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

Percentage Accurate: 95.5% → 99.7%

Time: 8.3s

Alternatives: 7

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: 95.5% 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: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+246}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z} \cdot \frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -2e+246)
   (/ (/ x z) (- t))
   (if (<= (* z t) 2e+251) (/ x (fma z (- t) y)) (* (/ -1.0 z) (/ x t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e+246) {
		tmp = (x / z) / -t;
	} else if ((z * t) <= 2e+251) {
		tmp = x / fma(z, -t, y);
	} else {
		tmp = (-1.0 / z) * (x / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -2e+246)
		tmp = Float64(Float64(x / z) / Float64(-t));
	elseif (Float64(z * t) <= 2e+251)
		tmp = Float64(x / fma(z, Float64(-t), y));
	else
		tmp = Float64(Float64(-1.0 / z) * Float64(x / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+246], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+251], N[(x / N[(z * (-t) + y), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / z), $MachinePrecision] * N[(x / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.00000000000000014e246

   1. Initial program 75.4%

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

   2. Taylor expanded in y around 0 75.4%

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

      1. *-commutative 75.4%

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

      2. associate-*r/ 75.4%

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

      3. neg-mul-1 75.4%

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

      4. *-commutative 75.4%

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

   4. Simplified 75.4%

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

      1. neg-mul-1 75.4%

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

      2. *-commutative 75.4%

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

      3. times-frac 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. frac-2neg 99.9%

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

      2. associate-*r/ 99.9%

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

      3. add-sqr-sqrt 60.5%

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

      4. sqrt-unprod 74.2%

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

      5. sqr-neg 74.2%

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

      6. sqrt-unprod 26.4%

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

      7. add-sqr-sqrt 58.2%

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

      8. associate-*l/ 58.2%

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

      9. neg-mul-1 58.2%

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

      10. add-sqr-sqrt 31.8%

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

      11. sqrt-unprod 53.8%

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

      12. sqr-neg 53.8%

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

      13. sqrt-unprod 39.1%

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

      14. add-sqr-sqrt 99.9%

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

   8. Applied egg-rr 99.9%

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

   if -2.00000000000000014e246 < (*.f64 z t) < 2.0000000000000001e251

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-rgt-neg-out 99.8%

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

      3. +-commutative 99.8%

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

      4. fma-def 99.9%

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

   3. Simplified 99.9%

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

   if 2.0000000000000001e251 < (*.f64 z t)

   1. Initial program 68.4%

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

   2. Taylor expanded in y around 0 68.4%

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

      1. *-commutative 68.4%

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

      2. associate-*r/ 68.4%

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

      3. neg-mul-1 68.4%

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

      4. *-commutative 68.4%

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

   4. Simplified 68.4%

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

      1. neg-mul-1 68.4%

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

      2. *-commutative 68.4%

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

      3. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+246}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z} \cdot \frac{x}{t}\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+246}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z} \cdot \frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -2e+246)
   (/ (/ x z) (- t))
   (if (<= (* z t) 2e+251) (/ x (- y (* z t))) (* (/ -1.0 z) (/ x t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e+246) {
		tmp = (x / z) / -t;
	} else if ((z * t) <= 2e+251) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-1.0 / z) * (x / 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 ((z * t) <= (-2d+246)) then
        tmp = (x / z) / -t
    else if ((z * t) <= 2d+251) then
        tmp = x / (y - (z * t))
    else
        tmp = ((-1.0d0) / z) * (x / t)
    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+246) {
		tmp = (x / z) / -t;
	} else if ((z * t) <= 2e+251) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-1.0 / z) * (x / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -2e+246:
		tmp = (x / z) / -t
	elif (z * t) <= 2e+251:
		tmp = x / (y - (z * t))
	else:
		tmp = (-1.0 / z) * (x / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -2e+246)
		tmp = Float64(Float64(x / z) / Float64(-t));
	elseif (Float64(z * t) <= 2e+251)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(-1.0 / z) * Float64(x / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -2e+246)
		tmp = (x / z) / -t;
	elseif ((z * t) <= 2e+251)
		tmp = x / (y - (z * t));
	else
		tmp = (-1.0 / z) * (x / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+246], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+251], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / z), $MachinePrecision] * N[(x / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.00000000000000014e246

   1. Initial program 75.4%

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

   2. Taylor expanded in y around 0 75.4%

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

      1. *-commutative 75.4%

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

      2. associate-*r/ 75.4%

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

      3. neg-mul-1 75.4%

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

      4. *-commutative 75.4%

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

   4. Simplified 75.4%

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

      1. neg-mul-1 75.4%

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

      2. *-commutative 75.4%

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

      3. times-frac 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. frac-2neg 99.9%

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

      2. associate-*r/ 99.9%

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

      3. add-sqr-sqrt 60.5%

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

      4. sqrt-unprod 74.2%

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

      5. sqr-neg 74.2%

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

      6. sqrt-unprod 26.4%

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

      7. add-sqr-sqrt 58.2%

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

      8. associate-*l/ 58.2%

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

      9. neg-mul-1 58.2%

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

      10. add-sqr-sqrt 31.8%

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

      11. sqrt-unprod 53.8%

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

      12. sqr-neg 53.8%

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

      13. sqrt-unprod 39.1%

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

      14. add-sqr-sqrt 99.9%

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

   8. Applied egg-rr 99.9%

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

   if -2.00000000000000014e246 < (*.f64 z t) < 2.0000000000000001e251

   1. Initial program 99.8%

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

   if 2.0000000000000001e251 < (*.f64 z t)

   1. Initial program 68.4%

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

   2. Taylor expanded in y around 0 68.4%

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

      1. *-commutative 68.4%

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

      2. associate-*r/ 68.4%

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

      3. neg-mul-1 68.4%

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

      4. *-commutative 68.4%

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

   4. Simplified 68.4%

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

      1. neg-mul-1 68.4%

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

      2. *-commutative 68.4%

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

      3. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+246}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z} \cdot \frac{x}{t}\\ \end{array} \]

Alternative 3: 67.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-19} \lor \neg \left(t \leq 1.8 \cdot 10^{-6} \lor \neg \left(t \leq 1750000\right) \land t \leq 3.1 \cdot 10^{+97}\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -9e-19)
         (not (or (<= t 1.8e-6) (and (not (<= t 1750000.0)) (<= t 3.1e+97)))))
   (/ (- x) (* z t))
   (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -9e-19) || !((t <= 1.8e-6) || (!(t <= 1750000.0) && (t <= 3.1e+97)))) {
		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 ((t <= (-9d-19)) .or. (.not. (t <= 1.8d-6) .or. (.not. (t <= 1750000.0d0)) .and. (t <= 3.1d+97))) 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 ((t <= -9e-19) || !((t <= 1.8e-6) || (!(t <= 1750000.0) && (t <= 3.1e+97)))) {
		tmp = -x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -9e-19) or not ((t <= 1.8e-6) or (not (t <= 1750000.0) and (t <= 3.1e+97))):
		tmp = -x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -9e-19) || !((t <= 1.8e-6) || (!(t <= 1750000.0) && (t <= 3.1e+97))))
		tmp = Float64(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 ((t <= -9e-19) || ~(((t <= 1.8e-6) || (~((t <= 1750000.0)) && (t <= 3.1e+97)))))
		tmp = -x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -9e-19], N[Not[Or[LessEqual[t, 1.8e-6], And[N[Not[LessEqual[t, 1750000.0]], $MachinePrecision], LessEqual[t, 3.1e+97]]]], $MachinePrecision]], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.00000000000000026e-19 or 1.79999999999999992e-6 < t < 1.75e6 or 3.09999999999999981e97 < t

   1. Initial program 88.0%

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

   2. Taylor expanded in y around 0 72.6%

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

      1. *-commutative 72.6%

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

      2. associate-*r/ 72.6%

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

      3. neg-mul-1 72.6%

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

      4. *-commutative 72.6%

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

   4. Simplified 72.6%

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

   if -9.00000000000000026e-19 < t < 1.79999999999999992e-6 or 1.75e6 < t < 3.09999999999999981e97

   1. Initial program 98.0%

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

   2. Taylor expanded in y around inf 74.1%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-19} \lor \neg \left(t \leq 1.8 \cdot 10^{-6} \lor \neg \left(t \leq 1750000\right) \land t \leq 3.1 \cdot 10^{+97}\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 70.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-18} \lor \neg \left(t \leq 6.4 \cdot 10^{-6} \lor \neg \left(t \leq 21500000\right) \land t \leq 8.5 \cdot 10^{+83}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.4e-18)
         (not (or (<= t 6.4e-6) (and (not (<= t 21500000.0)) (<= t 8.5e+83)))))
   (/ (/ x z) (- t))
   (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.4e-18) || !((t <= 6.4e-6) || (!(t <= 21500000.0) && (t <= 8.5e+83)))) {
		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 ((t <= (-1.4d-18)) .or. (.not. (t <= 6.4d-6) .or. (.not. (t <= 21500000.0d0)) .and. (t <= 8.5d+83))) 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 ((t <= -1.4e-18) || !((t <= 6.4e-6) || (!(t <= 21500000.0) && (t <= 8.5e+83)))) {
		tmp = (x / z) / -t;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.4e-18) or not ((t <= 6.4e-6) or (not (t <= 21500000.0) and (t <= 8.5e+83))):
		tmp = (x / z) / -t
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.4e-18) || !((t <= 6.4e-6) || (!(t <= 21500000.0) && (t <= 8.5e+83))))
		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 ((t <= -1.4e-18) || ~(((t <= 6.4e-6) || (~((t <= 21500000.0)) && (t <= 8.5e+83)))))
		tmp = (x / z) / -t;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.4e-18], N[Not[Or[LessEqual[t, 6.4e-6], And[N[Not[LessEqual[t, 21500000.0]], $MachinePrecision], LessEqual[t, 8.5e+83]]]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.40000000000000006e-18 or 6.3999999999999997e-6 < t < 2.15e7 or 8.4999999999999995e83 < t

   1. Initial program 86.5%

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

   2. Taylor expanded in y around 0 71.6%

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

      1. *-commutative 71.6%

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

      2. associate-*r/ 71.6%

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

      3. neg-mul-1 71.6%

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

      4. *-commutative 71.6%

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

   4. Simplified 71.6%

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

      1. neg-mul-1 71.6%

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

      2. *-commutative 71.6%

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

      3. times-frac 84.8%

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

   6. Applied egg-rr 84.8%

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

      1. frac-2neg 84.8%

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

      2. associate-*r/ 81.0%

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

      3. add-sqr-sqrt 42.8%

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

      4. sqrt-unprod 52.7%

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

      5. sqr-neg 52.7%

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

      6. sqrt-unprod 14.9%

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

      7. add-sqr-sqrt 33.2%

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

      8. associate-*l/ 33.2%

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

      9. neg-mul-1 33.2%

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

      10. add-sqr-sqrt 18.3%

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

      11. sqrt-unprod 46.4%

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

      12. sqr-neg 46.4%

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

      13. sqrt-unprod 37.9%

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

      14. add-sqr-sqrt 81.0%

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

   8. Applied egg-rr 81.0%

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

   if -1.40000000000000006e-18 < t < 6.3999999999999997e-6 or 2.15e7 < t < 8.4999999999999995e83

   1. Initial program 99.2%

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

   2. Taylor expanded in y around inf 75.6%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-18} \lor \neg \left(t \leq 6.4 \cdot 10^{-6} \lor \neg \left(t \leq 21500000\right) \land t \leq 8.5 \cdot 10^{+83}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 71.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-\frac{x}{t}}{z}\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t \leq 12000000:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{+84}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- (/ x t)) z)))
   (if (<= t -2.5e-20)
     t_1
     (if (<= t 2.15e-7)
       (/ x y)
       (if (<= t 12000000.0)
         (/ (/ x z) (- t))
         (if (<= t 1.95e+84) (/ x y) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -(x / t) / z;
	double tmp;
	if (t <= -2.5e-20) {
		tmp = t_1;
	} else if (t <= 2.15e-7) {
		tmp = x / y;
	} else if (t <= 12000000.0) {
		tmp = (x / z) / -t;
	} else if (t <= 1.95e+84) {
		tmp = x / y;
	} 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 = -(x / t) / z
    if (t <= (-2.5d-20)) then
        tmp = t_1
    else if (t <= 2.15d-7) then
        tmp = x / y
    else if (t <= 12000000.0d0) then
        tmp = (x / z) / -t
    else if (t <= 1.95d+84) then
        tmp = x / y
    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 = -(x / t) / z;
	double tmp;
	if (t <= -2.5e-20) {
		tmp = t_1;
	} else if (t <= 2.15e-7) {
		tmp = x / y;
	} else if (t <= 12000000.0) {
		tmp = (x / z) / -t;
	} else if (t <= 1.95e+84) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -(x / t) / z
	tmp = 0
	if t <= -2.5e-20:
		tmp = t_1
	elif t <= 2.15e-7:
		tmp = x / y
	elif t <= 12000000.0:
		tmp = (x / z) / -t
	elif t <= 1.95e+84:
		tmp = x / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-Float64(x / t)) / z)
	tmp = 0.0
	if (t <= -2.5e-20)
		tmp = t_1;
	elseif (t <= 2.15e-7)
		tmp = Float64(x / y);
	elseif (t <= 12000000.0)
		tmp = Float64(Float64(x / z) / Float64(-t));
	elseif (t <= 1.95e+84)
		tmp = Float64(x / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -(x / t) / z;
	tmp = 0.0;
	if (t <= -2.5e-20)
		tmp = t_1;
	elseif (t <= 2.15e-7)
		tmp = x / y;
	elseif (t <= 12000000.0)
		tmp = (x / z) / -t;
	elseif (t <= 1.95e+84)
		tmp = x / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-N[(x / t), $MachinePrecision]) / z), $MachinePrecision]}, If[LessEqual[t, -2.5e-20], t$95$1, If[LessEqual[t, 2.15e-7], N[(x / y), $MachinePrecision], If[LessEqual[t, 12000000.0], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision], If[LessEqual[t, 1.95e+84], N[(x / y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.4999999999999999e-20 or 1.95000000000000008e84 < t

   1. Initial program 85.7%

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

   2. Taylor expanded in y around 0 70.8%

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

      1. *-commutative 70.8%

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

      2. associate-*r/ 70.8%

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

      3. neg-mul-1 70.8%

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

      4. *-commutative 70.8%

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

   4. Simplified 70.8%

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

   5. Taylor expanded in x around 0 70.8%

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

      1. associate-/r* 84.9%

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

      2. mul-1-neg 84.9%

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

      3. distribute-neg-frac 84.9%

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

      4. distribute-neg-frac 84.9%

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

   7. Simplified 84.9%

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

   if -2.4999999999999999e-20 < t < 2.1500000000000001e-7 or 1.2e7 < t < 1.95000000000000008e84

   1. Initial program 99.2%

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

   2. Taylor expanded in y around inf 75.6%

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

   if 2.1500000000000001e-7 < t < 1.2e7

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 84.2%

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

      1. *-commutative 84.2%

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

      2. associate-*r/ 84.2%

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

      3. neg-mul-1 84.2%

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

      4. *-commutative 84.2%

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

   4. Simplified 84.2%

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

      1. neg-mul-1 84.2%

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

      2. *-commutative 84.2%

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

      3. times-frac 83.9%

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

   6. Applied egg-rr 83.9%

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

      1. frac-2neg 83.9%

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

      2. associate-*r/ 84.2%

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

      3. add-sqr-sqrt 67.5%

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

      4. sqrt-unprod 51.6%

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

      5. sqr-neg 51.6%

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

      6. sqrt-unprod 0.2%

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

      7. add-sqr-sqrt 2.5%

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

      8. associate-*l/ 2.5%

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

      9. neg-mul-1 2.5%

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

      10. add-sqr-sqrt 2.3%

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

      11. sqrt-unprod 19.2%

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

      12. sqr-neg 19.2%

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

      13. sqrt-unprod 16.7%

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

      14. add-sqr-sqrt 84.2%

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

   8. Applied egg-rr 84.2%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;t \leq 12000000:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{+84}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \end{array} \]

Alternative 6: 53.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-18} \lor \neg \left(t \leq 1.5 \cdot 10^{+110}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.4e-18) (not (<= t 1.5e+110))) (/ x (* z t)) (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.4e-18) || !(t <= 1.5e+110)) {
		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 ((t <= (-1.4d-18)) .or. (.not. (t <= 1.5d+110))) 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 ((t <= -1.4e-18) || !(t <= 1.5e+110)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.4e-18) or not (t <= 1.5e+110):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.4e-18) || !(t <= 1.5e+110))
		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 ((t <= -1.4e-18) || ~((t <= 1.5e+110)))
		tmp = x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.4e-18], N[Not[LessEqual[t, 1.5e+110]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.40000000000000006e-18 or 1.50000000000000004e110 < t

   1. Initial program 87.7%

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

   2. Taylor expanded in y around 0 72.7%

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

      1. *-commutative 72.7%

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

      2. associate-*r/ 72.7%

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

      3. neg-mul-1 72.7%

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

      4. *-commutative 72.7%

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

   4. Simplified 72.7%

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

      1. expm1-log1p-u 66.7%

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

      2. expm1-udef 43.7%

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

      3. add-sqr-sqrt 24.3%

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

      4. sqrt-unprod 39.0%

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

      5. sqr-neg 39.0%

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

      6. sqrt-unprod 17.3%

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

      7. add-sqr-sqrt 39.1%

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

      8. associate-/r* 39.1%

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

   6. Applied egg-rr 39.1%

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

      1. expm1-def 37.5%

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

      2. expm1-log1p 37.6%

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

      3. associate-/r* 36.7%

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

   8. Simplified 36.7%

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

   if -1.40000000000000006e-18 < t < 1.50000000000000004e110

   1. Initial program 97.5%

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

   2. Taylor expanded in y around inf 71.6%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-18} \lor \neg \left(t \leq 1.5 \cdot 10^{+110}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7: 53.4% 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 94.0%

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

2. Taylor expanded in y around inf 55.9%

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

3. Final simplification 55.9%

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

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 2023301 
(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))))