98.2%

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(z, -t, y\right)}} \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(z, -t, y\right)}} \]
   4. Add Preprocessing

   if 2.00000000000000013e265 < (*.f64 z t)

   1. Initial program 71.2%

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

   3. Taylor expanded in y around 0 71.2%

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

      1. associate-*r/ 71.2%

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

      2. neg-mul-1 71.2%

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

   5. Simplified 71.2%

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

      1. neg-mul-1 71.2%

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

      2. times-frac 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{-1}{t} \cdot \frac{x}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+265}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+101} \lor \neg \left(y \leq 160000000000\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -3.7e+101) (not (<= y 160000000000.0)))
   (/ x y)
   (/ (- x) (* z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -3.7e+101) || !(y <= 160000000000.0)) {
		tmp = x / y;
	} else {
		tmp = -x / (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
    real(8) :: tmp
    if ((y <= (-3.7d+101)) .or. (.not. (y <= 160000000000.0d0))) then
        tmp = x / y
    else
        tmp = -x / (z * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -3.7e+101) || !(y <= 160000000000.0)) {
		tmp = x / y;
	} else {
		tmp = -x / (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -3.7e+101) or not (y <= 160000000000.0):
		tmp = x / y
	else:
		tmp = -x / (z * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -3.7e+101) || !(y <= 160000000000.0))
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(-x) / Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -3.7e+101) || ~((y <= 160000000000.0)))
		tmp = x / y;
	else
		tmp = -x / (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -3.7e+101], N[Not[LessEqual[y, 160000000000.0]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.6999999999999997e101 or 1.6e11 < y

   1. Initial program 94.8%

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

   3. Taylor expanded in y around inf 85.3%

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

   if -3.6999999999999997e101 < y < 1.6e11

   1. Initial program 97.3%

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

   3. Taylor expanded in y around 0 76.4%

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

      1. associate-*r/ 76.4%

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

      2. neg-mul-1 76.4%

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

   5. Simplified 76.4%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+101} \lor \neg \left(y \leq 160000000000\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-171} \lor \neg \left(y \leq 6.6 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -5.5e-171) (not (<= y 6.6e-85))) (/ x y) (/ x (* z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -5.5e-171) || !(y <= 6.6e-85)) {
		tmp = x / y;
	} else {
		tmp = x / (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
    real(8) :: tmp
    if ((y <= (-5.5d-171)) .or. (.not. (y <= 6.6d-85))) then
        tmp = x / y
    else
        tmp = x / (z * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -5.5e-171) || !(y <= 6.6e-85)) {
		tmp = x / y;
	} else {
		tmp = x / (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -5.5e-171) or not (y <= 6.6e-85):
		tmp = x / y
	else:
		tmp = x / (z * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -5.5e-171) || !(y <= 6.6e-85))
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -5.5e-171) || ~((y <= 6.6e-85)))
		tmp = x / y;
	else
		tmp = x / (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -5.5e-171], N[Not[LessEqual[y, 6.6e-85]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.50000000000000037e-171 or 6.59999999999999945e-85 < y

   1. Initial program 96.5%

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

   3. Taylor expanded in y around inf 65.6%

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

   if -5.50000000000000037e-171 < y < 6.59999999999999945e-85

   1. Initial program 95.6%

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

   3. Taylor expanded in y around 0 91.2%

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

      1. associate-*r/ 91.2%

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

      2. neg-mul-1 91.2%

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

   5. Simplified 91.2%

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

      1. neg-mul-1 91.2%

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

      2. times-frac 89.0%

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

   7. Applied egg-rr 89.0%

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

      1. frac-times 91.2%

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

      2. *-commutative 91.2%

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

      3. frac-times 84.5%

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

      4. clear-num 84.5%

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

      5. frac-times 83.3%

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

      6. metadata-eval 83.3%

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

   9. Applied egg-rr 83.3%

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

       1. add-log-exp 44.0%

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

       2. *-un-lft-identity 44.0%

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

       3. log-prod 44.0%

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

       4. metadata-eval 44.0%

          \[\leadsto \color{blue}{0} + \log \left(e^{\frac{-1}{\frac{t}{x} \cdot z}}\right) \]

       5. add-log-exp 83.3%

          \[\leadsto 0 + \color{blue}{\frac{-1}{\frac{t}{x} \cdot z}} \]

       6. frac-2neg 83.3%

          \[\leadsto 0 + \color{blue}{\frac{--1}{-\frac{t}{x} \cdot z}} \]

       7. metadata-eval 83.3%

          \[\leadsto 0 + \frac{\color{blue}{1}}{-\frac{t}{x} \cdot z} \]

       8. distribute-rgt-neg-in 83.3%

          \[\leadsto 0 + \frac{1}{\color{blue}{\frac{t}{x} \cdot \left(-z\right)}} \]

       9. associate-/r/ 85.7%

          \[\leadsto 0 + \frac{1}{\color{blue}{\frac{t}{\frac{x}{-z}}}} \]

       10. clear-num 89.1%

           \[\leadsto 0 + \color{blue}{\frac{\frac{x}{-z}}{t}} \]

       11. div-inv 89.0%

           \[\leadsto 0 + \frac{\color{blue}{x \cdot \frac{1}{-z}}}{t} \]

       12. div-inv 89.1%

           \[\leadsto 0 + \frac{\color{blue}{\frac{x}{-z}}}{t} \]

       13. add-sqr-sqrt 45.7%

           \[\leadsto 0 + \frac{\frac{x}{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}}{t} \]

       14. sqrt-unprod 51.9%

           \[\leadsto 0 + \frac{\frac{x}{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}}{t} \]

       15. sqr-neg 51.9%

           \[\leadsto 0 + \frac{\frac{x}{\sqrt{\color{blue}{z \cdot z}}}}{t} \]

       16. sqrt-unprod 13.2%

           \[\leadsto 0 + \frac{\frac{x}{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}}{t} \]

       17. add-sqr-sqrt 31.2%

           \[\leadsto 0 + \frac{\frac{x}{\color{blue}{z}}}{t} \]

   11. Applied egg-rr 31.2%

       \[\leadsto \color{blue}{0 + \frac{\frac{x}{z}}{t}} \]
   12. Step-by-step derivation

       1. +-lft-identity 31.2%

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

       2. associate-/l/ 31.3%

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

   13. Simplified 31.3%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-171} \lor \neg \left(y \leq 6.6 \cdot 10^{-85}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-179} \lor \neg \left(y \leq 1.7 \cdot 10^{-86}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -4e-179) (not (<= y 1.7e-86))) (/ x y) (/ (/ x z) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4e-179) || !(y <= 1.7e-86)) {
		tmp = x / y;
	} else {
		tmp = (x / 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
    real(8) :: tmp
    if ((y <= (-4d-179)) .or. (.not. (y <= 1.7d-86))) then
        tmp = x / y
    else
        tmp = (x / z) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4e-179) || !(y <= 1.7e-86)) {
		tmp = x / y;
	} else {
		tmp = (x / z) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -4e-179) or not (y <= 1.7e-86):
		tmp = x / y
	else:
		tmp = (x / z) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -4e-179) || !(y <= 1.7e-86))
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / z) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -4e-179) || ~((y <= 1.7e-86)))
		tmp = x / y;
	else
		tmp = (x / z) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -4e-179], N[Not[LessEqual[y, 1.7e-86]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(N[(x / z), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.0000000000000001e-179 or 1.7e-86 < y

   1. Initial program 96.5%

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

   3. Taylor expanded in y around inf 65.6%

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

   if -4.0000000000000001e-179 < y < 1.7e-86

   1. Initial program 95.6%

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

   3. Taylor expanded in y around 0 91.2%

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

      1. associate-*r/ 91.2%

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

      2. neg-mul-1 91.2%

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

   5. Simplified 91.2%

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

      1. neg-mul-1 91.2%

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

      2. times-frac 89.0%

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

   7. Applied egg-rr 89.0%

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

      1. frac-times 91.2%

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

      2. *-commutative 91.2%

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

      3. frac-times 84.5%

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

      4. clear-num 84.5%

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

      5. frac-times 83.3%

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

      6. metadata-eval 83.3%

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

   9. Applied egg-rr 83.3%

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

       1. add-log-exp 44.0%

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

       2. *-un-lft-identity 44.0%

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

       3. log-prod 44.0%

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

       4. metadata-eval 44.0%

          \[\leadsto \color{blue}{0} + \log \left(e^{\frac{-1}{\frac{t}{x} \cdot z}}\right) \]

       5. add-log-exp 83.3%

          \[\leadsto 0 + \color{blue}{\frac{-1}{\frac{t}{x} \cdot z}} \]

       6. frac-2neg 83.3%

          \[\leadsto 0 + \color{blue}{\frac{--1}{-\frac{t}{x} \cdot z}} \]

       7. metadata-eval 83.3%

          \[\leadsto 0 + \frac{\color{blue}{1}}{-\frac{t}{x} \cdot z} \]

       8. distribute-rgt-neg-in 83.3%

          \[\leadsto 0 + \frac{1}{\color{blue}{\frac{t}{x} \cdot \left(-z\right)}} \]

       9. associate-/r/ 85.7%

          \[\leadsto 0 + \frac{1}{\color{blue}{\frac{t}{\frac{x}{-z}}}} \]

       10. clear-num 89.1%

           \[\leadsto 0 + \color{blue}{\frac{\frac{x}{-z}}{t}} \]

       11. div-inv 89.0%

           \[\leadsto 0 + \frac{\color{blue}{x \cdot \frac{1}{-z}}}{t} \]

       12. div-inv 89.1%

           \[\leadsto 0 + \frac{\color{blue}{\frac{x}{-z}}}{t} \]

       13. add-sqr-sqrt 45.7%

           \[\leadsto 0 + \frac{\frac{x}{\color{blue}{\sqrt{-z} \cdot \sqrt{-z}}}}{t} \]

       14. sqrt-unprod 51.9%

           \[\leadsto 0 + \frac{\frac{x}{\color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}}}{t} \]

       15. sqr-neg 51.9%

           \[\leadsto 0 + \frac{\frac{x}{\sqrt{\color{blue}{z \cdot z}}}}{t} \]

       16. sqrt-unprod 13.2%

           \[\leadsto 0 + \frac{\frac{x}{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}}{t} \]

       17. add-sqr-sqrt 31.2%

           \[\leadsto 0 + \frac{\frac{x}{\color{blue}{z}}}{t} \]

   11. Applied egg-rr 31.2%

       \[\leadsto \color{blue}{0 + \frac{\frac{x}{z}}{t}} \]
   12. Step-by-step derivation

       1. +-lft-identity 31.2%

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

   13. Simplified 31.2%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-179} \lor \neg \left(y \leq 1.7 \cdot 10^{-86}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+265}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) 2e+265) (/ x (- y (* z t))) (* (/ -1.0 t) (/ x z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 2e+265) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-1.0 / t) * (x / 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) <= 2d+265) then
        tmp = x / (y - (z * t))
    else
        tmp = ((-1.0d0) / t) * (x / 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) <= 2e+265) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-1.0 / t) * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= 2e+265:
		tmp = x / (y - (z * t))
	else:
		tmp = (-1.0 / t) * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= 2e+265)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(-1.0 / t) * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= 2e+265)
		tmp = x / (y - (z * t));
	else
		tmp = (-1.0 / t) * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], 2e+265], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / t), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 2.00000000000000013e265

   1. Initial program 98.2%

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

   if 2.00000000000000013e265 < (*.f64 z t)

   1. Initial program 71.2%

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

   3. Taylor expanded in y around 0 71.2%

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

      1. associate-*r/ 71.2%

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

      2. neg-mul-1 71.2%

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

   5. Simplified 71.2%

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

      1. neg-mul-1 71.2%

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

      2. times-frac 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{-1}{t} \cdot \frac{x}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+265}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.1% 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 96.2%

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

3. Taylor expanded in y around inf 51.5%

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

4. Final simplification 51.5%

   \[\leadsto \frac{x}{y} \]
5. Add Preprocessing

Developer target: 96.4% 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 2024027 
(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))))