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

Percentage Accurate: 95.7% → 99.5%

Time: 5.6s

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: 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: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 10^{+198}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) (- INFINITY)) (not (<= (* z t) 1e+198)))
   (/ (/ (- x) z) t)
   (/ x (fma z (- t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -((double) INFINITY)) || !((z * t) <= 1e+198)) {
		tmp = (-x / z) / t;
	} 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)) || !(Float64(z * t) <= 1e+198))
		tmp = Float64(Float64(Float64(-x) / z) / t);
	else
		tmp = Float64(x / fma(z, Float64(-t), y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+198]], $MachinePrecision]], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision], N[(x / N[(z * (-t) + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 1.00000000000000002e198 < (*.f64 z t)

   1. Initial program 73.8%

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

      1. add-sqr-sqrt 38.0%

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

      2. add-sqr-sqrt 27.6%

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

      3. difference-of-squares 27.6%

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

   3. Applied egg-rr 27.6%

      \[\leadsto \frac{x}{\color{blue}{\left(\sqrt{y} + \sqrt{z \cdot t}\right) \cdot \left(\sqrt{y} - \sqrt{z \cdot t}\right)}} \]
   4. Step-by-step derivation

      1. difference-of-squares 27.6%

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

      2. add-sqr-sqrt 55.5%

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

      3. add-sqr-sqrt 73.8%

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

      4. *-commutative 73.8%

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

      5. flip-- 8.1%

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

      6. pow2 8.1%

         \[\leadsto \frac{x}{\frac{\color{blue}{{y}^{2}} - \left(t \cdot z\right) \cdot \left(t \cdot z\right)}{y + t \cdot z}} \]

      7. pow2 8.1%

         \[\leadsto \frac{x}{\frac{{y}^{2} - \color{blue}{{\left(t \cdot z\right)}^{2}}}{y + t \cdot z}} \]

   5. Applied egg-rr 8.1%

      \[\leadsto \frac{x}{\color{blue}{\frac{{y}^{2} - {\left(t \cdot z\right)}^{2}}{y + t \cdot z}}} \]

   6. Taylor expanded in y around 0 73.8%

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

      1. mul-1-neg 73.8%

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

      2. distribute-frac-neg 73.8%

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

      3. *-commutative 73.8%

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

      4. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

   if -inf.0 < (*.f64 z t) < 1.00000000000000002e198

   1. Initial program 99.9%

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

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-lft-neg-out 99.9%

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

      4. distribute-rgt-neg-out 99.9%

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

      5. fma-def 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 10^{+198}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 10^{+198}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) (- INFINITY)) (not (<= (* z t) 1e+198)))
   (/ (/ (- x) z) t)
   (/ x (- y (* z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -((double) INFINITY)) || !((z * t) <= 1e+198)) {
		tmp = (-x / z) / t;
	} 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) || !((z * t) <= 1e+198)) {
		tmp = (-x / z) / t;
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -math.inf) or not ((z * t) <= 1e+198):
		tmp = (-x / z) / t
	else:
		tmp = x / (y - (z * t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= Float64(-Inf)) || !(Float64(z * t) <= 1e+198))
		tmp = Float64(Float64(Float64(-x) / z) / t);
	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) || ~(((z * t) <= 1e+198)))
		tmp = (-x / z) / t;
	else
		tmp = x / (y - (z * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+198]], $MachinePrecision]], N[(N[((-x) / z), $MachinePrecision] / t), $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 1.00000000000000002e198 < (*.f64 z t)

   1. Initial program 73.8%

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

      1. add-sqr-sqrt 38.0%

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

      2. add-sqr-sqrt 27.6%

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

      3. difference-of-squares 27.6%

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

   3. Applied egg-rr 27.6%

      \[\leadsto \frac{x}{\color{blue}{\left(\sqrt{y} + \sqrt{z \cdot t}\right) \cdot \left(\sqrt{y} - \sqrt{z \cdot t}\right)}} \]
   4. Step-by-step derivation

      1. difference-of-squares 27.6%

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

      2. add-sqr-sqrt 55.5%

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

      3. add-sqr-sqrt 73.8%

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

      4. *-commutative 73.8%

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

      5. flip-- 8.1%

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

      6. pow2 8.1%

         \[\leadsto \frac{x}{\frac{\color{blue}{{y}^{2}} - \left(t \cdot z\right) \cdot \left(t \cdot z\right)}{y + t \cdot z}} \]

      7. pow2 8.1%

         \[\leadsto \frac{x}{\frac{{y}^{2} - \color{blue}{{\left(t \cdot z\right)}^{2}}}{y + t \cdot z}} \]

   5. Applied egg-rr 8.1%

      \[\leadsto \frac{x}{\color{blue}{\frac{{y}^{2} - {\left(t \cdot z\right)}^{2}}{y + t \cdot z}}} \]

   6. Taylor expanded in y around 0 73.8%

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

      1. mul-1-neg 73.8%

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

      2. distribute-frac-neg 73.8%

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

      3. *-commutative 73.8%

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

      4. associate-/r* 99.9%

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

   8. Simplified 99.9%

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

   if -inf.0 < (*.f64 z t) < 1.00000000000000002e198

   1. Initial program 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 10^{+198}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

Alternative 3: 72.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00086 \lor \neg \left(y \leq 2.8 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -0.00086) (not (<= y 2.8e-16))) (/ x y) (/ (- x) (* z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -0.00086) || !(y <= 2.8e-16)) {
		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 <= (-0.00086d0)) .or. (.not. (y <= 2.8d-16))) 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 <= -0.00086) || !(y <= 2.8e-16)) {
		tmp = x / y;
	} else {
		tmp = -x / (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -0.00086) or not (y <= 2.8e-16):
		tmp = x / y
	else:
		tmp = -x / (z * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -0.00086) || !(y <= 2.8e-16))
		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 <= -0.00086) || ~((y <= 2.8e-16)))
		tmp = x / y;
	else
		tmp = -x / (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -0.00086], N[Not[LessEqual[y, 2.8e-16]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.59999999999999979e-4 or 2.8000000000000001e-16 < y

   1. Initial program 95.2%

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

   2. Taylor expanded in y around inf 75.8%

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

   if -8.59999999999999979e-4 < y < 2.8000000000000001e-16

   1. Initial program 96.5%

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

   2. Taylor expanded in y around 0 77.1%

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

      1. associate-*r/ 77.1%

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

      2. neg-mul-1 77.1%

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

   4. Simplified 77.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00086 \lor \neg \left(y \leq 2.8 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \]

Alternative 4: 56.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+206} \lor \neg \left(z \leq 8.2 \cdot 10^{+39}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.3e+206) (not (<= z 8.2e+39))) (/ x (* z t)) (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.3e+206) || !(z <= 8.2e+39)) {
		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 <= (-1.3d+206)) .or. (.not. (z <= 8.2d+39))) 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 <= -1.3e+206) || !(z <= 8.2e+39)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.3e+206) or not (z <= 8.2e+39):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.3e+206) || !(z <= 8.2e+39))
		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 <= -1.3e+206) || ~((z <= 8.2e+39)))
		tmp = x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.3e+206], N[Not[LessEqual[z, 8.2e+39]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.29999999999999994e206 or 8.20000000000000008e39 < z

   1. Initial program 89.4%

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

      1. add-sqr-sqrt 53.5%

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

      2. add-sqr-sqrt 21.9%

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

      3. difference-of-squares 21.9%

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

   3. Applied egg-rr 21.9%

      \[\leadsto \frac{x}{\color{blue}{\left(\sqrt{y} + \sqrt{z \cdot t}\right) \cdot \left(\sqrt{y} - \sqrt{z \cdot t}\right)}} \]
   4. Step-by-step derivation

      1. difference-of-squares 21.9%

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

      2. add-sqr-sqrt 34.4%

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

      3. add-sqr-sqrt 89.4%

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

      4. *-commutative 89.4%

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

      5. flip-- 40.3%

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

      6. pow2 40.3%

         \[\leadsto \frac{x}{\frac{\color{blue}{{y}^{2}} - \left(t \cdot z\right) \cdot \left(t \cdot z\right)}{y + t \cdot z}} \]

      7. pow2 40.3%

         \[\leadsto \frac{x}{\frac{{y}^{2} - \color{blue}{{\left(t \cdot z\right)}^{2}}}{y + t \cdot z}} \]

   5. Applied egg-rr 40.3%

      \[\leadsto \frac{x}{\color{blue}{\frac{{y}^{2} - {\left(t \cdot z\right)}^{2}}{y + t \cdot z}}} \]

   6. Taylor expanded in y around 0 74.3%

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

      1. mul-1-neg 74.3%

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

      2. distribute-frac-neg 74.3%

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

      3. *-commutative 74.3%

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

      4. associate-/r* 77.8%

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

   8. Simplified 77.8%

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

      1. expm1-log1p-u 71.7%

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

      2. expm1-udef 50.2%

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

      3. div-inv 50.2%

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

      4. associate-*l/ 46.6%

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

      5. add-sqr-sqrt 26.7%

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

      6. sqrt-unprod 42.3%

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

      7. sqr-neg 42.3%

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

      8. sqrt-unprod 17.2%

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

      9. add-sqr-sqrt 42.6%

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

      10. div-inv 42.6%

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

   10. Applied egg-rr 42.6%

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

       1. expm1-def 39.6%

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

       2. expm1-log1p 39.7%

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

       3. associate-/r* 39.9%

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

   12. Simplified 39.9%

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

   if -1.29999999999999994e206 < z < 8.20000000000000008e39

   1. Initial program 98.3%

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

   2. Taylor expanded in y around inf 62.2%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+206} \lor \neg \left(z \leq 8.2 \cdot 10^{+39}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 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 95.8%

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

2. Taylor expanded in y around inf 51.8%

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

3. Final simplification 51.8%

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

Developer target: 96.3% 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 2023306 
(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))))