Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, A

Percentage Accurate: 88.7% → 97.0%

Time: 10.9s

Alternatives: 11

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \end{array} \]

FPCore

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

C

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

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) - x) / ((t * z) - x))) / (x + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \end{array} \]

FPCore

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

C

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

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) - x) / ((t * z) - x))) / (x + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{+156} \lor \neg \left(z \leq 1.05 \cdot 10^{+124}\right):\\ \;\;\;\;\frac{x + \frac{y}{t - \frac{x}{z}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -4.7e+156) (not (<= z 1.05e+124)))
   (/ (+ x (/ y (- t (/ x z)))) (+ x 1.0))
   (/ (+ x (/ (- (* z y) x) (- (* z t) x))) (+ x 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.7e+156) || !(z <= 1.05e+124)) {
		tmp = (x + (y / (t - (x / z)))) / (x + 1.0);
	} else {
		tmp = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	}
	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 <= (-4.7d+156)) .or. (.not. (z <= 1.05d+124))) then
        tmp = (x + (y / (t - (x / z)))) / (x + 1.0d0)
    else
        tmp = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.7e+156) || !(z <= 1.05e+124)) {
		tmp = (x + (y / (t - (x / z)))) / (x + 1.0);
	} else {
		tmp = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -4.7e+156) or not (z <= 1.05e+124):
		tmp = (x + (y / (t - (x / z)))) / (x + 1.0)
	else:
		tmp = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -4.7e+156) || !(z <= 1.05e+124))
		tmp = Float64(Float64(x + Float64(y / Float64(t - Float64(x / z)))) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x + Float64(Float64(Float64(z * y) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -4.7e+156) || ~((z <= 1.05e+124)))
		tmp = (x + (y / (t - (x / z)))) / (x + 1.0);
	else
		tmp = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -4.7e+156], N[Not[LessEqual[z, 1.05e+124]], $MachinePrecision]], N[(N[(x + N[(y / N[(t - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.7e156 or 1.05000000000000006e124 < z

   1. Initial program 76.3%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 76.3%

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

   3. Simplified 76.3%

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

   4. Taylor expanded in y around inf 76.3%

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

      1. associate-/l* 91.7%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in t around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   9. Simplified 100.0%

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

   if -4.7e156 < z < 1.05000000000000006e124

   1. Initial program 99.4%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{+156} \lor \neg \left(z \leq 1.05 \cdot 10^{+124}\right):\\ \;\;\;\;\frac{x + \frac{y}{t - \frac{x}{z}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1}\\ \end{array} \]

Alternative 2: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+122} \lor \neg \left(z \leq -11 \lor \neg \left(z \leq -1.4 \cdot 10^{-39}\right) \land z \leq 1.8 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{z \cdot y}{x + 1}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.7e+122)
         (not (or (<= z -11.0) (and (not (<= z -1.4e-39)) (<= z 1.8e-30)))))
   (/ (+ x (/ y t)) (+ x 1.0))
   (- 1.0 (/ (/ (* z y) (+ x 1.0)) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.7e+122) || !((z <= -11.0) || (!(z <= -1.4e-39) && (z <= 1.8e-30)))) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0 - (((z * y) / (x + 1.0)) / x);
	}
	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 <= (-2.7d+122)) .or. (.not. (z <= (-11.0d0)) .or. (.not. (z <= (-1.4d-39))) .and. (z <= 1.8d-30))) then
        tmp = (x + (y / t)) / (x + 1.0d0)
    else
        tmp = 1.0d0 - (((z * y) / (x + 1.0d0)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.7e+122) || !((z <= -11.0) || (!(z <= -1.4e-39) && (z <= 1.8e-30)))) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0 - (((z * y) / (x + 1.0)) / x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.7e+122) or not ((z <= -11.0) or (not (z <= -1.4e-39) and (z <= 1.8e-30))):
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = 1.0 - (((z * y) / (x + 1.0)) / x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.7e+122) || !((z <= -11.0) || (!(z <= -1.4e-39) && (z <= 1.8e-30))))
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(z * y) / Float64(x + 1.0)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2.7e+122) || ~(((z <= -11.0) || (~((z <= -1.4e-39)) && (z <= 1.8e-30)))))
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = 1.0 - (((z * y) / (x + 1.0)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.7e+122], N[Not[Or[LessEqual[z, -11.0], And[N[Not[LessEqual[z, -1.4e-39]], $MachinePrecision], LessEqual[z, 1.8e-30]]]], $MachinePrecision]], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(z * y), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.6999999999999998e122 or -11 < z < -1.4000000000000001e-39 or 1.8000000000000002e-30 < z

   1. Initial program 85.2%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 85.2%

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

   3. Simplified 85.2%

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

   4. Taylor expanded in z around inf 89.6%

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

   if -2.6999999999999998e122 < z < -11 or -1.4000000000000001e-39 < z < 1.8000000000000002e-30

   1. Initial program 99.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 82.5%

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

      1. mul-1-neg 82.5%

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

      2. unsub-neg 82.5%

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

      3. *-commutative 82.5%

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

      4. +-commutative 82.5%

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

   6. Simplified 82.5%

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

   7. Taylor expanded in z around 0 82.5%

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

      1. mul-1-neg 82.5%

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

      2. unsub-neg 82.5%

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

      3. *-commutative 82.5%

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

      4. associate-/r* 82.5%

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

      5. *-commutative 82.5%

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

      6. +-commutative 82.5%

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

   9. Simplified 82.5%

      \[\leadsto \color{blue}{1 - \frac{\frac{z \cdot y}{x + 1}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+122} \lor \neg \left(z \leq -11 \lor \neg \left(z \leq -1.4 \cdot 10^{-39}\right) \land z \leq 1.8 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{z \cdot y}{x + 1}}{x}\\ \end{array} \]

Alternative 3: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{t \cdot \left(x + 1\right)}\\ t_2 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-9}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-119}:\\ \;\;\;\;1 - \frac{z \cdot y}{x}\\ \mathbf{elif}\;x \leq -1.32 \cdot 10^{-173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-241}:\\ \;\;\;\;x + x \cdot \frac{-1}{z \cdot t}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-107}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (* t (+ x 1.0)))) (t_2 (/ x (+ x 1.0))))
   (if (<= x -1.7e-9)
     t_2
     (if (<= x -2.7e-119)
       (- 1.0 (/ (* z y) x))
       (if (<= x -1.32e-173)
         t_1
         (if (<= x -1.2e-241)
           (+ x (* x (/ -1.0 (* z t))))
           (if (<= x 4.2e-107) t_1 t_2)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / (t * (x + 1.0));
	double t_2 = x / (x + 1.0);
	double tmp;
	if (x <= -1.7e-9) {
		tmp = t_2;
	} else if (x <= -2.7e-119) {
		tmp = 1.0 - ((z * y) / x);
	} else if (x <= -1.32e-173) {
		tmp = t_1;
	} else if (x <= -1.2e-241) {
		tmp = x + (x * (-1.0 / (z * t)));
	} else if (x <= 4.2e-107) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = y / (t * (x + 1.0d0))
    t_2 = x / (x + 1.0d0)
    if (x <= (-1.7d-9)) then
        tmp = t_2
    else if (x <= (-2.7d-119)) then
        tmp = 1.0d0 - ((z * y) / x)
    else if (x <= (-1.32d-173)) then
        tmp = t_1
    else if (x <= (-1.2d-241)) then
        tmp = x + (x * ((-1.0d0) / (z * t)))
    else if (x <= 4.2d-107) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y / (t * (x + 1.0));
	double t_2 = x / (x + 1.0);
	double tmp;
	if (x <= -1.7e-9) {
		tmp = t_2;
	} else if (x <= -2.7e-119) {
		tmp = 1.0 - ((z * y) / x);
	} else if (x <= -1.32e-173) {
		tmp = t_1;
	} else if (x <= -1.2e-241) {
		tmp = x + (x * (-1.0 / (z * t)));
	} else if (x <= 4.2e-107) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y / (t * (x + 1.0))
	t_2 = x / (x + 1.0)
	tmp = 0
	if x <= -1.7e-9:
		tmp = t_2
	elif x <= -2.7e-119:
		tmp = 1.0 - ((z * y) / x)
	elif x <= -1.32e-173:
		tmp = t_1
	elif x <= -1.2e-241:
		tmp = x + (x * (-1.0 / (z * t)))
	elif x <= 4.2e-107:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(t * Float64(x + 1.0)))
	t_2 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.7e-9)
		tmp = t_2;
	elseif (x <= -2.7e-119)
		tmp = Float64(1.0 - Float64(Float64(z * y) / x));
	elseif (x <= -1.32e-173)
		tmp = t_1;
	elseif (x <= -1.2e-241)
		tmp = Float64(x + Float64(x * Float64(-1.0 / Float64(z * t))));
	elseif (x <= 4.2e-107)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y / (t * (x + 1.0));
	t_2 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -1.7e-9)
		tmp = t_2;
	elseif (x <= -2.7e-119)
		tmp = 1.0 - ((z * y) / x);
	elseif (x <= -1.32e-173)
		tmp = t_1;
	elseif (x <= -1.2e-241)
		tmp = x + (x * (-1.0 / (z * t)));
	elseif (x <= 4.2e-107)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.7e-9], t$95$2, If[LessEqual[x, -2.7e-119], N[(1.0 - N[(N[(z * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.32e-173], t$95$1, If[LessEqual[x, -1.2e-241], N[(x + N[(x * N[(-1.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e-107], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.6999999999999999e-9 or 4.1999999999999998e-107 < x

   1. Initial program 94.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in t around inf 85.3%

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

      1. +-commutative 85.3%

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

   6. Simplified 85.3%

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

   if -1.6999999999999999e-9 < x < -2.70000000000000027e-119

   1. Initial program 99.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 58.3%

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

      1. mul-1-neg 58.3%

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

      2. unsub-neg 58.3%

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

      3. *-commutative 58.3%

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

      4. +-commutative 58.3%

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

   6. Simplified 58.3%

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

   7. Taylor expanded in z around 0 58.3%

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

      1. mul-1-neg 58.3%

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

      2. unsub-neg 58.3%

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

      3. *-commutative 58.3%

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

      4. associate-/r* 58.3%

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

      5. *-commutative 58.3%

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

      6. +-commutative 58.3%

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

   9. Simplified 58.3%

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

   10. Taylor expanded in x around 0 58.3%

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

   if -2.70000000000000027e-119 < x < -1.32e-173 or -1.2e-241 < x < 4.1999999999999998e-107

   1. Initial program 86.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 86.1%

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

   3. Simplified 86.1%

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

   4. Taylor expanded in z around inf 78.6%

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

   5. Taylor expanded in y around inf 63.6%

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

      1. +-commutative 63.6%

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

   7. Simplified 63.6%

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

   if -1.32e-173 < x < -1.2e-241

   1. Initial program 93.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 93.6%

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

   3. Simplified 93.6%

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

   4. Taylor expanded in y around 0 72.6%

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

      1. +-commutative 72.6%

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

   6. Simplified 72.6%

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

   7. Taylor expanded in x around 0 65.4%

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

      1. sub-neg 65.4%

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

      2. distribute-lft-in 65.4%

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

      3. *-rgt-identity 65.4%

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

      4. distribute-neg-frac 65.4%

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

      5. metadata-eval 65.4%

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

   9. Simplified 65.4%

      \[\leadsto \color{blue}{x + x \cdot \frac{-1}{t \cdot z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-119}:\\ \;\;\;\;1 - \frac{z \cdot y}{x}\\ \mathbf{elif}\;x \leq -1.32 \cdot 10^{-173}:\\ \;\;\;\;\frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-241}:\\ \;\;\;\;x + x \cdot \frac{-1}{z \cdot t}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-107}:\\ \;\;\;\;\frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]

Alternative 4: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-194} \lor \neg \left(y \leq 300000000000\right):\\ \;\;\;\;\frac{x + \frac{y}{t - \frac{x}{z}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \frac{x}{z \cdot t - x}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.9e-194) (not (<= y 300000000000.0)))
   (/ (+ x (/ y (- t (/ x z)))) (+ x 1.0))
   (/ (- x (/ x (- (* z t) x))) (+ x 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.9e-194) || !(y <= 300000000000.0)) {
		tmp = (x + (y / (t - (x / z)))) / (x + 1.0);
	} else {
		tmp = (x - (x / ((z * t) - x))) / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-1.9d-194)) .or. (.not. (y <= 300000000000.0d0))) then
        tmp = (x + (y / (t - (x / z)))) / (x + 1.0d0)
    else
        tmp = (x - (x / ((z * t) - x))) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.9e-194) || !(y <= 300000000000.0)) {
		tmp = (x + (y / (t - (x / z)))) / (x + 1.0);
	} else {
		tmp = (x - (x / ((z * t) - x))) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1.9e-194) or not (y <= 300000000000.0):
		tmp = (x + (y / (t - (x / z)))) / (x + 1.0)
	else:
		tmp = (x - (x / ((z * t) - x))) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.9e-194) || !(y <= 300000000000.0))
		tmp = Float64(Float64(x + Float64(y / Float64(t - Float64(x / z)))) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x - Float64(x / Float64(Float64(z * t) - x))) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.9e-194) || ~((y <= 300000000000.0)))
		tmp = (x + (y / (t - (x / z)))) / (x + 1.0);
	else
		tmp = (x - (x / ((z * t) - x))) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.9e-194], N[Not[LessEqual[y, 300000000000.0]], $MachinePrecision]], N[(N[(x + N[(y / N[(t - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(x / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.9000000000000001e-194 or 3e11 < y

   1. Initial program 88.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 88.9%

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

   3. Simplified 88.9%

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

   4. Taylor expanded in y around inf 86.9%

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

      1. associate-/l* 93.7%

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

   6. Simplified 93.7%

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

   7. Taylor expanded in t around 0 97.9%

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

      1. mul-1-neg 97.9%

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

      2. unsub-neg 97.9%

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

   9. Simplified 97.9%

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

   if -1.9000000000000001e-194 < y < 3e11

   1. Initial program 100.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 93.3%

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

      1. +-commutative 93.3%

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

   6. Simplified 93.3%

      \[\leadsto \color{blue}{\frac{x - \frac{x}{t \cdot z - x}}{x + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-194} \lor \neg \left(y \leq 300000000000\right):\\ \;\;\;\;\frac{x + \frac{y}{t - \frac{x}{z}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \frac{x}{z \cdot t - x}}{x + 1}\\ \end{array} \]

Alternative 5: 69.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-116}:\\ \;\;\;\;1 - \frac{z \cdot y}{x}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-104}:\\ \;\;\;\;\frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (+ x 1.0))))
   (if (<= x -1.05e-9)
     t_1
     (if (<= x -1e-116)
       (- 1.0 (/ (* z y) x))
       (if (<= x 1.9e-104) (/ y (* t (+ x 1.0))) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / (x + 1.0);
	double tmp;
	if (x <= -1.05e-9) {
		tmp = t_1;
	} else if (x <= -1e-116) {
		tmp = 1.0 - ((z * y) / x);
	} else if (x <= 1.9e-104) {
		tmp = y / (t * (x + 1.0));
	} 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 / (x + 1.0d0)
    if (x <= (-1.05d-9)) then
        tmp = t_1
    else if (x <= (-1d-116)) then
        tmp = 1.0d0 - ((z * y) / x)
    else if (x <= 1.9d-104) then
        tmp = y / (t * (x + 1.0d0))
    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 / (x + 1.0);
	double tmp;
	if (x <= -1.05e-9) {
		tmp = t_1;
	} else if (x <= -1e-116) {
		tmp = 1.0 - ((z * y) / x);
	} else if (x <= 1.9e-104) {
		tmp = y / (t * (x + 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / (x + 1.0)
	tmp = 0
	if x <= -1.05e-9:
		tmp = t_1
	elif x <= -1e-116:
		tmp = 1.0 - ((z * y) / x)
	elif x <= 1.9e-104:
		tmp = y / (t * (x + 1.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.05e-9)
		tmp = t_1;
	elseif (x <= -1e-116)
		tmp = Float64(1.0 - Float64(Float64(z * y) / x));
	elseif (x <= 1.9e-104)
		tmp = Float64(y / Float64(t * Float64(x + 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -1.05e-9)
		tmp = t_1;
	elseif (x <= -1e-116)
		tmp = 1.0 - ((z * y) / x);
	elseif (x <= 1.9e-104)
		tmp = y / (t * (x + 1.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05e-9], t$95$1, If[LessEqual[x, -1e-116], N[(1.0 - N[(N[(z * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e-104], N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.0500000000000001e-9 or 1.9e-104 < x

   1. Initial program 94.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in t around inf 85.3%

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

      1. +-commutative 85.3%

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

   6. Simplified 85.3%

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

   if -1.0500000000000001e-9 < x < -9.9999999999999999e-117

   1. Initial program 99.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 58.3%

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

      1. mul-1-neg 58.3%

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

      2. unsub-neg 58.3%

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

      3. *-commutative 58.3%

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

      4. +-commutative 58.3%

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

   6. Simplified 58.3%

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

   7. Taylor expanded in z around 0 58.3%

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

      1. mul-1-neg 58.3%

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

      2. unsub-neg 58.3%

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

      3. *-commutative 58.3%

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

      4. associate-/r* 58.3%

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

      5. *-commutative 58.3%

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

      6. +-commutative 58.3%

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

   9. Simplified 58.3%

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

   10. Taylor expanded in x around 0 58.3%

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

   if -9.9999999999999999e-117 < x < 1.9e-104

   1. Initial program 87.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in z around inf 78.8%

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

   5. Taylor expanded in y around inf 57.3%

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

      1. +-commutative 57.3%

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

   7. Simplified 57.3%

      \[\leadsto \color{blue}{\frac{y}{t \cdot \left(x + 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-116}:\\ \;\;\;\;1 - \frac{z \cdot y}{x}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-104}:\\ \;\;\;\;\frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]

Alternative 6: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{-40} \lor \neg \left(z \leq 6.8 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{z \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -4.9e-40) (not (<= z 6.8e-32)))
   (/ (+ x (/ y t)) (+ x 1.0))
   (- 1.0 (/ (* z y) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.9e-40) || !(z <= 6.8e-32)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0 - ((z * y) / x);
	}
	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 <= (-4.9d-40)) .or. (.not. (z <= 6.8d-32))) then
        tmp = (x + (y / t)) / (x + 1.0d0)
    else
        tmp = 1.0d0 - ((z * y) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.9e-40) || !(z <= 6.8e-32)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0 - ((z * y) / x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -4.9e-40) or not (z <= 6.8e-32):
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = 1.0 - ((z * y) / x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -4.9e-40) || !(z <= 6.8e-32))
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	else
		tmp = Float64(1.0 - Float64(Float64(z * y) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -4.9e-40) || ~((z <= 6.8e-32)))
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = 1.0 - ((z * y) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -4.9e-40], N[Not[LessEqual[z, 6.8e-32]], $MachinePrecision]], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(z * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.8999999999999997e-40 or 6.79999999999999956e-32 < z

   1. Initial program 88.2%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 88.2%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in z around inf 84.5%

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

   if -4.8999999999999997e-40 < z < 6.79999999999999956e-32

   1. Initial program 99.9%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 82.1%

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

      1. mul-1-neg 82.1%

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

      2. unsub-neg 82.1%

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

      3. *-commutative 82.1%

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

      4. +-commutative 82.1%

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

   6. Simplified 82.1%

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

   7. Taylor expanded in z around 0 82.1%

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

      1. mul-1-neg 82.1%

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

      2. unsub-neg 82.1%

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

      3. *-commutative 82.1%

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

      4. associate-/r* 82.1%

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

      5. *-commutative 82.1%

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

      6. +-commutative 82.1%

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

   9. Simplified 82.1%

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

   10. Taylor expanded in x around 0 78.6%

       \[\leadsto 1 - \color{blue}{\frac{y \cdot z}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{-40} \lor \neg \left(z \leq 6.8 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{z \cdot y}{x}\\ \end{array} \]

Alternative 7: 88.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{x + \frac{y}{t - \frac{x}{z}}}{x + 1} \end{array} \]

FPCore

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

C

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

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 / (t - (x / z)))) / (x + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.1%

   \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
2. Step-by-step derivation

   1. *-commutative 93.1%

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

3. Simplified 93.1%

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

4. Taylor expanded in y around inf 83.7%

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

   1. associate-/l* 88.0%

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

6. Simplified 88.0%

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

7. Taylor expanded in t around 0 90.6%

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

   1. mul-1-neg 90.6%

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

   2. unsub-neg 90.6%

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

9. Simplified 90.6%

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

10. Final simplification 90.6%

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

Alternative 8: 63.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-71} \lor \neg \left(t \leq 4 \cdot 10^{-97}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - y \cdot \frac{z}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -2.9e-71) (not (<= t 4e-97)))
   (/ x (+ x 1.0))
   (- 1.0 (* y (/ z x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.9e-71) || !(t <= 4e-97)) {
		tmp = x / (x + 1.0);
	} else {
		tmp = 1.0 - (y * (z / x));
	}
	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 <= (-2.9d-71)) .or. (.not. (t <= 4d-97))) then
        tmp = x / (x + 1.0d0)
    else
        tmp = 1.0d0 - (y * (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.9e-71) || !(t <= 4e-97)) {
		tmp = x / (x + 1.0);
	} else {
		tmp = 1.0 - (y * (z / x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -2.9e-71) or not (t <= 4e-97):
		tmp = x / (x + 1.0)
	else:
		tmp = 1.0 - (y * (z / x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -2.9e-71) || !(t <= 4e-97))
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = Float64(1.0 - Float64(y * Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -2.9e-71) || ~((t <= 4e-97)))
		tmp = x / (x + 1.0);
	else
		tmp = 1.0 - (y * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -2.9e-71], N[Not[LessEqual[t, 4e-97]], $MachinePrecision]], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.8999999999999999e-71 or 4.00000000000000014e-97 < t

   1. Initial program 92.6%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 92.6%

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

   3. Simplified 92.6%

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

   4. Taylor expanded in t around inf 70.4%

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

      1. +-commutative 70.4%

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

   6. Simplified 70.4%

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

   if -2.8999999999999999e-71 < t < 4.00000000000000014e-97

   1. Initial program 94.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in t around 0 78.1%

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

      1. mul-1-neg 78.1%

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

      2. unsub-neg 78.1%

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

      3. *-commutative 78.1%

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

      4. +-commutative 78.1%

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

   6. Simplified 78.1%

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

   7. Taylor expanded in z around 0 78.0%

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

      1. mul-1-neg 78.0%

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

      2. unsub-neg 78.0%

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

      3. *-commutative 78.0%

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

      4. associate-/r* 78.1%

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

      5. *-commutative 78.1%

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

      6. +-commutative 78.1%

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

   9. Simplified 78.1%

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

   10. Taylor expanded in x around 0 71.8%

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

       1. *-commutative 71.8%

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

       2. associate-*l/ 71.8%

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

       3. *-commutative 71.8%

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

   12. Simplified 71.8%

       \[\leadsto 1 - \color{blue}{y \cdot \frac{z}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-71} \lor \neg \left(t \leq 4 \cdot 10^{-97}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - y \cdot \frac{z}{x}\\ \end{array} \]

Alternative 9: 60.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{+153} \lor \neg \left(t \leq 8.8 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -6.4e+153) (not (<= t 8.8e+61))) (/ x (+ x 1.0)) 1.0))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -6.4e+153) || !(t <= 8.8e+61)) {
		tmp = x / (x + 1.0);
	} else {
		tmp = 1.0;
	}
	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 <= (-6.4d+153)) .or. (.not. (t <= 8.8d+61))) then
        tmp = x / (x + 1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -6.4e+153) || !(t <= 8.8e+61)) {
		tmp = x / (x + 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -6.4e+153) or not (t <= 8.8e+61):
		tmp = x / (x + 1.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -6.4e+153) || !(t <= 8.8e+61))
		tmp = Float64(x / Float64(x + 1.0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -6.4e+153) || ~((t <= 8.8e+61)))
		tmp = x / (x + 1.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -6.4e+153], N[Not[LessEqual[t, 8.8e+61]], $MachinePrecision]], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -6.4000000000000003e153 or 8.8000000000000001e61 < t

   1. Initial program 91.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in t around inf 80.4%

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

      1. +-commutative 80.4%

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

   6. Simplified 80.4%

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

   if -6.4000000000000003e153 < t < 8.8000000000000001e61

   1. Initial program 94.2%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in y around inf 82.2%

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

      1. associate-/l* 87.3%

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

   6. Simplified 87.3%

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

   7. Taylor expanded in x around inf 61.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{+153} \lor \neg \left(t \leq 8.8 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 55.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-100}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-19}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -7.5e-100) 1.0 (if (<= x 5.2e-19) (- x (* x x)) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -7.5e-100) {
		tmp = 1.0;
	} else if (x <= 5.2e-19) {
		tmp = x - (x * x);
	} else {
		tmp = 1.0;
	}
	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 (x <= (-7.5d-100)) then
        tmp = 1.0d0
    else if (x <= 5.2d-19) then
        tmp = x - (x * x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -7.5e-100) {
		tmp = 1.0;
	} else if (x <= 5.2e-19) {
		tmp = x - (x * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -7.5e-100:
		tmp = 1.0
	elif x <= 5.2e-19:
		tmp = x - (x * x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -7.5e-100)
		tmp = 1.0;
	elseif (x <= 5.2e-19)
		tmp = Float64(x - Float64(x * x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -7.5e-100)
		tmp = 1.0;
	elseif (x <= 5.2e-19)
		tmp = x - (x * x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -7.5e-100], 1.0, If[LessEqual[x, 5.2e-19], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -7.50000000000000015e-100 or 5.20000000000000026e-19 < x

   1. Initial program 94.7%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 94.7%

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

   3. Simplified 94.7%

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

   4. Taylor expanded in y around inf 87.8%

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

      1. associate-/l* 91.9%

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

   6. Simplified 91.9%

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

   7. Taylor expanded in x around inf 81.4%

      \[\leadsto \color{blue}{1} \]

   if -7.50000000000000015e-100 < x < 5.20000000000000026e-19

   1. Initial program 90.0%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Step-by-step derivation

      1. *-commutative 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in t around inf 29.2%

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

      1. +-commutative 29.2%

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

   6. Simplified 29.2%

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

   7. Taylor expanded in x around 0 29.2%

      \[\leadsto \color{blue}{x + -1 \cdot {x}^{2}} \]
   8. Step-by-step derivation

      1. mul-1-neg 29.2%

         \[\leadsto x + \color{blue}{\left(-{x}^{2}\right)} \]

      2. unsub-neg 29.2%

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      3. unpow2 29.2%

         \[\leadsto x - \color{blue}{x \cdot x} \]

   9. Simplified 29.2%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-100}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-19}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 53.6% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := 1.0

Derivation

1. Initial program 93.1%

   \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
2. Step-by-step derivation

   1. *-commutative 93.1%

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

3. Simplified 93.1%

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

4. Taylor expanded in y around inf 83.7%

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

   1. associate-/l* 88.0%

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

6. Simplified 88.0%

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

7. Taylor expanded in x around inf 57.9%

   \[\leadsto \color{blue}{1} \]

8. Final simplification 57.9%

   \[\leadsto 1 \]

Developer target: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1} \end{array} \]

FPCore

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

C

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

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 / (t - (x / z))) - (x / ((t * z) - x)))) / (x + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023271 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))