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

Percentage Accurate: 96.0% → 97.9%

Time: 7.3s

Alternatives: 6

Speedup: 0.6×

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: 96.0% 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: 97.9% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY)) (/ (/ x t) (- z)) (/ x (fma (- z) t y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = (x / t) / -z;
	} else {
		tmp = x / fma(-z, t, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(Float64(x / t) / Float64(-z));
	else
		tmp = Float64(x / fma(Float64(-z), t, y));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 65.0%

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

      1. clear-num 65.0%

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

      2. associate-/r/ 65.0%

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

   3. Applied egg-rr 65.0%

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

   4. Taylor expanded in y around 0 65.0%

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

      1. associate-*l/ 65.0%

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

      2. neg-mul-1 65.0%

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

      3. associate-/r* 100.0%

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

      4. remove-double-neg 100.0%

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

      5. frac-2neg 100.0%

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

      6. frac-2neg 100.0%

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

      7. distribute-neg-frac 100.0%

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

      8. add-sqr-sqrt 47.2%

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

      9. sqrt-unprod 70.1%

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

      10. sqr-neg 70.1%

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

      11. sqrt-unprod 37.2%

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

      12. add-sqr-sqrt 64.4%

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

      13. remove-double-neg 64.4%

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

      14. frac-2neg 64.4%

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

      15. add-sqr-sqrt 27.2%

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

      16. sqrt-unprod 64.9%

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

      17. sqr-neg 64.9%

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

      18. sqrt-unprod 52.6%

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

      19. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

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

   1. Initial program 99.0%

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

      1. sub-neg 99.0%

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

      2. +-commutative 99.0%

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

      3. distribute-lft-neg-in 99.0%

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

      4. fma-def 99.0%

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

   3. Applied egg-rr 99.0%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(-z, t, y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \end{array} \]

Alternative 2: 62.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+258} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+104}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -5e+258) (not (<= (* z t) 2e+104)))
   (/ x (* z t))
   (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e+258) || !((z * t) <= 2e+104)) {
		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+258)) .or. (.not. ((z * t) <= 2d+104))) 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+258) || !((z * t) <= 2e+104)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -5e+258) or not ((z * t) <= 2e+104):
		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+258) || !(Float64(z * t) <= 2e+104))
		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+258) || ~(((z * t) <= 2e+104)))
		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+258], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+104]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5e258 or 2e104 < (*.f64 z t)

   1. Initial program 85.1%

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

      1. clear-num 85.1%

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

      2. associate-/r/ 85.0%

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

   3. Applied egg-rr 85.0%

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

   4. Taylor expanded in y around 0 80.0%

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

      1. *-commutative 80.0%

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

      2. frac-2neg 80.0%

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

      3. metadata-eval 80.0%

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

      4. *-commutative 80.0%

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

      5. un-div-inv 80.0%

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

      6. distribute-rgt-neg-in 80.0%

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

      7. add-sqr-sqrt 30.1%

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

      8. sqrt-unprod 62.1%

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

      9. sqr-neg 62.1%

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

      10. sqrt-unprod 33.6%

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

      11. add-sqr-sqrt 58.7%

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

   6. Applied egg-rr 58.7%

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

   if -5e258 < (*.f64 z t) < 2e104

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 65.3%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+258} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+104}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3: 97.9% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY)) (/ (/ x t) (- z)) (/ x (- y (* z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = (x / t) / -z;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = (x / t) / -z;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = (x / t) / -z
	else:
		tmp = x / (y - (z * t))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], 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

   1. Initial program 65.0%

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

      1. clear-num 65.0%

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

      2. associate-/r/ 65.0%

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

   3. Applied egg-rr 65.0%

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

   4. Taylor expanded in y around 0 65.0%

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

      1. associate-*l/ 65.0%

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

      2. neg-mul-1 65.0%

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

      3. associate-/r* 100.0%

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

      4. remove-double-neg 100.0%

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

      5. frac-2neg 100.0%

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

      6. frac-2neg 100.0%

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

      7. distribute-neg-frac 100.0%

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

      8. add-sqr-sqrt 47.2%

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

      9. sqrt-unprod 70.1%

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

      10. sqr-neg 70.1%

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

      11. sqrt-unprod 37.2%

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

      12. add-sqr-sqrt 64.4%

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

      13. remove-double-neg 64.4%

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

      14. frac-2neg 64.4%

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

      15. add-sqr-sqrt 27.2%

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

      16. sqrt-unprod 64.9%

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

      17. sqr-neg 64.9%

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

      18. sqrt-unprod 52.6%

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

      19. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

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

   1. Initial program 99.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

Alternative 4: 71.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-139} \lor \neg \left(y \leq 2.5 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.8e-139) (not (<= y 2.5e-28)))
   (/ 1.0 (/ y x))
   (/ (- x) (* z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.8e-139) || !(y <= 2.5e-28)) {
		tmp = 1.0 / (y / x);
	} 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 <= (-1.8d-139)) .or. (.not. (y <= 2.5d-28))) then
        tmp = 1.0d0 / (y / x)
    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 <= -1.8e-139) || !(y <= 2.5e-28)) {
		tmp = 1.0 / (y / x);
	} else {
		tmp = -x / (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1.8e-139) or not (y <= 2.5e-28):
		tmp = 1.0 / (y / x)
	else:
		tmp = -x / (z * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.8e-139) || !(y <= 2.5e-28))
		tmp = Float64(1.0 / Float64(y / x));
	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 <= -1.8e-139) || ~((y <= 2.5e-28)))
		tmp = 1.0 / (y / x);
	else
		tmp = -x / (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.8e-139], N[Not[LessEqual[y, 2.5e-28]], $MachinePrecision]], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.80000000000000002e-139 or 2.5000000000000001e-28 < y

   1. Initial program 97.0%

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

      1. clear-num 96.6%

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

      2. associate-/r/ 96.8%

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

   3. Applied egg-rr 96.8%

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

   4. Taylor expanded in y around inf 77.0%

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

      1. associate-*l/ 77.2%

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

      2. associate-/l* 77.5%

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

   6. Applied egg-rr 77.5%

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

   if -1.80000000000000002e-139 < y < 2.5000000000000001e-28

   1. Initial program 95.5%

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

   2. Taylor expanded in y around 0 83.0%

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

      1. associate-*r/ 83.0%

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

      2. neg-mul-1 83.0%

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

   4. Simplified 83.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-139} \lor \neg \left(y \leq 2.5 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \]

Alternative 5: 53.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{y}{x}} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ 1.0 (/ y x)))

C

double code(double x, double y, double z, double t) {
	return 1.0 / (y / x);
}

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 = 1.0d0 / (y / x)
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 / (y / x);
}

Python

def code(x, y, z, t):
	return 1.0 / (y / x)

Julia

function code(x, y, z, t)
	return Float64(1.0 / Float64(y / x))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 / (y / x);
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.5%

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

   1. clear-num 96.0%

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

   2. associate-/r/ 96.2%

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

3. Applied egg-rr 96.2%

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

4. Taylor expanded in y around inf 54.7%

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

   1. associate-*l/ 54.9%

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

   2. associate-/l* 55.1%

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

6. Applied egg-rr 55.1%

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

7. Final simplification 55.1%

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

Alternative 6: 54.1% 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 96.5%

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

2. Taylor expanded in y around inf 54.9%

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

3. Final simplification 54.9%

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

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 2023230 
(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))))