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

Percentage Accurate: 95.8% → 97.6%

Time: 7.2s

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: 95.8% 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: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) 2e+261) (/ x (fma z (- t) y)) (/ (/ (- x) t) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 2e+261) {
		tmp = x / fma(z, -t, y);
	} else {
		tmp = (-x / t) / z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= 2e+261)
		tmp = Float64(x / fma(z, Float64(-t), y));
	else
		tmp = Float64(Float64(Float64(-x) / t) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], 2e+261], N[(x / N[(z * (-t) + y), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 1.9999999999999999e261

   1. Initial program 99.0%

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

      1. cancel-sign-sub-inv 99.0%

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

      2. +-commutative 99.0%

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

      3. distribute-lft-neg-out 99.0%

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

      4. distribute-rgt-neg-out 99.0%

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

      5. fma-def 99.1%

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

   3. Simplified 99.1%

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

   if 1.9999999999999999e261 < (*.f64 z t)

   1. Initial program 53.7%

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

   2. Taylor expanded in y around 0 53.7%

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

      1. associate-*r/ 53.7%

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

      2. neg-mul-1 53.7%

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

   4. Simplified 53.7%

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

      1. neg-mul-1 53.7%

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

      2. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in t around 0 53.7%

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

      1. mul-1-neg 53.7%

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

      2. associate-/r* 99.8%

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

   9. Simplified 99.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternative 2: 79.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+23} \lor \neg \left(z \cdot t \leq 10^{-32}\right):\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -5e+23) (not (<= (* z t) 1e-32)))
   (/ (/ (- x) t) z)
   (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e+23) || !((z * t) <= 1e-32)) {
		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+23)) .or. (.not. ((z * t) <= 1d-32))) 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+23) || !((z * t) <= 1e-32)) {
		tmp = (-x / t) / z;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -5e+23) or not ((z * t) <= 1e-32):
		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+23) || !(Float64(z * t) <= 1e-32))
		tmp = Float64(Float64(Float64(-x) / t) / 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+23) || ~(((z * t) <= 1e-32)))
		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+23], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e-32]], $MachinePrecision]], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.9999999999999999e23 or 1.00000000000000006e-32 < (*.f64 z t)

   1. Initial program 91.6%

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

   2. Taylor expanded in y around 0 71.9%

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

      1. associate-*r/ 71.9%

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

      2. neg-mul-1 71.9%

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

   4. Simplified 71.9%

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

      1. neg-mul-1 71.9%

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

      2. times-frac 77.5%

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

   6. Applied egg-rr 77.5%

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

   7. Taylor expanded in t around 0 71.9%

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

      1. mul-1-neg 71.9%

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

      2. associate-/r* 77.7%

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

   9. Simplified 77.7%

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

   if -4.9999999999999999e23 < (*.f64 z t) < 1.00000000000000006e-32

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 83.7%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+23} \lor \neg \left(z \cdot t \leq 10^{-32}\right):\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3: 79.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+23} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -5e+23) (not (<= (* z t) 2e-18)))
   (/ (/ x z) (- t))
   (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e+23) || !((z * t) <= 2e-18)) {
		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+23)) .or. (.not. ((z * t) <= 2d-18))) 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+23) || !((z * t) <= 2e-18)) {
		tmp = (x / z) / -t;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -5e+23) or not ((z * t) <= 2e-18):
		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+23) || !(Float64(z * t) <= 2e-18))
		tmp = Float64(Float64(x / z) / Float64(-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+23) || ~(((z * t) <= 2e-18)))
		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+23], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e-18]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.9999999999999999e23 or 2.0000000000000001e-18 < (*.f64 z t)

   1. Initial program 91.5%

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

      1. sub-neg 91.5%

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

      2. +-commutative 91.5%

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

      3. distribute-rgt-neg-in 91.5%

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

      4. fma-udef 91.5%

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

      5. frac-2neg 91.5%

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

      6. div-inv 91.4%

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

      7. fma-udef 91.4%

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

      8. distribute-rgt-neg-in 91.4%

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

      9. distribute-neg-in 91.4%

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

   3. Applied egg-rr 91.4%

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

   4. Taylor expanded in z around inf 72.2%

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

      1. *-commutative 72.2%

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

      2. associate-/r* 73.0%

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

   6. Simplified 73.0%

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

      1. associate-*r/ 78.1%

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

      2. frac-2neg 78.1%

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

      3. add-sqr-sqrt 41.6%

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

      4. sqrt-unprod 44.6%

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

      5. sqr-neg 44.6%

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

      6. sqrt-unprod 16.4%

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

      7. add-sqr-sqrt 35.8%

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

      8. distribute-lft-neg-out 35.8%

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

      9. un-div-inv 35.8%

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

      10. add-sqr-sqrt 19.5%

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

      11. sqrt-unprod 49.8%

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

      12. sqr-neg 49.8%

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

      13. sqrt-unprod 36.3%

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

      14. add-sqr-sqrt 78.1%

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

   8. Applied egg-rr 78.1%

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

   if -4.9999999999999999e23 < (*.f64 z t) < 2.0000000000000001e-18

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 83.3%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+23} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 63.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+197} \lor \neg \left(z \cdot t \leq 4 \cdot 10^{+179}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -4e+197) (not (<= (* z t) 4e+179)))
   (/ x (* z t))
   (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -4e+197) || !((z * t) <= 4e+179)) {
		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) <= (-4d+197)) .or. (.not. ((z * t) <= 4d+179))) 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) <= -4e+197) || !((z * t) <= 4e+179)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -4e+197) or not ((z * t) <= 4e+179):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -4e+197) || !(Float64(z * t) <= 4e+179))
		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) <= -4e+197) || ~(((z * t) <= 4e+179)))
		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], -4e+197], N[Not[LessEqual[N[(z * t), $MachinePrecision], 4e+179]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -3.9999999999999998e197 or 3.99999999999999992e179 < (*.f64 z t)

   1. Initial program 81.5%

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

      1. sub-neg 81.5%

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

      2. +-commutative 81.5%

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

      3. distribute-rgt-neg-in 81.5%

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

      4. fma-udef 81.5%

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

      5. frac-2neg 81.5%

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

      6. div-inv 81.4%

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

      7. fma-udef 81.4%

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

      8. distribute-rgt-neg-in 81.4%

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

      9. distribute-neg-in 81.4%

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

   3. Applied egg-rr 81.4%

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

   4. Taylor expanded in z around inf 76.8%

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

      1. *-commutative 76.8%

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

      2. associate-/r* 78.6%

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

   6. Simplified 78.6%

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

      1. associate-/l/ 76.8%

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

      2. div-inv 76.9%

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

      3. add-sqr-sqrt 38.2%

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

      4. *-commutative 38.2%

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

      5. sqrt-unprod 62.5%

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

      6. sqr-neg 62.5%

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

      7. sqrt-unprod 29.3%

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

      8. add-sqr-sqrt 57.9%

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

   8. Applied egg-rr 57.9%

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

   if -3.9999999999999998e197 < (*.f64 z t) < 3.99999999999999992e179

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 69.1%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+197} \lor \neg \left(z \cdot t \leq 4 \cdot 10^{+179}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) 2e+261) (/ x (- y (* z t))) (/ (/ (- x) t) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 2e+261) {
		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) <= 2d+261) 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) <= 2e+261) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-x / t) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= 2e+261:
		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) <= 2e+261)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(Float64(-x) / t) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= 2e+261)
		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], 2e+261], 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) < 1.9999999999999999e261

   1. Initial program 99.0%

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

   if 1.9999999999999999e261 < (*.f64 z t)

   1. Initial program 53.7%

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

   2. Taylor expanded in y around 0 53.7%

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

      1. associate-*r/ 53.7%

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

      2. neg-mul-1 53.7%

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

   4. Simplified 53.7%

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

      1. neg-mul-1 53.7%

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

      2. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in t around 0 53.7%

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

      1. mul-1-neg 53.7%

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

      2. associate-/r* 99.8%

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

   9. Simplified 99.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternative 6: 54.7% 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 96.0%

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

2. Taylor expanded in y around inf 57.9%

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

3. Final simplification 57.9%

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

Developer target: 96.6% 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 2023271 
(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))))