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

Percentage Accurate: 95.9% → 97.8%

Time: 10.0s

Alternatives: 9

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: 95.9% 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: 97.8% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY)) (* (/ -1.0 t) (/ x z)) (/ x (fma (- z) t y))))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 75.9%

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

   3. Taylor expanded in y around 0 75.9%

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

      1. mul-1-neg 75.9%

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

      2. associate-/r* 99.9%

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

      3. distribute-neg-frac2 99.9%

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

   5. Simplified 99.9%

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

      1. div-inv 99.9%

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

      2. associate-/l* 78.8%

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

      3. add-sqr-sqrt 47.3%

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

      4. sqrt-unprod 75.8%

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

      5. sqr-neg 75.8%

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

      6. sqrt-unprod 28.5%

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

      7. add-sqr-sqrt 75.8%

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

   7. Applied egg-rr 75.8%

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

      1. associate-/l/ 75.9%

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

      2. associate-*r/ 75.9%

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

      3. *-commutative 75.9%

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

      4. *-commutative 75.9%

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

      5. *-lft-identity 75.9%

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

   9. Simplified 75.9%

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

       1. add-sqr-sqrt 75.9%

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

       2. sqrt-unprod 75.9%

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

       3. sqr-neg 75.9%

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

       4. distribute-frac-neg 75.9%

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

       5. distribute-frac-neg 75.9%

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

       6. sqrt-unprod 75.9%

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

       7. add-sqr-sqrt 75.9%

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

       8. neg-mul-1 75.9%

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

       9. times-frac 99.9%

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

   11. Applied egg-rr 99.9%

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

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

   1. Initial program 98.3%

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

      1. sub-neg 98.3%

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

      2. +-commutative 98.3%

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

      3. distribute-lft-neg-in 98.3%

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

      4. fma-define 98.4%

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

   4. Applied egg-rr 98.4%

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

4. Final simplification 98.5%

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

Alternative 2: 72.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-73} \lor \neg \left(y \leq 1.12 \cdot 10^{-29}\right) \land y \leq 1.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -9.2e+59)
   (/ 1.0 (/ y x))
   (if (or (<= y 1.35e-73) (and (not (<= y 1.12e-29)) (<= y 1.2e+26)))
     (/ x (* t (- z)))
     (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9.2e+59) {
		tmp = 1.0 / (y / x);
	} else if ((y <= 1.35e-73) || (!(y <= 1.12e-29) && (y <= 1.2e+26))) {
		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 (y <= (-9.2d+59)) then
        tmp = 1.0d0 / (y / x)
    else if ((y <= 1.35d-73) .or. (.not. (y <= 1.12d-29)) .and. (y <= 1.2d+26)) 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 (y <= -9.2e+59) {
		tmp = 1.0 / (y / x);
	} else if ((y <= 1.35e-73) || (!(y <= 1.12e-29) && (y <= 1.2e+26))) {
		tmp = x / (t * -z);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -9.2e+59:
		tmp = 1.0 / (y / x)
	elif (y <= 1.35e-73) or (not (y <= 1.12e-29) and (y <= 1.2e+26)):
		tmp = x / (t * -z)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -9.2e+59)
		tmp = Float64(1.0 / Float64(y / x));
	elseif ((y <= 1.35e-73) || (!(y <= 1.12e-29) && (y <= 1.2e+26)))
		tmp = Float64(x / Float64(t * Float64(-z)));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -9.2e+59)
		tmp = 1.0 / (y / x);
	elseif ((y <= 1.35e-73) || (~((y <= 1.12e-29)) && (y <= 1.2e+26)))
		tmp = x / (t * -z);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -9.2e+59], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 1.35e-73], And[N[Not[LessEqual[y, 1.12e-29]], $MachinePrecision], LessEqual[y, 1.2e+26]]], N[(x / N[(t * (-z)), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.20000000000000032e59

   1. Initial program 93.2%

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

      1. clear-num 91.9%

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

      2. inv-pow 91.9%

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

   4. Applied egg-rr 91.9%

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

   5. Taylor expanded in y around 0 89.2%

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

      1. unpow-1 89.2%

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

      2. +-commutative 89.2%

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

      3. mul-1-neg 89.2%

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

      4. unsub-neg 89.2%

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

      5. div-sub 91.9%

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

      6. *-commutative 91.9%

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

   7. Applied egg-rr 91.9%

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

   8. Taylor expanded in y around inf 83.5%

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

   if -9.20000000000000032e59 < y < 1.34999999999999997e-73 or 1.11999999999999995e-29 < y < 1.20000000000000002e26

   1. Initial program 97.8%

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

   3. Taylor expanded in y around 0 76.6%

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

      1. associate-*r/ 76.6%

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

      2. neg-mul-1 76.6%

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

   5. Simplified 76.6%

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

   if 1.34999999999999997e-73 < y < 1.11999999999999995e-29 or 1.20000000000000002e26 < y

   1. Initial program 95.1%

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

   3. Taylor expanded in y around inf 83.5%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-73} \lor \neg \left(y \leq 1.12 \cdot 10^{-29}\right) \land y \leq 1.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{t \cdot \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-137} \lor \neg \left(t \leq 1.08 \cdot 10^{+71}\right):\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -5.5e-137) (not (<= t 1.08e+71))) (/ (/ x t) (- z)) (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -5.5e-137) || !(t <= 1.08e+71)) {
		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 ((t <= (-5.5d-137)) .or. (.not. (t <= 1.08d+71))) 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 ((t <= -5.5e-137) || !(t <= 1.08e+71)) {
		tmp = (x / t) / -z;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -5.5e-137) or not (t <= 1.08e+71):
		tmp = (x / t) / -z
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -5.5e-137) || !(t <= 1.08e+71))
		tmp = Float64(Float64(x / t) / Float64(-z));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -5.5e-137) || ~((t <= 1.08e+71)))
		tmp = (x / t) / -z;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -5.5e-137], N[Not[LessEqual[t, 1.08e+71]], $MachinePrecision]], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.5000000000000003e-137 or 1.08e71 < t

   1. Initial program 93.4%

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

   3. Taylor expanded in y around 0 67.8%

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

      1. mul-1-neg 67.8%

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

      2. associate-/r* 75.3%

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

      3. distribute-neg-frac2 75.3%

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

   5. Simplified 75.3%

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

   if -5.5000000000000003e-137 < t < 1.08e71

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 72.2%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-137} \lor \neg \left(t \leq 1.08 \cdot 10^{+71}\right):\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-130}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-1}{z}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+69}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -9e-130)
   (* (/ x t) (/ -1.0 z))
   (if (<= t 3.4e+69) (/ x y) (/ (/ x t) (- z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9e-130) {
		tmp = (x / t) * (-1.0 / z);
	} else if (t <= 3.4e+69) {
		tmp = x / y;
	} 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 (t <= (-9d-130)) then
        tmp = (x / t) * ((-1.0d0) / z)
    else if (t <= 3.4d+69) then
        tmp = x / y
    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 (t <= -9e-130) {
		tmp = (x / t) * (-1.0 / z);
	} else if (t <= 3.4e+69) {
		tmp = x / y;
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -9e-130:
		tmp = (x / t) * (-1.0 / z)
	elif t <= 3.4e+69:
		tmp = x / y
	else:
		tmp = (x / t) / -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -9e-130)
		tmp = Float64(Float64(x / t) * Float64(-1.0 / z));
	elseif (t <= 3.4e+69)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / t) / Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -9e-130)
		tmp = (x / t) * (-1.0 / z);
	elseif (t <= 3.4e+69)
		tmp = x / y;
	else
		tmp = (x / t) / -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -9e-130], N[(N[(x / t), $MachinePrecision] * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.4e+69], N[(x / y), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9e-130

   1. Initial program 91.1%

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

   3. Taylor expanded in y around 0 61.8%

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

      1. mul-1-neg 61.8%

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

      2. associate-/r* 67.8%

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

      3. distribute-neg-frac2 67.8%

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

   5. Simplified 67.8%

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

      1. div-inv 67.8%

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

      2. frac-2neg 67.8%

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

      3. metadata-eval 67.8%

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

      4. remove-double-neg 67.8%

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

   7. Applied egg-rr 67.8%

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

   if -9e-130 < t < 3.39999999999999986e69

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 72.2%

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

   if 3.39999999999999986e69 < t

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0 76.6%

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

      1. mul-1-neg 76.6%

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

      2. associate-/r* 86.4%

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

      3. distribute-neg-frac2 86.4%

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

   5. Simplified 86.4%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-130}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-1}{z}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+69}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+171} \lor \neg \left(z \leq 1.9 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -7.6e+171) (not (<= z 1.9e+55))) (/ x (* z t)) (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7.6e+171) || !(z <= 1.9e+55)) {
		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 <= (-7.6d+171)) .or. (.not. (z <= 1.9d+55))) 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 <= -7.6e+171) || !(z <= 1.9e+55)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -7.6e+171) or not (z <= 1.9e+55):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -7.6e+171) || !(z <= 1.9e+55))
		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 <= -7.6e+171) || ~((z <= 1.9e+55)))
		tmp = x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -7.6e+171], N[Not[LessEqual[z, 1.9e+55]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.6000000000000004e171 or 1.9e55 < z

   1. Initial program 92.2%

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

   3. Taylor expanded in y around 0 68.5%

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

      1. mul-1-neg 68.5%

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

      2. associate-/r* 73.1%

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

      3. distribute-neg-frac2 73.1%

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

   5. Simplified 73.1%

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

      1. div-inv 73.1%

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

      2. associate-/l* 69.2%

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

      3. add-sqr-sqrt 25.0%

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

      4. sqrt-unprod 45.2%

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

      5. sqr-neg 45.2%

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

      6. sqrt-unprod 23.3%

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

      7. add-sqr-sqrt 43.1%

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

   7. Applied egg-rr 43.1%

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

      1. associate-/l/ 43.1%

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

      2. associate-*r/ 43.1%

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

      3. *-commutative 43.1%

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

      4. *-commutative 43.1%

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

      5. *-lft-identity 43.1%

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

   9. Simplified 43.1%

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

   if -7.6000000000000004e171 < z < 1.9e55

   1. Initial program 98.7%

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

   3. Taylor expanded in y around inf 62.3%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+171} \lor \neg \left(z \leq 1.9 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{x}{z}}{t}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -7e+169) (/ (/ x z) t) (if (<= z 1.15e+55) (/ x y) (/ x (* z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7e+169) {
		tmp = (x / z) / t;
	} else if (z <= 1.15e+55) {
		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 (z <= (-7d+169)) then
        tmp = (x / z) / t
    else if (z <= 1.15d+55) 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 (z <= -7e+169) {
		tmp = (x / z) / t;
	} else if (z <= 1.15e+55) {
		tmp = x / y;
	} else {
		tmp = x / (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -7e+169:
		tmp = (x / z) / t
	elif z <= 1.15e+55:
		tmp = x / y
	else:
		tmp = x / (z * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -7e+169)
		tmp = Float64(Float64(x / z) / t);
	elseif (z <= 1.15e+55)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -7e+169)
		tmp = (x / z) / t;
	elseif (z <= 1.15e+55)
		tmp = x / y;
	else
		tmp = x / (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -7e+169], N[(N[(x / z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 1.15e+55], N[(x / y), $MachinePrecision], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.00000000000000038e169

   1. Initial program 93.2%

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

   3. Taylor expanded in y around 0 75.0%

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

      1. mul-1-neg 75.0%

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

      2. associate-/r* 78.6%

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

      3. distribute-neg-frac2 78.6%

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

   5. Simplified 78.6%

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

      1. div-inv 78.6%

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

      2. associate-/l* 75.0%

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

      3. add-sqr-sqrt 74.9%

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

      4. sqrt-unprod 62.2%

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

      5. sqr-neg 62.2%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 59.3%

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

   7. Applied egg-rr 59.3%

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

      1. associate-/l/ 59.3%

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

      2. associate-*r/ 59.3%

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

      3. *-commutative 59.3%

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

      4. *-commutative 59.3%

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

      5. *-lft-identity 59.3%

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

   9. Simplified 59.3%

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

       1. associate-/r* 59.1%

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

       2. div-inv 59.1%

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

       3. div-inv 59.1%

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

       4. associate-*r* 59.3%

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

       5. associate-*l/ 59.3%

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

       6. *-un-lft-identity 59.3%

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

   11. Applied egg-rr 59.3%

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

       1. associate-*r/ 62.2%

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

       2. *-commutative 62.2%

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

       3. associate-*l/ 62.2%

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

       4. associate-*r/ 62.2%

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

       5. *-lft-identity 62.2%

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

   13. Simplified 62.2%

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

   if -7.00000000000000038e169 < z < 1.14999999999999994e55

   1. Initial program 98.7%

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

   3. Taylor expanded in y around inf 62.3%

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

   if 1.14999999999999994e55 < z

   1. Initial program 91.7%

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

   3. Taylor expanded in y around 0 65.2%

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

      1. mul-1-neg 65.2%

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

      2. associate-/r* 70.3%

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

      3. distribute-neg-frac2 70.3%

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

   5. Simplified 70.3%

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

      1. div-inv 70.3%

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

      2. associate-/l* 66.3%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 36.7%

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

      5. sqr-neg 36.7%

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

      6. sqrt-unprod 35.0%

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

      7. add-sqr-sqrt 35.0%

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

   7. Applied egg-rr 35.0%

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

      1. associate-/l/ 35.0%

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

      2. associate-*r/ 35.0%

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

      3. *-commutative 35.0%

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

      4. *-commutative 35.0%

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

      5. *-lft-identity 35.0%

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

   9. Simplified 35.0%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{x}{z}}{t}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY)) (* (/ -1.0 t) (/ x z)) (/ x (- y (* z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = (-1.0 / t) * (x / z);
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = (-1.0 / t) * (x / z);
	} else {
		tmp = x / (y - (z * t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = (-1.0 / t) * (x / z)
	else:
		tmp = x / (y - (z * t))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 75.9%

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

   3. Taylor expanded in y around 0 75.9%

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

      1. mul-1-neg 75.9%

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

      2. associate-/r* 99.9%

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

      3. distribute-neg-frac2 99.9%

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

   5. Simplified 99.9%

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

      1. div-inv 99.9%

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

      2. associate-/l* 78.8%

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

      3. add-sqr-sqrt 47.3%

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

      4. sqrt-unprod 75.8%

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

      5. sqr-neg 75.8%

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

      6. sqrt-unprod 28.5%

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

      7. add-sqr-sqrt 75.8%

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

   7. Applied egg-rr 75.8%

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

      1. associate-/l/ 75.9%

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

      2. associate-*r/ 75.9%

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

      3. *-commutative 75.9%

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

      4. *-commutative 75.9%

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

      5. *-lft-identity 75.9%

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

   9. Simplified 75.9%

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

       1. add-sqr-sqrt 75.9%

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

       2. sqrt-unprod 75.9%

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

       3. sqr-neg 75.9%

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

       4. distribute-frac-neg 75.9%

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

       5. distribute-frac-neg 75.9%

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

       6. sqrt-unprod 75.9%

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

       7. add-sqr-sqrt 75.9%

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

       8. neg-mul-1 75.9%

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

       9. times-frac 99.9%

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

   11. Applied egg-rr 99.9%

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

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

   1. Initial program 98.3%

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

4. Final simplification 98.5%

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

Alternative 8: 54.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 / (y / x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.3%

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

   1. clear-num 95.8%

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

   2. inv-pow 95.8%

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

4. Applied egg-rr 95.8%

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

5. Taylor expanded in y around 0 92.3%

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

   1. unpow-1 92.3%

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

   2. +-commutative 92.3%

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

   3. mul-1-neg 92.3%

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

   4. unsub-neg 92.3%

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

   5. div-sub 95.8%

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

   6. *-commutative 95.8%

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

7. Applied egg-rr 95.8%

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

8. Taylor expanded in y around inf 51.9%

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

9. Final simplification 51.9%

   \[\leadsto \frac{1}{\frac{y}{x}} \]
10. Add Preprocessing

Alternative 9: 54.5% 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 96.3%

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

3. Taylor expanded in y around inf 51.8%

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

4. Final simplification 51.8%

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

Developer target: 96.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{if}\;x < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x < 2.1378306434876444 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (- (/ y x) (* (/ z x) t)))))
   (if (< x -1.618195973607049e+50)
     t_1
     (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / ((y / x) - ((z / x) * t))
    if (x < (-1.618195973607049d+50)) then
        tmp = t_1
    else if (x < 2.1378306434876444d+131) then
        tmp = x / (y - (z * t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 / ((y / x) - ((z / x) * t))
	tmp = 0
	if x < -1.618195973607049e+50:
		tmp = t_1
	elif x < 2.1378306434876444e+131:
		tmp = x / (y - (z * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(Float64(y / x) - Float64(Float64(z / x) * t)))
	tmp = 0.0
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 / ((y / x) - ((z / x) * t));
	tmp = 0.0;
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = x / (y - (z * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 / N[(N[(y / x), $MachinePrecision] - N[(N[(z / x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[x, -1.618195973607049e+50], t$95$1, If[Less[x, 2.1378306434876444e+131], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024044 
(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))))