Optimisation.CirclePacking:place from circle-packing-0.1.0.4, B

Percentage Accurate: 100.0% → 100.0%

Time: 8.2s

Alternatives: 13

Speedup: 1.0×

• 26.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x + y\right) - z}{t \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (- (+ x y) z) (* t 2.0)))

C

double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

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) / (t * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Python

def code(x, y, z, t):
	return ((x + y) - z) / (t * 2.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x + y) - z) / Float64(t * 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x + y) - z) / (t * 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x + y\right) - z}{t \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (- (+ x y) z) (* t 2.0)))

C

double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

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) / (t * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Python

def code(x, y, z, t):
	return ((x + y) - z) / (t * 2.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x + y) - z) / Float64(t * 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x + y) - z) / (t * 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x + y\right) - z}{t \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (- (+ x y) z) (* t 2.0)))

C

double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

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) / (t * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x + y) - z) / (t * 2.0);
}

Python

def code(x, y, z, t):
	return ((x + y) - z) / (t * 2.0)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x + y) - z) / Float64(t * 2.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x + y) - z) / (t * 2.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+86} \lor \neg \left(z \leq 2.4 \cdot 10^{+60}\right):\\ \;\;\;\;\left(y - z\right) \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.3e+86) (not (<= z 2.4e+60)))
   (* (- y z) (/ 0.5 t))
   (* (+ x y) (/ 0.5 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.3e+86) || !(z <= 2.4e+60)) {
		tmp = (y - z) * (0.5 / t);
	} else {
		tmp = (x + y) * (0.5 / 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 <= (-1.3d+86)) .or. (.not. (z <= 2.4d+60))) then
        tmp = (y - z) * (0.5d0 / t)
    else
        tmp = (x + y) * (0.5d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.3e+86) || !(z <= 2.4e+60)) {
		tmp = (y - z) * (0.5 / t);
	} else {
		tmp = (x + y) * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.3e+86) or not (z <= 2.4e+60):
		tmp = (y - z) * (0.5 / t)
	else:
		tmp = (x + y) * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.3e+86) || !(z <= 2.4e+60))
		tmp = Float64(Float64(y - z) * Float64(0.5 / t));
	else
		tmp = Float64(Float64(x + y) * Float64(0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.3e+86) || ~((z <= 2.4e+60)))
		tmp = (y - z) * (0.5 / t);
	else
		tmp = (x + y) * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.3e+86], N[Not[LessEqual[z, 2.4e+60]], $MachinePrecision]], N[(N[(y - z), $MachinePrecision] * N[(0.5 / t), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2999999999999999e86 or 2.4e60 < z

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 93.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y - z}{t}} \]
   4. Step-by-step derivation

      1. *-commutative 93.4%

         \[\leadsto \color{blue}{\frac{y - z}{t} \cdot 0.5} \]

      2. associate-*l/ 93.4%

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 0.5}{t}} \]

      3. associate-*r/ 93.1%

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{0.5}{t}} \]

   5. Simplified 93.1%

      \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{0.5}{t}} \]

   if -1.2999999999999999e86 < z < 2.4e60

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 92.2%

      \[\leadsto \frac{\color{blue}{x + y}}{t \cdot 2} \]
   4. Step-by-step derivation

      1. +-commutative 92.2%

         \[\leadsto \frac{\color{blue}{y + x}}{t \cdot 2} \]

   5. Simplified 92.2%

      \[\leadsto \frac{\color{blue}{y + x}}{t \cdot 2} \]
   6. Step-by-step derivation

      1. clear-num 92.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{t \cdot 2}{y + x}}} \]

      2. associate-/r/ 92.0%

         \[\leadsto \color{blue}{\frac{1}{t \cdot 2} \cdot \left(y + x\right)} \]

      3. *-commutative 92.0%

         \[\leadsto \frac{1}{\color{blue}{2 \cdot t}} \cdot \left(y + x\right) \]

      4. associate-/r* 92.0%

         \[\leadsto \color{blue}{\frac{\frac{1}{2}}{t}} \cdot \left(y + x\right) \]

      5. metadata-eval 92.0%

         \[\leadsto \frac{\color{blue}{0.5}}{t} \cdot \left(y + x\right) \]

   7. Applied egg-rr 92.0%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(y + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+86} \lor \neg \left(z \leq 2.4 \cdot 10^{+60}\right):\\ \;\;\;\;\left(y - z\right) \cdot \frac{0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{0.5}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{t \cdot 2}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-221}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.85e+48)
   (/ x (* t 2.0))
   (if (<= x -2.1e-221) (/ (* z -0.5) t) (/ y (* t 2.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.85e+48) {
		tmp = x / (t * 2.0);
	} else if (x <= -2.1e-221) {
		tmp = (z * -0.5) / t;
	} else {
		tmp = y / (t * 2.0);
	}
	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 <= (-1.85d+48)) then
        tmp = x / (t * 2.0d0)
    else if (x <= (-2.1d-221)) then
        tmp = (z * (-0.5d0)) / t
    else
        tmp = y / (t * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.85e+48) {
		tmp = x / (t * 2.0);
	} else if (x <= -2.1e-221) {
		tmp = (z * -0.5) / t;
	} else {
		tmp = y / (t * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.85e+48:
		tmp = x / (t * 2.0)
	elif x <= -2.1e-221:
		tmp = (z * -0.5) / t
	else:
		tmp = y / (t * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.85e+48)
		tmp = Float64(x / Float64(t * 2.0));
	elseif (x <= -2.1e-221)
		tmp = Float64(Float64(z * -0.5) / t);
	else
		tmp = Float64(y / Float64(t * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.85e+48)
		tmp = x / (t * 2.0);
	elseif (x <= -2.1e-221)
		tmp = (z * -0.5) / t;
	else
		tmp = y / (t * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.85e+48], N[(x / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.1e-221], N[(N[(z * -0.5), $MachinePrecision] / t), $MachinePrecision], N[(y / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.85e48

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 68.4%

      \[\leadsto \frac{\color{blue}{x}}{t \cdot 2} \]

   if -1.85e48 < x < -2.1e-221

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 51.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. associate-*r/ 51.3%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot z}{t}} \]

   5. Simplified 51.3%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot z}{t}} \]

   if -2.1e-221 < x

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 43.4%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{t \cdot 2}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-221}:\\ \;\;\;\;\frac{z \cdot -0.5}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0009:\\ \;\;\;\;\frac{x - z}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -0.0009) (/ (- x z) (* t 2.0)) (/ (- y z) (* t 2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -0.0009) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (y - z) / (t * 2.0);
	}
	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 <= (-0.0009d0)) then
        tmp = (x - z) / (t * 2.0d0)
    else
        tmp = (y - z) / (t * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -0.0009) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (y - z) / (t * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -0.0009:
		tmp = (x - z) / (t * 2.0)
	else:
		tmp = (y - z) / (t * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -0.0009)
		tmp = Float64(Float64(x - z) / Float64(t * 2.0));
	else
		tmp = Float64(Float64(y - z) / Float64(t * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -0.0009)
		tmp = (x - z) / (t * 2.0);
	else
		tmp = (y - z) / (t * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -0.0009], N[(N[(x - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.9999999999999998e-4

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 86.0%

      \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]

   if -8.9999999999999998e-4 < x

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 76.8%

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

Alternative 5: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{-24}:\\ \;\;\;\;\frac{x - z}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.05e-24) (/ (- x z) (* t 2.0)) (* (- y z) (/ 0.5 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.05e-24) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (y - z) * (0.5 / 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 (x <= (-3.05d-24)) then
        tmp = (x - z) / (t * 2.0d0)
    else
        tmp = (y - z) * (0.5d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.05e-24) {
		tmp = (x - z) / (t * 2.0);
	} else {
		tmp = (y - z) * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -3.05e-24:
		tmp = (x - z) / (t * 2.0)
	else:
		tmp = (y - z) * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3.05e-24)
		tmp = Float64(Float64(x - z) / Float64(t * 2.0));
	else
		tmp = Float64(Float64(y - z) * Float64(0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -3.05e-24)
		tmp = (x - z) / (t * 2.0);
	else
		tmp = (y - z) * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -3.05e-24], N[(N[(x - z), $MachinePrecision] / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.05000000000000018e-24

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 85.1%

      \[\leadsto \frac{\color{blue}{x - z}}{t \cdot 2} \]

   if -3.05000000000000018e-24 < x

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 76.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y - z}{t}} \]
   4. Step-by-step derivation

      1. *-commutative 76.9%

         \[\leadsto \color{blue}{\frac{y - z}{t} \cdot 0.5} \]

      2. associate-*l/ 76.9%

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 0.5}{t}} \]

      3. associate-*r/ 76.7%

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{0.5}{t}} \]

   5. Simplified 76.7%

      \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{0.5}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-24}:\\ \;\;\;\;\frac{0.5}{\frac{t}{x - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3e-24) (/ 0.5 (/ t (- x z))) (* (- y z) (/ 0.5 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3e-24) {
		tmp = 0.5 / (t / (x - z));
	} else {
		tmp = (y - z) * (0.5 / 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 (x <= (-3d-24)) then
        tmp = 0.5d0 / (t / (x - z))
    else
        tmp = (y - z) * (0.5d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3e-24) {
		tmp = 0.5 / (t / (x - z));
	} else {
		tmp = (y - z) * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -3e-24:
		tmp = 0.5 / (t / (x - z))
	else:
		tmp = (y - z) * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3e-24)
		tmp = Float64(0.5 / Float64(t / Float64(x - z)));
	else
		tmp = Float64(Float64(y - z) * Float64(0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -3e-24)
		tmp = 0.5 / (t / (x - z));
	else
		tmp = (y - z) * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -3e-24], N[(0.5 / N[(t / N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.99999999999999995e-24

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 96.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t} + 0.5 \cdot \frac{y - z}{t}} \]
   4. Step-by-step derivation

      1. associate-*r/ 96.4%

         \[\leadsto \color{blue}{\frac{0.5 \cdot x}{t}} + 0.5 \cdot \frac{y - z}{t} \]

      2. associate-*l/ 96.3%

         \[\leadsto \color{blue}{\frac{0.5}{t} \cdot x} + 0.5 \cdot \frac{y - z}{t} \]

      3. associate-*r/ 96.3%

         \[\leadsto \frac{0.5}{t} \cdot x + \color{blue}{\frac{0.5 \cdot \left(y - z\right)}{t}} \]

      4. associate-*l/ 96.2%

         \[\leadsto \frac{0.5}{t} \cdot x + \color{blue}{\frac{0.5}{t} \cdot \left(y - z\right)} \]

      5. distribute-lft-in 99.8%

         \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(x + \left(y - z\right)\right)} \]

      6. +-commutative 99.8%

         \[\leadsto \frac{0.5}{t} \cdot \color{blue}{\left(\left(y - z\right) + x\right)} \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(\left(y - z\right) + x\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\left(y - z\right) + x\right)}{t}} \]

      2. clear-num 99.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{t}{0.5 \cdot \left(\left(y - z\right) + x\right)}}} \]

      3. associate-+l- 99.6%

         \[\leadsto \frac{1}{\frac{t}{0.5 \cdot \color{blue}{\left(y - \left(z - x\right)\right)}}} \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}}} \]
   8. Step-by-step derivation

      1. frac-2neg 99.6%

         \[\leadsto \color{blue}{\frac{-1}{-\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}}} \]

      2. metadata-eval 99.6%

         \[\leadsto \frac{\color{blue}{-1}}{-\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}} \]

      3. div-inv 99.6%

         \[\leadsto \color{blue}{-1 \cdot \frac{1}{-\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}}} \]

      4. associate-/r* 99.6%

         \[\leadsto -1 \cdot \frac{1}{-\color{blue}{\frac{\frac{t}{0.5}}{y - \left(z - x\right)}}} \]

      5. div-inv 99.6%

         \[\leadsto -1 \cdot \frac{1}{-\frac{\color{blue}{t \cdot \frac{1}{0.5}}}{y - \left(z - x\right)}} \]

      6. metadata-eval 99.6%

         \[\leadsto -1 \cdot \frac{1}{-\frac{t \cdot \color{blue}{2}}{y - \left(z - x\right)}} \]

      7. distribute-neg-frac 99.6%

         \[\leadsto -1 \cdot \frac{1}{\color{blue}{\frac{-t \cdot 2}{y - \left(z - x\right)}}} \]

      8. distribute-rgt-neg-in 99.6%

         \[\leadsto -1 \cdot \frac{1}{\frac{\color{blue}{t \cdot \left(-2\right)}}{y - \left(z - x\right)}} \]

      9. metadata-eval 99.6%

         \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot \color{blue}{-2}}{y - \left(z - x\right)}} \]

      10. associate--r- 99.6%

          \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{\color{blue}{\left(y - z\right) + x}}} \]

      11. sub-neg 99.6%

          \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{\color{blue}{\left(y + \left(-z\right)\right)} + x}} \]

      12. associate-+l+ 99.6%

          \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{\color{blue}{y + \left(\left(-z\right) + x\right)}}} \]

      13. +-commutative 99.6%

          \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{y + \color{blue}{\left(x + \left(-z\right)\right)}}} \]

      14. sub-neg 99.6%

          \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{y + \color{blue}{\left(x - z\right)}}} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{1}{\frac{t \cdot -2}{y + \left(x - z\right)}}} \]
   10. Step-by-step derivation

       1. mul-1-neg 99.6%

          \[\leadsto \color{blue}{-\frac{1}{\frac{t \cdot -2}{y + \left(x - z\right)}}} \]

       2. distribute-neg-frac2 99.6%

          \[\leadsto \color{blue}{\frac{1}{-\frac{t \cdot -2}{y + \left(x - z\right)}}} \]

       3. *-commutative 99.6%

          \[\leadsto \frac{1}{-\frac{\color{blue}{-2 \cdot t}}{y + \left(x - z\right)}} \]

       4. distribute-frac-neg 99.6%

          \[\leadsto \frac{1}{\color{blue}{\frac{--2 \cdot t}{y + \left(x - z\right)}}} \]

       5. distribute-lft-neg-in 99.6%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(--2\right) \cdot t}}{y + \left(x - z\right)}} \]

       6. metadata-eval 99.6%

          \[\leadsto \frac{1}{\frac{\color{blue}{2} \cdot t}{y + \left(x - z\right)}} \]

       7. associate-*r/ 99.6%

          \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{t}{y + \left(x - z\right)}}} \]

       8. metadata-eval 99.6%

          \[\leadsto \frac{\color{blue}{--1}}{2 \cdot \frac{t}{y + \left(x - z\right)}} \]

       9. distribute-neg-frac 99.6%

          \[\leadsto \color{blue}{-\frac{-1}{2 \cdot \frac{t}{y + \left(x - z\right)}}} \]

       10. associate-/r* 99.6%

           \[\leadsto -\color{blue}{\frac{\frac{-1}{2}}{\frac{t}{y + \left(x - z\right)}}} \]

       11. metadata-eval 99.6%

           \[\leadsto -\frac{\color{blue}{-0.5}}{\frac{t}{y + \left(x - z\right)}} \]

       12. distribute-neg-frac 99.6%

           \[\leadsto \color{blue}{\frac{--0.5}{\frac{t}{y + \left(x - z\right)}}} \]

       13. metadata-eval 99.6%

           \[\leadsto \frac{\color{blue}{0.5}}{\frac{t}{y + \left(x - z\right)}} \]

       14. associate-+r- 99.6%

           \[\leadsto \frac{0.5}{\frac{t}{\color{blue}{\left(y + x\right) - z}}} \]

       15. +-commutative 99.6%

           \[\leadsto \frac{0.5}{\frac{t}{\color{blue}{\left(x + y\right)} - z}} \]

       16. associate-+r- 99.6%

           \[\leadsto \frac{0.5}{\frac{t}{\color{blue}{x + \left(y - z\right)}}} \]

   11. Simplified 99.6%

       \[\leadsto \color{blue}{\frac{0.5}{\frac{t}{x + \left(y - z\right)}}} \]

   12. Taylor expanded in y around 0 84.8%

       \[\leadsto \frac{0.5}{\color{blue}{\frac{t}{x - z}}} \]

   if -2.99999999999999995e-24 < x

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 76.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y - z}{t}} \]
   4. Step-by-step derivation

      1. *-commutative 76.9%

         \[\leadsto \color{blue}{\frac{y - z}{t} \cdot 0.5} \]

      2. associate-*l/ 76.9%

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 0.5}{t}} \]

      3. associate-*r/ 76.7%

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{0.5}{t}} \]

   5. Simplified 76.7%

      \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{0.5}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{0.5}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4.1e+48) (/ x (* t 2.0)) (* (- y z) (/ 0.5 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.1e+48) {
		tmp = x / (t * 2.0);
	} else {
		tmp = (y - z) * (0.5 / 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 (x <= (-4.1d+48)) then
        tmp = x / (t * 2.0d0)
    else
        tmp = (y - z) * (0.5d0 / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.1e+48) {
		tmp = x / (t * 2.0);
	} else {
		tmp = (y - z) * (0.5 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -4.1e+48:
		tmp = x / (t * 2.0)
	else:
		tmp = (y - z) * (0.5 / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4.1e+48)
		tmp = Float64(x / Float64(t * 2.0));
	else
		tmp = Float64(Float64(y - z) * Float64(0.5 / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -4.1e+48)
		tmp = x / (t * 2.0);
	else
		tmp = (y - z) * (0.5 / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4.1e+48], N[(x / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.1000000000000003e48

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 68.4%

      \[\leadsto \frac{\color{blue}{x}}{t \cdot 2} \]

   if -4.1000000000000003e48 < x

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 77.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{y - z}{t}} \]
   4. Step-by-step derivation

      1. *-commutative 77.3%

         \[\leadsto \color{blue}{\frac{y - z}{t} \cdot 0.5} \]

      2. associate-*l/ 77.3%

         \[\leadsto \color{blue}{\frac{\left(y - z\right) \cdot 0.5}{t}} \]

      3. associate-*r/ 77.1%

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{0.5}{t}} \]

   5. Simplified 77.1%

      \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{0.5}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 48.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.049:\\ \;\;\;\;\frac{x}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{t \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -0.049) (/ x (* t 2.0)) (/ y (* t 2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -0.049) {
		tmp = x / (t * 2.0);
	} else {
		tmp = y / (t * 2.0);
	}
	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 <= (-0.049d0)) then
        tmp = x / (t * 2.0d0)
    else
        tmp = y / (t * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -0.049) {
		tmp = x / (t * 2.0);
	} else {
		tmp = y / (t * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -0.049:
		tmp = x / (t * 2.0)
	else:
		tmp = y / (t * 2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -0.049)
		tmp = Float64(x / Float64(t * 2.0));
	else
		tmp = Float64(y / Float64(t * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -0.049)
		tmp = x / (t * 2.0);
	else
		tmp = y / (t * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -0.049], N[(x / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], N[(y / N[(t * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.049000000000000002

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 60.5%

      \[\leadsto \frac{\color{blue}{x}}{t \cdot 2} \]

   if -0.049000000000000002 < x

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 44.5%

      \[\leadsto \frac{\color{blue}{y}}{t \cdot 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 48.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.048:\\ \;\;\;\;\frac{x}{t \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{t}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -0.048) (/ x (* t 2.0)) (/ 0.5 (/ t y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -0.048) {
		tmp = x / (t * 2.0);
	} else {
		tmp = 0.5 / (t / 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 (x <= (-0.048d0)) then
        tmp = x / (t * 2.0d0)
    else
        tmp = 0.5d0 / (t / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -0.048) {
		tmp = x / (t * 2.0);
	} else {
		tmp = 0.5 / (t / y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -0.048:
		tmp = x / (t * 2.0)
	else:
		tmp = 0.5 / (t / y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -0.048)
		tmp = Float64(x / Float64(t * 2.0));
	else
		tmp = Float64(0.5 / Float64(t / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -0.048)
		tmp = x / (t * 2.0);
	else
		tmp = 0.5 / (t / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -0.048], N[(x / N[(t * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(t / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.048000000000000001

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 60.5%

      \[\leadsto \frac{\color{blue}{x}}{t \cdot 2} \]

   if -0.048000000000000001 < x

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t} + 0.5 \cdot \frac{y - z}{t}} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot x}{t}} + 0.5 \cdot \frac{y - z}{t} \]

      2. associate-*l/ 98.9%

         \[\leadsto \color{blue}{\frac{0.5}{t} \cdot x} + 0.5 \cdot \frac{y - z}{t} \]

      3. associate-*r/ 98.9%

         \[\leadsto \frac{0.5}{t} \cdot x + \color{blue}{\frac{0.5 \cdot \left(y - z\right)}{t}} \]

      4. associate-*l/ 98.7%

         \[\leadsto \frac{0.5}{t} \cdot x + \color{blue}{\frac{0.5}{t} \cdot \left(y - z\right)} \]

      5. distribute-lft-in 99.7%

         \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(x + \left(y - z\right)\right)} \]

      6. +-commutative 99.7%

         \[\leadsto \frac{0.5}{t} \cdot \color{blue}{\left(\left(y - z\right) + x\right)} \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(\left(y - z\right) + x\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\left(y - z\right) + x\right)}{t}} \]

      2. clear-num 99.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{t}{0.5 \cdot \left(\left(y - z\right) + x\right)}}} \]

      3. associate-+l- 99.8%

         \[\leadsto \frac{1}{\frac{t}{0.5 \cdot \color{blue}{\left(y - \left(z - x\right)\right)}}} \]

   7. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}}} \]
   8. Step-by-step derivation

      1. frac-2neg 99.8%

         \[\leadsto \color{blue}{\frac{-1}{-\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}}} \]

      2. metadata-eval 99.8%

         \[\leadsto \frac{\color{blue}{-1}}{-\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}} \]

      3. div-inv 99.8%

         \[\leadsto \color{blue}{-1 \cdot \frac{1}{-\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}}} \]

      4. associate-/r* 99.8%

         \[\leadsto -1 \cdot \frac{1}{-\color{blue}{\frac{\frac{t}{0.5}}{y - \left(z - x\right)}}} \]

      5. div-inv 99.8%

         \[\leadsto -1 \cdot \frac{1}{-\frac{\color{blue}{t \cdot \frac{1}{0.5}}}{y - \left(z - x\right)}} \]

      6. metadata-eval 99.8%

         \[\leadsto -1 \cdot \frac{1}{-\frac{t \cdot \color{blue}{2}}{y - \left(z - x\right)}} \]

      7. distribute-neg-frac 99.8%

         \[\leadsto -1 \cdot \frac{1}{\color{blue}{\frac{-t \cdot 2}{y - \left(z - x\right)}}} \]

      8. distribute-rgt-neg-in 99.8%

         \[\leadsto -1 \cdot \frac{1}{\frac{\color{blue}{t \cdot \left(-2\right)}}{y - \left(z - x\right)}} \]

      9. metadata-eval 99.8%

         \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot \color{blue}{-2}}{y - \left(z - x\right)}} \]

      10. associate--r- 99.8%

          \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{\color{blue}{\left(y - z\right) + x}}} \]

      11. sub-neg 99.8%

          \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{\color{blue}{\left(y + \left(-z\right)\right)} + x}} \]

      12. associate-+l+ 99.8%

          \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{\color{blue}{y + \left(\left(-z\right) + x\right)}}} \]

      13. +-commutative 99.8%

          \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{y + \color{blue}{\left(x + \left(-z\right)\right)}}} \]

      14. sub-neg 99.8%

          \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{y + \color{blue}{\left(x - z\right)}}} \]

   9. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{1}{\frac{t \cdot -2}{y + \left(x - z\right)}}} \]
   10. Step-by-step derivation

       1. mul-1-neg 99.8%

          \[\leadsto \color{blue}{-\frac{1}{\frac{t \cdot -2}{y + \left(x - z\right)}}} \]

       2. distribute-neg-frac2 99.8%

          \[\leadsto \color{blue}{\frac{1}{-\frac{t \cdot -2}{y + \left(x - z\right)}}} \]

       3. *-commutative 99.8%

          \[\leadsto \frac{1}{-\frac{\color{blue}{-2 \cdot t}}{y + \left(x - z\right)}} \]

       4. distribute-frac-neg 99.8%

          \[\leadsto \frac{1}{\color{blue}{\frac{--2 \cdot t}{y + \left(x - z\right)}}} \]

       5. distribute-lft-neg-in 99.8%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(--2\right) \cdot t}}{y + \left(x - z\right)}} \]

       6. metadata-eval 99.8%

          \[\leadsto \frac{1}{\frac{\color{blue}{2} \cdot t}{y + \left(x - z\right)}} \]

       7. associate-*r/ 99.8%

          \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{t}{y + \left(x - z\right)}}} \]

       8. metadata-eval 99.8%

          \[\leadsto \frac{\color{blue}{--1}}{2 \cdot \frac{t}{y + \left(x - z\right)}} \]

       9. distribute-neg-frac 99.8%

          \[\leadsto \color{blue}{-\frac{-1}{2 \cdot \frac{t}{y + \left(x - z\right)}}} \]

       10. associate-/r* 99.8%

           \[\leadsto -\color{blue}{\frac{\frac{-1}{2}}{\frac{t}{y + \left(x - z\right)}}} \]

       11. metadata-eval 99.8%

           \[\leadsto -\frac{\color{blue}{-0.5}}{\frac{t}{y + \left(x - z\right)}} \]

       12. distribute-neg-frac 99.8%

           \[\leadsto \color{blue}{\frac{--0.5}{\frac{t}{y + \left(x - z\right)}}} \]

       13. metadata-eval 99.8%

           \[\leadsto \frac{\color{blue}{0.5}}{\frac{t}{y + \left(x - z\right)}} \]

       14. associate-+r- 99.8%

           \[\leadsto \frac{0.5}{\frac{t}{\color{blue}{\left(y + x\right) - z}}} \]

       15. +-commutative 99.8%

           \[\leadsto \frac{0.5}{\frac{t}{\color{blue}{\left(x + y\right)} - z}} \]

       16. associate-+r- 99.8%

           \[\leadsto \frac{0.5}{\frac{t}{\color{blue}{x + \left(y - z\right)}}} \]

   11. Simplified 99.8%

       \[\leadsto \color{blue}{\frac{0.5}{\frac{t}{x + \left(y - z\right)}}} \]

   12. Taylor expanded in y around inf 44.5%

       \[\leadsto \frac{0.5}{\color{blue}{\frac{t}{y}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 47.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0265:\\ \;\;\;\;\frac{0.5}{\frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{t}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -0.0265) (/ 0.5 (/ t x)) (/ 0.5 (/ t y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -0.0265) {
		tmp = 0.5 / (t / x);
	} else {
		tmp = 0.5 / (t / 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 (x <= (-0.0265d0)) then
        tmp = 0.5d0 / (t / x)
    else
        tmp = 0.5d0 / (t / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -0.0265) {
		tmp = 0.5 / (t / x);
	} else {
		tmp = 0.5 / (t / y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -0.0265:
		tmp = 0.5 / (t / x)
	else:
		tmp = 0.5 / (t / y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -0.0265)
		tmp = Float64(0.5 / Float64(t / x));
	else
		tmp = Float64(0.5 / Float64(t / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -0.0265)
		tmp = 0.5 / (t / x);
	else
		tmp = 0.5 / (t / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -0.0265], N[(0.5 / N[(t / x), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(t / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0264999999999999993

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 96.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t} + 0.5 \cdot \frac{y - z}{t}} \]
   4. Step-by-step derivation

      1. associate-*r/ 96.2%

         \[\leadsto \color{blue}{\frac{0.5 \cdot x}{t}} + 0.5 \cdot \frac{y - z}{t} \]

      2. associate-*l/ 96.1%

         \[\leadsto \color{blue}{\frac{0.5}{t} \cdot x} + 0.5 \cdot \frac{y - z}{t} \]

      3. associate-*r/ 96.1%

         \[\leadsto \frac{0.5}{t} \cdot x + \color{blue}{\frac{0.5 \cdot \left(y - z\right)}{t}} \]

      4. associate-*l/ 96.0%

         \[\leadsto \frac{0.5}{t} \cdot x + \color{blue}{\frac{0.5}{t} \cdot \left(y - z\right)} \]

      5. distribute-lft-in 99.8%

         \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(x + \left(y - z\right)\right)} \]

      6. +-commutative 99.8%

         \[\leadsto \frac{0.5}{t} \cdot \color{blue}{\left(\left(y - z\right) + x\right)} \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(\left(y - z\right) + x\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\left(y - z\right) + x\right)}{t}} \]

      2. clear-num 99.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{t}{0.5 \cdot \left(\left(y - z\right) + x\right)}}} \]

      3. associate-+l- 99.6%

         \[\leadsto \frac{1}{\frac{t}{0.5 \cdot \color{blue}{\left(y - \left(z - x\right)\right)}}} \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}}} \]
   8. Step-by-step derivation

      1. frac-2neg 99.6%

         \[\leadsto \color{blue}{\frac{-1}{-\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}}} \]

      2. metadata-eval 99.6%

         \[\leadsto \frac{\color{blue}{-1}}{-\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}} \]

      3. div-inv 99.6%

         \[\leadsto \color{blue}{-1 \cdot \frac{1}{-\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}}} \]

      4. associate-/r* 99.6%

         \[\leadsto -1 \cdot \frac{1}{-\color{blue}{\frac{\frac{t}{0.5}}{y - \left(z - x\right)}}} \]

      5. div-inv 99.6%

         \[\leadsto -1 \cdot \frac{1}{-\frac{\color{blue}{t \cdot \frac{1}{0.5}}}{y - \left(z - x\right)}} \]

      6. metadata-eval 99.6%

         \[\leadsto -1 \cdot \frac{1}{-\frac{t \cdot \color{blue}{2}}{y - \left(z - x\right)}} \]

      7. distribute-neg-frac 99.6%

         \[\leadsto -1 \cdot \frac{1}{\color{blue}{\frac{-t \cdot 2}{y - \left(z - x\right)}}} \]

      8. distribute-rgt-neg-in 99.6%

         \[\leadsto -1 \cdot \frac{1}{\frac{\color{blue}{t \cdot \left(-2\right)}}{y - \left(z - x\right)}} \]

      9. metadata-eval 99.6%

         \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot \color{blue}{-2}}{y - \left(z - x\right)}} \]

      10. associate--r- 99.6%

          \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{\color{blue}{\left(y - z\right) + x}}} \]

      11. sub-neg 99.6%

          \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{\color{blue}{\left(y + \left(-z\right)\right)} + x}} \]

      12. associate-+l+ 99.6%

          \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{\color{blue}{y + \left(\left(-z\right) + x\right)}}} \]

      13. +-commutative 99.6%

          \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{y + \color{blue}{\left(x + \left(-z\right)\right)}}} \]

      14. sub-neg 99.6%

          \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{y + \color{blue}{\left(x - z\right)}}} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{1}{\frac{t \cdot -2}{y + \left(x - z\right)}}} \]
   10. Step-by-step derivation

       1. mul-1-neg 99.6%

          \[\leadsto \color{blue}{-\frac{1}{\frac{t \cdot -2}{y + \left(x - z\right)}}} \]

       2. distribute-neg-frac2 99.6%

          \[\leadsto \color{blue}{\frac{1}{-\frac{t \cdot -2}{y + \left(x - z\right)}}} \]

       3. *-commutative 99.6%

          \[\leadsto \frac{1}{-\frac{\color{blue}{-2 \cdot t}}{y + \left(x - z\right)}} \]

       4. distribute-frac-neg 99.6%

          \[\leadsto \frac{1}{\color{blue}{\frac{--2 \cdot t}{y + \left(x - z\right)}}} \]

       5. distribute-lft-neg-in 99.6%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(--2\right) \cdot t}}{y + \left(x - z\right)}} \]

       6. metadata-eval 99.6%

          \[\leadsto \frac{1}{\frac{\color{blue}{2} \cdot t}{y + \left(x - z\right)}} \]

       7. associate-*r/ 99.6%

          \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{t}{y + \left(x - z\right)}}} \]

       8. metadata-eval 99.6%

          \[\leadsto \frac{\color{blue}{--1}}{2 \cdot \frac{t}{y + \left(x - z\right)}} \]

       9. distribute-neg-frac 99.6%

          \[\leadsto \color{blue}{-\frac{-1}{2 \cdot \frac{t}{y + \left(x - z\right)}}} \]

       10. associate-/r* 99.6%

           \[\leadsto -\color{blue}{\frac{\frac{-1}{2}}{\frac{t}{y + \left(x - z\right)}}} \]

       11. metadata-eval 99.6%

           \[\leadsto -\frac{\color{blue}{-0.5}}{\frac{t}{y + \left(x - z\right)}} \]

       12. distribute-neg-frac 99.6%

           \[\leadsto \color{blue}{\frac{--0.5}{\frac{t}{y + \left(x - z\right)}}} \]

       13. metadata-eval 99.6%

           \[\leadsto \frac{\color{blue}{0.5}}{\frac{t}{y + \left(x - z\right)}} \]

       14. associate-+r- 99.6%

           \[\leadsto \frac{0.5}{\frac{t}{\color{blue}{\left(y + x\right) - z}}} \]

       15. +-commutative 99.6%

           \[\leadsto \frac{0.5}{\frac{t}{\color{blue}{\left(x + y\right)} - z}} \]

       16. associate-+r- 99.6%

           \[\leadsto \frac{0.5}{\frac{t}{\color{blue}{x + \left(y - z\right)}}} \]

   11. Simplified 99.6%

       \[\leadsto \color{blue}{\frac{0.5}{\frac{t}{x + \left(y - z\right)}}} \]

   12. Taylor expanded in x around inf 60.4%

       \[\leadsto \frac{0.5}{\color{blue}{\frac{t}{x}}} \]

   if -0.0264999999999999993 < x

   1. Initial program 100.0%

      \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t} + 0.5 \cdot \frac{y - z}{t}} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot x}{t}} + 0.5 \cdot \frac{y - z}{t} \]

      2. associate-*l/ 98.9%

         \[\leadsto \color{blue}{\frac{0.5}{t} \cdot x} + 0.5 \cdot \frac{y - z}{t} \]

      3. associate-*r/ 98.9%

         \[\leadsto \frac{0.5}{t} \cdot x + \color{blue}{\frac{0.5 \cdot \left(y - z\right)}{t}} \]

      4. associate-*l/ 98.7%

         \[\leadsto \frac{0.5}{t} \cdot x + \color{blue}{\frac{0.5}{t} \cdot \left(y - z\right)} \]

      5. distribute-lft-in 99.7%

         \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(x + \left(y - z\right)\right)} \]

      6. +-commutative 99.7%

         \[\leadsto \frac{0.5}{t} \cdot \color{blue}{\left(\left(y - z\right) + x\right)} \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(\left(y - z\right) + x\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\left(y - z\right) + x\right)}{t}} \]

      2. clear-num 99.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{t}{0.5 \cdot \left(\left(y - z\right) + x\right)}}} \]

      3. associate-+l- 99.8%

         \[\leadsto \frac{1}{\frac{t}{0.5 \cdot \color{blue}{\left(y - \left(z - x\right)\right)}}} \]

   7. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}}} \]
   8. Step-by-step derivation

      1. frac-2neg 99.8%

         \[\leadsto \color{blue}{\frac{-1}{-\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}}} \]

      2. metadata-eval 99.8%

         \[\leadsto \frac{\color{blue}{-1}}{-\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}} \]

      3. div-inv 99.8%

         \[\leadsto \color{blue}{-1 \cdot \frac{1}{-\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}}} \]

      4. associate-/r* 99.8%

         \[\leadsto -1 \cdot \frac{1}{-\color{blue}{\frac{\frac{t}{0.5}}{y - \left(z - x\right)}}} \]

      5. div-inv 99.8%

         \[\leadsto -1 \cdot \frac{1}{-\frac{\color{blue}{t \cdot \frac{1}{0.5}}}{y - \left(z - x\right)}} \]

      6. metadata-eval 99.8%

         \[\leadsto -1 \cdot \frac{1}{-\frac{t \cdot \color{blue}{2}}{y - \left(z - x\right)}} \]

      7. distribute-neg-frac 99.8%

         \[\leadsto -1 \cdot \frac{1}{\color{blue}{\frac{-t \cdot 2}{y - \left(z - x\right)}}} \]

      8. distribute-rgt-neg-in 99.8%

         \[\leadsto -1 \cdot \frac{1}{\frac{\color{blue}{t \cdot \left(-2\right)}}{y - \left(z - x\right)}} \]

      9. metadata-eval 99.8%

         \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot \color{blue}{-2}}{y - \left(z - x\right)}} \]

      10. associate--r- 99.8%

          \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{\color{blue}{\left(y - z\right) + x}}} \]

      11. sub-neg 99.8%

          \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{\color{blue}{\left(y + \left(-z\right)\right)} + x}} \]

      12. associate-+l+ 99.8%

          \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{\color{blue}{y + \left(\left(-z\right) + x\right)}}} \]

      13. +-commutative 99.8%

          \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{y + \color{blue}{\left(x + \left(-z\right)\right)}}} \]

      14. sub-neg 99.8%

          \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{y + \color{blue}{\left(x - z\right)}}} \]

   9. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{1}{\frac{t \cdot -2}{y + \left(x - z\right)}}} \]
   10. Step-by-step derivation

       1. mul-1-neg 99.8%

          \[\leadsto \color{blue}{-\frac{1}{\frac{t \cdot -2}{y + \left(x - z\right)}}} \]

       2. distribute-neg-frac2 99.8%

          \[\leadsto \color{blue}{\frac{1}{-\frac{t \cdot -2}{y + \left(x - z\right)}}} \]

       3. *-commutative 99.8%

          \[\leadsto \frac{1}{-\frac{\color{blue}{-2 \cdot t}}{y + \left(x - z\right)}} \]

       4. distribute-frac-neg 99.8%

          \[\leadsto \frac{1}{\color{blue}{\frac{--2 \cdot t}{y + \left(x - z\right)}}} \]

       5. distribute-lft-neg-in 99.8%

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(--2\right) \cdot t}}{y + \left(x - z\right)}} \]

       6. metadata-eval 99.8%

          \[\leadsto \frac{1}{\frac{\color{blue}{2} \cdot t}{y + \left(x - z\right)}} \]

       7. associate-*r/ 99.8%

          \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{t}{y + \left(x - z\right)}}} \]

       8. metadata-eval 99.8%

          \[\leadsto \frac{\color{blue}{--1}}{2 \cdot \frac{t}{y + \left(x - z\right)}} \]

       9. distribute-neg-frac 99.8%

          \[\leadsto \color{blue}{-\frac{-1}{2 \cdot \frac{t}{y + \left(x - z\right)}}} \]

       10. associate-/r* 99.8%

           \[\leadsto -\color{blue}{\frac{\frac{-1}{2}}{\frac{t}{y + \left(x - z\right)}}} \]

       11. metadata-eval 99.8%

           \[\leadsto -\frac{\color{blue}{-0.5}}{\frac{t}{y + \left(x - z\right)}} \]

       12. distribute-neg-frac 99.8%

           \[\leadsto \color{blue}{\frac{--0.5}{\frac{t}{y + \left(x - z\right)}}} \]

       13. metadata-eval 99.8%

           \[\leadsto \frac{\color{blue}{0.5}}{\frac{t}{y + \left(x - z\right)}} \]

       14. associate-+r- 99.8%

           \[\leadsto \frac{0.5}{\frac{t}{\color{blue}{\left(y + x\right) - z}}} \]

       15. +-commutative 99.8%

           \[\leadsto \frac{0.5}{\frac{t}{\color{blue}{\left(x + y\right)} - z}} \]

       16. associate-+r- 99.8%

           \[\leadsto \frac{0.5}{\frac{t}{\color{blue}{x + \left(y - z\right)}}} \]

   11. Simplified 99.8%

       \[\leadsto \color{blue}{\frac{0.5}{\frac{t}{x + \left(y - z\right)}}} \]

   12. Taylor expanded in y around inf 44.5%

       \[\leadsto \frac{0.5}{\color{blue}{\frac{t}{y}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{t}{x + \left(y - z\right)}} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ 0.5 (/ t (+ x (- y z)))))

C

double code(double x, double y, double z, double t) {
	return 0.5 / (t / (x + (y - z)));
}

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 = 0.5d0 / (t / (x + (y - z)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.5 / (t / (x + (y - z)));
}

Python

def code(x, y, z, t):
	return 0.5 / (t / (x + (y - z)))

Julia

function code(x, y, z, t)
	return Float64(0.5 / Float64(t / Float64(x + Float64(y - z))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.5 / (t / (x + (y - z)));
end

Wolfram

code[x_, y_, z_, t_] := N[(0.5 / N[(t / N[(x + N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 98.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t} + 0.5 \cdot \frac{y - z}{t}} \]
4. Step-by-step derivation

   1. associate-*r/ 98.4%

      \[\leadsto \color{blue}{\frac{0.5 \cdot x}{t}} + 0.5 \cdot \frac{y - z}{t} \]

   2. associate-*l/ 98.3%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot x} + 0.5 \cdot \frac{y - z}{t} \]

   3. associate-*r/ 98.3%

      \[\leadsto \frac{0.5}{t} \cdot x + \color{blue}{\frac{0.5 \cdot \left(y - z\right)}{t}} \]

   4. associate-*l/ 98.2%

      \[\leadsto \frac{0.5}{t} \cdot x + \color{blue}{\frac{0.5}{t} \cdot \left(y - z\right)} \]

   5. distribute-lft-in 99.7%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(x + \left(y - z\right)\right)} \]

   6. +-commutative 99.7%

      \[\leadsto \frac{0.5}{t} \cdot \color{blue}{\left(\left(y - z\right) + x\right)} \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(\left(y - z\right) + x\right)} \]
6. Step-by-step derivation

   1. associate-*l/ 100.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\left(y - z\right) + x\right)}{t}} \]

   2. clear-num 99.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{t}{0.5 \cdot \left(\left(y - z\right) + x\right)}}} \]

   3. associate-+l- 99.7%

      \[\leadsto \frac{1}{\frac{t}{0.5 \cdot \color{blue}{\left(y - \left(z - x\right)\right)}}} \]

7. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}}} \]
8. Step-by-step derivation

   1. frac-2neg 99.7%

      \[\leadsto \color{blue}{\frac{-1}{-\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}}} \]

   2. metadata-eval 99.7%

      \[\leadsto \frac{\color{blue}{-1}}{-\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}} \]

   3. div-inv 99.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{1}{-\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}}} \]

   4. associate-/r* 99.7%

      \[\leadsto -1 \cdot \frac{1}{-\color{blue}{\frac{\frac{t}{0.5}}{y - \left(z - x\right)}}} \]

   5. div-inv 99.7%

      \[\leadsto -1 \cdot \frac{1}{-\frac{\color{blue}{t \cdot \frac{1}{0.5}}}{y - \left(z - x\right)}} \]

   6. metadata-eval 99.7%

      \[\leadsto -1 \cdot \frac{1}{-\frac{t \cdot \color{blue}{2}}{y - \left(z - x\right)}} \]

   7. distribute-neg-frac 99.7%

      \[\leadsto -1 \cdot \frac{1}{\color{blue}{\frac{-t \cdot 2}{y - \left(z - x\right)}}} \]

   8. distribute-rgt-neg-in 99.7%

      \[\leadsto -1 \cdot \frac{1}{\frac{\color{blue}{t \cdot \left(-2\right)}}{y - \left(z - x\right)}} \]

   9. metadata-eval 99.7%

      \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot \color{blue}{-2}}{y - \left(z - x\right)}} \]

   10. associate--r- 99.7%

       \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{\color{blue}{\left(y - z\right) + x}}} \]

   11. sub-neg 99.7%

       \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{\color{blue}{\left(y + \left(-z\right)\right)} + x}} \]

   12. associate-+l+ 99.7%

       \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{\color{blue}{y + \left(\left(-z\right) + x\right)}}} \]

   13. +-commutative 99.7%

       \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{y + \color{blue}{\left(x + \left(-z\right)\right)}}} \]

   14. sub-neg 99.7%

       \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{y + \color{blue}{\left(x - z\right)}}} \]

9. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{1}{\frac{t \cdot -2}{y + \left(x - z\right)}}} \]
10. Step-by-step derivation

    1. mul-1-neg 99.7%

       \[\leadsto \color{blue}{-\frac{1}{\frac{t \cdot -2}{y + \left(x - z\right)}}} \]

    2. distribute-neg-frac2 99.7%

       \[\leadsto \color{blue}{\frac{1}{-\frac{t \cdot -2}{y + \left(x - z\right)}}} \]

    3. *-commutative 99.7%

       \[\leadsto \frac{1}{-\frac{\color{blue}{-2 \cdot t}}{y + \left(x - z\right)}} \]

    4. distribute-frac-neg 99.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{--2 \cdot t}{y + \left(x - z\right)}}} \]

    5. distribute-lft-neg-in 99.7%

       \[\leadsto \frac{1}{\frac{\color{blue}{\left(--2\right) \cdot t}}{y + \left(x - z\right)}} \]

    6. metadata-eval 99.7%

       \[\leadsto \frac{1}{\frac{\color{blue}{2} \cdot t}{y + \left(x - z\right)}} \]

    7. associate-*r/ 99.7%

       \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{t}{y + \left(x - z\right)}}} \]

    8. metadata-eval 99.7%

       \[\leadsto \frac{\color{blue}{--1}}{2 \cdot \frac{t}{y + \left(x - z\right)}} \]

    9. distribute-neg-frac 99.7%

       \[\leadsto \color{blue}{-\frac{-1}{2 \cdot \frac{t}{y + \left(x - z\right)}}} \]

    10. associate-/r* 99.7%

        \[\leadsto -\color{blue}{\frac{\frac{-1}{2}}{\frac{t}{y + \left(x - z\right)}}} \]

    11. metadata-eval 99.7%

        \[\leadsto -\frac{\color{blue}{-0.5}}{\frac{t}{y + \left(x - z\right)}} \]

    12. distribute-neg-frac 99.7%

        \[\leadsto \color{blue}{\frac{--0.5}{\frac{t}{y + \left(x - z\right)}}} \]

    13. metadata-eval 99.7%

        \[\leadsto \frac{\color{blue}{0.5}}{\frac{t}{y + \left(x - z\right)}} \]

    14. associate-+r- 99.7%

        \[\leadsto \frac{0.5}{\frac{t}{\color{blue}{\left(y + x\right) - z}}} \]

    15. +-commutative 99.7%

        \[\leadsto \frac{0.5}{\frac{t}{\color{blue}{\left(x + y\right)} - z}} \]

    16. associate-+r- 99.7%

        \[\leadsto \frac{0.5}{\frac{t}{\color{blue}{x + \left(y - z\right)}}} \]

11. Simplified 99.7%

    \[\leadsto \color{blue}{\frac{0.5}{\frac{t}{x + \left(y - z\right)}}} \]
12. Add Preprocessing

Alternative 12: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + \left(y - z\right)\right) \cdot \frac{0.5}{t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (+ x (- y z)) (/ 0.5 t)))

C

double code(double x, double y, double z, double t) {
	return (x + (y - z)) * (0.5 / 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)) * (0.5d0 / t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x + (y - z)) * (0.5 / t);
}

Python

def code(x, y, z, t):
	return (x + (y - z)) * (0.5 / t)

Julia

function code(x, y, z, t)
	return Float64(Float64(x + Float64(y - z)) * Float64(0.5 / t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x + (y - z)) * (0.5 / t);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x + N[(y - z), $MachinePrecision]), $MachinePrecision] * N[(0.5 / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 98.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t} + 0.5 \cdot \frac{y - z}{t}} \]
4. Step-by-step derivation

   1. associate-*r/ 98.4%

      \[\leadsto \color{blue}{\frac{0.5 \cdot x}{t}} + 0.5 \cdot \frac{y - z}{t} \]

   2. associate-*l/ 98.3%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot x} + 0.5 \cdot \frac{y - z}{t} \]

   3. associate-*r/ 98.3%

      \[\leadsto \frac{0.5}{t} \cdot x + \color{blue}{\frac{0.5 \cdot \left(y - z\right)}{t}} \]

   4. associate-*l/ 98.2%

      \[\leadsto \frac{0.5}{t} \cdot x + \color{blue}{\frac{0.5}{t} \cdot \left(y - z\right)} \]

   5. distribute-lft-in 99.7%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(x + \left(y - z\right)\right)} \]

   6. +-commutative 99.7%

      \[\leadsto \frac{0.5}{t} \cdot \color{blue}{\left(\left(y - z\right) + x\right)} \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(\left(y - z\right) + x\right)} \]

6. Final simplification 99.7%

   \[\leadsto \left(x + \left(y - z\right)\right) \cdot \frac{0.5}{t} \]
7. Add Preprocessing

Alternative 13: 36.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{t}{x}} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ 0.5 (/ t x)))

C

double code(double x, double y, double z, double t) {
	return 0.5 / (t / 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 = 0.5d0 / (t / x)
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.5 / (t / x);
}

Python

def code(x, y, z, t):
	return 0.5 / (t / x)

Julia

function code(x, y, z, t)
	return Float64(0.5 / Float64(t / x))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.5 / (t / x);
end

Wolfram

code[x_, y_, z_, t_] := N[(0.5 / N[(t / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{\left(x + y\right) - z}{t \cdot 2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 98.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{x}{t} + 0.5 \cdot \frac{y - z}{t}} \]
4. Step-by-step derivation

   1. associate-*r/ 98.4%

      \[\leadsto \color{blue}{\frac{0.5 \cdot x}{t}} + 0.5 \cdot \frac{y - z}{t} \]

   2. associate-*l/ 98.3%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot x} + 0.5 \cdot \frac{y - z}{t} \]

   3. associate-*r/ 98.3%

      \[\leadsto \frac{0.5}{t} \cdot x + \color{blue}{\frac{0.5 \cdot \left(y - z\right)}{t}} \]

   4. associate-*l/ 98.2%

      \[\leadsto \frac{0.5}{t} \cdot x + \color{blue}{\frac{0.5}{t} \cdot \left(y - z\right)} \]

   5. distribute-lft-in 99.7%

      \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(x + \left(y - z\right)\right)} \]

   6. +-commutative 99.7%

      \[\leadsto \frac{0.5}{t} \cdot \color{blue}{\left(\left(y - z\right) + x\right)} \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{0.5}{t} \cdot \left(\left(y - z\right) + x\right)} \]
6. Step-by-step derivation

   1. associate-*l/ 100.0%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\left(y - z\right) + x\right)}{t}} \]

   2. clear-num 99.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{t}{0.5 \cdot \left(\left(y - z\right) + x\right)}}} \]

   3. associate-+l- 99.7%

      \[\leadsto \frac{1}{\frac{t}{0.5 \cdot \color{blue}{\left(y - \left(z - x\right)\right)}}} \]

7. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}}} \]
8. Step-by-step derivation

   1. frac-2neg 99.7%

      \[\leadsto \color{blue}{\frac{-1}{-\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}}} \]

   2. metadata-eval 99.7%

      \[\leadsto \frac{\color{blue}{-1}}{-\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}} \]

   3. div-inv 99.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{1}{-\frac{t}{0.5 \cdot \left(y - \left(z - x\right)\right)}}} \]

   4. associate-/r* 99.7%

      \[\leadsto -1 \cdot \frac{1}{-\color{blue}{\frac{\frac{t}{0.5}}{y - \left(z - x\right)}}} \]

   5. div-inv 99.7%

      \[\leadsto -1 \cdot \frac{1}{-\frac{\color{blue}{t \cdot \frac{1}{0.5}}}{y - \left(z - x\right)}} \]

   6. metadata-eval 99.7%

      \[\leadsto -1 \cdot \frac{1}{-\frac{t \cdot \color{blue}{2}}{y - \left(z - x\right)}} \]

   7. distribute-neg-frac 99.7%

      \[\leadsto -1 \cdot \frac{1}{\color{blue}{\frac{-t \cdot 2}{y - \left(z - x\right)}}} \]

   8. distribute-rgt-neg-in 99.7%

      \[\leadsto -1 \cdot \frac{1}{\frac{\color{blue}{t \cdot \left(-2\right)}}{y - \left(z - x\right)}} \]

   9. metadata-eval 99.7%

      \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot \color{blue}{-2}}{y - \left(z - x\right)}} \]

   10. associate--r- 99.7%

       \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{\color{blue}{\left(y - z\right) + x}}} \]

   11. sub-neg 99.7%

       \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{\color{blue}{\left(y + \left(-z\right)\right)} + x}} \]

   12. associate-+l+ 99.7%

       \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{\color{blue}{y + \left(\left(-z\right) + x\right)}}} \]

   13. +-commutative 99.7%

       \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{y + \color{blue}{\left(x + \left(-z\right)\right)}}} \]

   14. sub-neg 99.7%

       \[\leadsto -1 \cdot \frac{1}{\frac{t \cdot -2}{y + \color{blue}{\left(x - z\right)}}} \]

9. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{1}{\frac{t \cdot -2}{y + \left(x - z\right)}}} \]
10. Step-by-step derivation

    1. mul-1-neg 99.7%

       \[\leadsto \color{blue}{-\frac{1}{\frac{t \cdot -2}{y + \left(x - z\right)}}} \]

    2. distribute-neg-frac2 99.7%

       \[\leadsto \color{blue}{\frac{1}{-\frac{t \cdot -2}{y + \left(x - z\right)}}} \]

    3. *-commutative 99.7%

       \[\leadsto \frac{1}{-\frac{\color{blue}{-2 \cdot t}}{y + \left(x - z\right)}} \]

    4. distribute-frac-neg 99.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{--2 \cdot t}{y + \left(x - z\right)}}} \]

    5. distribute-lft-neg-in 99.7%

       \[\leadsto \frac{1}{\frac{\color{blue}{\left(--2\right) \cdot t}}{y + \left(x - z\right)}} \]

    6. metadata-eval 99.7%

       \[\leadsto \frac{1}{\frac{\color{blue}{2} \cdot t}{y + \left(x - z\right)}} \]

    7. associate-*r/ 99.7%

       \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{t}{y + \left(x - z\right)}}} \]

    8. metadata-eval 99.7%

       \[\leadsto \frac{\color{blue}{--1}}{2 \cdot \frac{t}{y + \left(x - z\right)}} \]

    9. distribute-neg-frac 99.7%

       \[\leadsto \color{blue}{-\frac{-1}{2 \cdot \frac{t}{y + \left(x - z\right)}}} \]

    10. associate-/r* 99.7%

        \[\leadsto -\color{blue}{\frac{\frac{-1}{2}}{\frac{t}{y + \left(x - z\right)}}} \]

    11. metadata-eval 99.7%

        \[\leadsto -\frac{\color{blue}{-0.5}}{\frac{t}{y + \left(x - z\right)}} \]

    12. distribute-neg-frac 99.7%

        \[\leadsto \color{blue}{\frac{--0.5}{\frac{t}{y + \left(x - z\right)}}} \]

    13. metadata-eval 99.7%

        \[\leadsto \frac{\color{blue}{0.5}}{\frac{t}{y + \left(x - z\right)}} \]

    14. associate-+r- 99.7%

        \[\leadsto \frac{0.5}{\frac{t}{\color{blue}{\left(y + x\right) - z}}} \]

    15. +-commutative 99.7%

        \[\leadsto \frac{0.5}{\frac{t}{\color{blue}{\left(x + y\right)} - z}} \]

    16. associate-+r- 99.7%

        \[\leadsto \frac{0.5}{\frac{t}{\color{blue}{x + \left(y - z\right)}}} \]

11. Simplified 99.7%

    \[\leadsto \color{blue}{\frac{0.5}{\frac{t}{x + \left(y - z\right)}}} \]

12. Taylor expanded in x around inf 36.2%

    \[\leadsto \frac{0.5}{\color{blue}{\frac{t}{x}}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, B"
  :precision binary64
  (/ (- (+ x y) z) (* t 2.0)))