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

Percentage Accurate: 93.0% → 97.8%

Time: 7.8s

Alternatives: 14

Speedup: 1.0×

• 25.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z - x, \frac{y}{t}, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.7%

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

   1. +-commutative 94.7%

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

   2. *-commutative 94.7%

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

   3. associate-/l* 98.8%

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

   4. fma-define 98.8%

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

4. Applied egg-rr 98.8%

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

Alternative 2: 53.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t}\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{+115}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-126}:\\ \;\;\;\;z \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-298}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{-144}:\\ \;\;\;\;x \cdot \frac{y}{-t}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z y) t)))
   (if (<= t -1.6e+115)
     x
     (if (<= t -5.5e-88)
       t_1
       (if (<= t -1.1e-126)
         (* z (/ x z))
         (if (<= t 1.85e-298)
           (* z (/ y t))
           (if (<= t 2.45e-144)
             (* x (/ y (- t)))
             (if (<= t 8.5e-26) t_1 x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * y) / t;
	double tmp;
	if (t <= -1.6e+115) {
		tmp = x;
	} else if (t <= -5.5e-88) {
		tmp = t_1;
	} else if (t <= -1.1e-126) {
		tmp = z * (x / z);
	} else if (t <= 1.85e-298) {
		tmp = z * (y / t);
	} else if (t <= 2.45e-144) {
		tmp = x * (y / -t);
	} else if (t <= 8.5e-26) {
		tmp = t_1;
	} else {
		tmp = 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) :: t_1
    real(8) :: tmp
    t_1 = (z * y) / t
    if (t <= (-1.6d+115)) then
        tmp = x
    else if (t <= (-5.5d-88)) then
        tmp = t_1
    else if (t <= (-1.1d-126)) then
        tmp = z * (x / z)
    else if (t <= 1.85d-298) then
        tmp = z * (y / t)
    else if (t <= 2.45d-144) then
        tmp = x * (y / -t)
    else if (t <= 8.5d-26) then
        tmp = t_1
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * y) / t;
	double tmp;
	if (t <= -1.6e+115) {
		tmp = x;
	} else if (t <= -5.5e-88) {
		tmp = t_1;
	} else if (t <= -1.1e-126) {
		tmp = z * (x / z);
	} else if (t <= 1.85e-298) {
		tmp = z * (y / t);
	} else if (t <= 2.45e-144) {
		tmp = x * (y / -t);
	} else if (t <= 8.5e-26) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * y) / t
	tmp = 0
	if t <= -1.6e+115:
		tmp = x
	elif t <= -5.5e-88:
		tmp = t_1
	elif t <= -1.1e-126:
		tmp = z * (x / z)
	elif t <= 1.85e-298:
		tmp = z * (y / t)
	elif t <= 2.45e-144:
		tmp = x * (y / -t)
	elif t <= 8.5e-26:
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * y) / t)
	tmp = 0.0
	if (t <= -1.6e+115)
		tmp = x;
	elseif (t <= -5.5e-88)
		tmp = t_1;
	elseif (t <= -1.1e-126)
		tmp = Float64(z * Float64(x / z));
	elseif (t <= 1.85e-298)
		tmp = Float64(z * Float64(y / t));
	elseif (t <= 2.45e-144)
		tmp = Float64(x * Float64(y / Float64(-t)));
	elseif (t <= 8.5e-26)
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * y) / t;
	tmp = 0.0;
	if (t <= -1.6e+115)
		tmp = x;
	elseif (t <= -5.5e-88)
		tmp = t_1;
	elseif (t <= -1.1e-126)
		tmp = z * (x / z);
	elseif (t <= 1.85e-298)
		tmp = z * (y / t);
	elseif (t <= 2.45e-144)
		tmp = x * (y / -t);
	elseif (t <= 8.5e-26)
		tmp = t_1;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[t, -1.6e+115], x, If[LessEqual[t, -5.5e-88], t$95$1, If[LessEqual[t, -1.1e-126], N[(z * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.85e-298], N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.45e-144], N[(x * N[(y / (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.5e-26], t$95$1, x]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.6e115 or 8.50000000000000004e-26 < t

   1. Initial program 89.0%

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

   3. Taylor expanded in y around 0 72.3%

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

   if -1.6e115 < t < -5.49999999999999971e-88 or 2.45000000000000005e-144 < t < 8.50000000000000004e-26

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/l* 99.7%

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

      4. fma-define 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in z around inf 58.9%

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

   if -5.49999999999999971e-88 < t < -1.10000000000000007e-126

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 75.1%

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

   4. Taylor expanded in z around inf 74.8%

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

   5. Taylor expanded in x around inf 75.6%

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

   if -1.10000000000000007e-126 < t < 1.8499999999999999e-298

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 69.7%

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

   4. Taylor expanded in z around inf 73.5%

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

   5. Taylor expanded in x around 0 66.8%

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

   if 1.8499999999999999e-298 < t < 2.45000000000000005e-144

   1. Initial program 96.1%

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

      1. +-commutative 96.1%

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

      2. *-commutative 96.1%

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

      3. associate-/l* 96.6%

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

      4. fma-define 96.6%

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

   4. Applied egg-rr 96.6%

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

   5. Taylor expanded in y around inf 81.1%

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

      1. div-sub 85.0%

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

      2. associate-/l* 92.6%

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

      3. associate-*l/ 92.6%

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

   7. Simplified 92.6%

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

   8. Taylor expanded in z around 0 66.9%

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

      1. mul-1-neg 66.9%

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

      2. associate-/l* 70.1%

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

      3. distribute-lft-neg-out 70.1%

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

      4. *-commutative 70.1%

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

   10. Simplified 70.1%

       \[\leadsto \color{blue}{\frac{y}{t} \cdot \left(-x\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+115}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-126}:\\ \;\;\;\;z \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-298}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{-144}:\\ \;\;\;\;x \cdot \frac{y}{-t}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t}\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{+115}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-126}:\\ \;\;\;\;z \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{-297}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{-145}:\\ \;\;\;\;\frac{x}{t} \cdot \left(-y\right)\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* z y) t)))
   (if (<= t -1.6e+115)
     x
     (if (<= t -1.4e-82)
       t_1
       (if (<= t -1.2e-126)
         (* z (/ x z))
         (if (<= t 4.9e-297)
           (* z (/ y t))
           (if (<= t 1.22e-145)
             (* (/ x t) (- y))
             (if (<= t 3.4e-26) t_1 x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * y) / t;
	double tmp;
	if (t <= -1.6e+115) {
		tmp = x;
	} else if (t <= -1.4e-82) {
		tmp = t_1;
	} else if (t <= -1.2e-126) {
		tmp = z * (x / z);
	} else if (t <= 4.9e-297) {
		tmp = z * (y / t);
	} else if (t <= 1.22e-145) {
		tmp = (x / t) * -y;
	} else if (t <= 3.4e-26) {
		tmp = t_1;
	} else {
		tmp = 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) :: t_1
    real(8) :: tmp
    t_1 = (z * y) / t
    if (t <= (-1.6d+115)) then
        tmp = x
    else if (t <= (-1.4d-82)) then
        tmp = t_1
    else if (t <= (-1.2d-126)) then
        tmp = z * (x / z)
    else if (t <= 4.9d-297) then
        tmp = z * (y / t)
    else if (t <= 1.22d-145) then
        tmp = (x / t) * -y
    else if (t <= 3.4d-26) then
        tmp = t_1
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * y) / t;
	double tmp;
	if (t <= -1.6e+115) {
		tmp = x;
	} else if (t <= -1.4e-82) {
		tmp = t_1;
	} else if (t <= -1.2e-126) {
		tmp = z * (x / z);
	} else if (t <= 4.9e-297) {
		tmp = z * (y / t);
	} else if (t <= 1.22e-145) {
		tmp = (x / t) * -y;
	} else if (t <= 3.4e-26) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * y) / t
	tmp = 0
	if t <= -1.6e+115:
		tmp = x
	elif t <= -1.4e-82:
		tmp = t_1
	elif t <= -1.2e-126:
		tmp = z * (x / z)
	elif t <= 4.9e-297:
		tmp = z * (y / t)
	elif t <= 1.22e-145:
		tmp = (x / t) * -y
	elif t <= 3.4e-26:
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * y) / t)
	tmp = 0.0
	if (t <= -1.6e+115)
		tmp = x;
	elseif (t <= -1.4e-82)
		tmp = t_1;
	elseif (t <= -1.2e-126)
		tmp = Float64(z * Float64(x / z));
	elseif (t <= 4.9e-297)
		tmp = Float64(z * Float64(y / t));
	elseif (t <= 1.22e-145)
		tmp = Float64(Float64(x / t) * Float64(-y));
	elseif (t <= 3.4e-26)
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * y) / t;
	tmp = 0.0;
	if (t <= -1.6e+115)
		tmp = x;
	elseif (t <= -1.4e-82)
		tmp = t_1;
	elseif (t <= -1.2e-126)
		tmp = z * (x / z);
	elseif (t <= 4.9e-297)
		tmp = z * (y / t);
	elseif (t <= 1.22e-145)
		tmp = (x / t) * -y;
	elseif (t <= 3.4e-26)
		tmp = t_1;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[t, -1.6e+115], x, If[LessEqual[t, -1.4e-82], t$95$1, If[LessEqual[t, -1.2e-126], N[(z * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.9e-297], N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.22e-145], N[(N[(x / t), $MachinePrecision] * (-y)), $MachinePrecision], If[LessEqual[t, 3.4e-26], t$95$1, x]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.6e115 or 3.40000000000000013e-26 < t

   1. Initial program 89.0%

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

   3. Taylor expanded in y around 0 72.3%

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

   if -1.6e115 < t < -1.40000000000000012e-82 or 1.2199999999999999e-145 < t < 3.40000000000000013e-26

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/l* 99.7%

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

      4. fma-define 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in z around inf 58.9%

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

   if -1.40000000000000012e-82 < t < -1.20000000000000003e-126

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 75.1%

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

   4. Taylor expanded in z around inf 74.8%

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

   5. Taylor expanded in x around inf 75.6%

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

   if -1.20000000000000003e-126 < t < 4.89999999999999997e-297

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 69.7%

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

   4. Taylor expanded in z around inf 73.5%

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

   5. Taylor expanded in x around 0 66.8%

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

   if 4.89999999999999997e-297 < t < 1.2199999999999999e-145

   1. Initial program 96.1%

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

      1. +-commutative 96.1%

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

      2. *-commutative 96.1%

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

      3. associate-/l* 96.6%

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

      4. fma-define 96.6%

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

   4. Applied egg-rr 96.6%

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

   5. Taylor expanded in y around inf 81.1%

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

      1. div-sub 85.0%

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

      2. associate-/l* 92.6%

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

      3. associate-*l/ 92.6%

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

   7. Simplified 92.6%

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

   8. Taylor expanded in z around 0 66.9%

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

      1. mul-1-neg 66.9%

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

      2. associate-/l* 70.1%

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

      3. distribute-rgt-neg-in 70.1%

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

      4. distribute-frac-neg 70.1%

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

      5. associate-*r/ 66.9%

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

   10. Simplified 66.9%

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

   11. Taylor expanded in x around 0 66.9%

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

       1. mul-1-neg 66.9%

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

       2. associate-*r/ 70.1%

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

       3. *-commutative 70.1%

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

       4. associate-*l/ 66.9%

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

       5. distribute-frac-neg2 66.9%

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

       6. associate-/l* 70.0%

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

   13. Simplified 70.0%

       \[\leadsto \color{blue}{y \cdot \frac{x}{-t}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+115}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-82}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-126}:\\ \;\;\;\;z \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{-297}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{-145}:\\ \;\;\;\;\frac{x}{t} \cdot \left(-y\right)\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-26}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{y}{t}\\ t_2 := x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.26 \cdot 10^{+110}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+123}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (/ y t))) (t_2 (* x (- 1.0 (/ y t)))))
   (if (<= z -3.2e+165)
     t_1
     (if (<= z -1.26e+110)
       t_2
       (if (<= z -5.2e+46) (/ (* z y) t) (if (<= z 1.8e+123) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (y / t);
	double t_2 = x * (1.0 - (y / t));
	double tmp;
	if (z <= -3.2e+165) {
		tmp = t_1;
	} else if (z <= -1.26e+110) {
		tmp = t_2;
	} else if (z <= -5.2e+46) {
		tmp = (z * y) / t;
	} else if (z <= 1.8e+123) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = z * (y / t)
    t_2 = x * (1.0d0 - (y / t))
    if (z <= (-3.2d+165)) then
        tmp = t_1
    else if (z <= (-1.26d+110)) then
        tmp = t_2
    else if (z <= (-5.2d+46)) then
        tmp = (z * y) / t
    else if (z <= 1.8d+123) then
        tmp = t_2
    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 = z * (y / t);
	double t_2 = x * (1.0 - (y / t));
	double tmp;
	if (z <= -3.2e+165) {
		tmp = t_1;
	} else if (z <= -1.26e+110) {
		tmp = t_2;
	} else if (z <= -5.2e+46) {
		tmp = (z * y) / t;
	} else if (z <= 1.8e+123) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * (y / t)
	t_2 = x * (1.0 - (y / t))
	tmp = 0
	if z <= -3.2e+165:
		tmp = t_1
	elif z <= -1.26e+110:
		tmp = t_2
	elif z <= -5.2e+46:
		tmp = (z * y) / t
	elif z <= 1.8e+123:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(y / t))
	t_2 = Float64(x * Float64(1.0 - Float64(y / t)))
	tmp = 0.0
	if (z <= -3.2e+165)
		tmp = t_1;
	elseif (z <= -1.26e+110)
		tmp = t_2;
	elseif (z <= -5.2e+46)
		tmp = Float64(Float64(z * y) / t);
	elseif (z <= 1.8e+123)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = z * (y / t);
	t_2 = x * (1.0 - (y / t));
	tmp = 0.0;
	if (z <= -3.2e+165)
		tmp = t_1;
	elseif (z <= -1.26e+110)
		tmp = t_2;
	elseif (z <= -5.2e+46)
		tmp = (z * y) / t;
	elseif (z <= 1.8e+123)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.2e+165], t$95$1, If[LessEqual[z, -1.26e+110], t$95$2, If[LessEqual[z, -5.2e+46], N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 1.8e+123], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.2e165 or 1.79999999999999999e123 < z

   1. Initial program 91.0%

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

   3. Taylor expanded in z around inf 88.0%

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

   4. Taylor expanded in z around inf 92.1%

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

   5. Taylor expanded in x around 0 78.3%

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

   if -3.2e165 < z < -1.25999999999999992e110 or -5.20000000000000027e46 < z < 1.79999999999999999e123

   1. Initial program 95.5%

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

   3. Taylor expanded in x around inf 81.9%

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

      1. mul-1-neg 81.9%

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

      2. unsub-neg 81.9%

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

   5. Simplified 81.9%

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

   if -1.25999999999999992e110 < z < -5.20000000000000027e46

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/l* 99.8%

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

      4. fma-define 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in z around inf 67.8%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+165}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq -1.26 \cdot 10^{+110}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+123}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{+116}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -2.9 \cdot 10^{-85}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-126}:\\ \;\;\;\;z \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-26}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.45e+116)
   x
   (if (<= t -2.9e-85)
     (/ (* z y) t)
     (if (<= t -1.2e-126) (* z (/ x z)) (if (<= t 4.4e-26) (* z (/ y t)) x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.45e+116) {
		tmp = x;
	} else if (t <= -2.9e-85) {
		tmp = (z * y) / t;
	} else if (t <= -1.2e-126) {
		tmp = z * (x / z);
	} else if (t <= 4.4e-26) {
		tmp = z * (y / t);
	} else {
		tmp = 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 <= (-1.45d+116)) then
        tmp = x
    else if (t <= (-2.9d-85)) then
        tmp = (z * y) / t
    else if (t <= (-1.2d-126)) then
        tmp = z * (x / z)
    else if (t <= 4.4d-26) then
        tmp = z * (y / t)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.45e+116) {
		tmp = x;
	} else if (t <= -2.9e-85) {
		tmp = (z * y) / t;
	} else if (t <= -1.2e-126) {
		tmp = z * (x / z);
	} else if (t <= 4.4e-26) {
		tmp = z * (y / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.45e+116:
		tmp = x
	elif t <= -2.9e-85:
		tmp = (z * y) / t
	elif t <= -1.2e-126:
		tmp = z * (x / z)
	elif t <= 4.4e-26:
		tmp = z * (y / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.45e+116)
		tmp = x;
	elseif (t <= -2.9e-85)
		tmp = Float64(Float64(z * y) / t);
	elseif (t <= -1.2e-126)
		tmp = Float64(z * Float64(x / z));
	elseif (t <= 4.4e-26)
		tmp = Float64(z * Float64(y / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.45e+116)
		tmp = x;
	elseif (t <= -2.9e-85)
		tmp = (z * y) / t;
	elseif (t <= -1.2e-126)
		tmp = z * (x / z);
	elseif (t <= 4.4e-26)
		tmp = z * (y / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.45e+116], x, If[LessEqual[t, -2.9e-85], N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, -1.2e-126], N[(z * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.4e-26], N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision], x]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.4500000000000001e116 or 4.4000000000000002e-26 < t

   1. Initial program 89.0%

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

   3. Taylor expanded in y around 0 72.3%

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

   if -1.4500000000000001e116 < t < -2.9000000000000002e-85

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/l* 99.6%

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

      4. fma-define 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in z around inf 60.9%

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

   if -2.9000000000000002e-85 < t < -1.20000000000000003e-126

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 75.1%

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

   4. Taylor expanded in z around inf 74.8%

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

   5. Taylor expanded in x around inf 75.6%

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

   if -1.20000000000000003e-126 < t < 4.4000000000000002e-26

   1. Initial program 98.9%

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

   3. Taylor expanded in z around inf 66.0%

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

   4. Taylor expanded in z around inf 66.8%

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

   5. Taylor expanded in x around 0 56.8%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{+116}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -2.9 \cdot 10^{-85}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-126}:\\ \;\;\;\;z \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-26}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{+115}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -9.8 \cdot 10^{-88}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-126}:\\ \;\;\;\;z \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-26}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.65e+115)
   x
   (if (<= t -9.8e-88)
     (* y (/ z t))
     (if (<= t -1.2e-126) (* z (/ x z)) (if (<= t 6.5e-26) (* z (/ y t)) x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.65e+115) {
		tmp = x;
	} else if (t <= -9.8e-88) {
		tmp = y * (z / t);
	} else if (t <= -1.2e-126) {
		tmp = z * (x / z);
	} else if (t <= 6.5e-26) {
		tmp = z * (y / t);
	} else {
		tmp = 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 <= (-1.65d+115)) then
        tmp = x
    else if (t <= (-9.8d-88)) then
        tmp = y * (z / t)
    else if (t <= (-1.2d-126)) then
        tmp = z * (x / z)
    else if (t <= 6.5d-26) then
        tmp = z * (y / t)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.65e+115) {
		tmp = x;
	} else if (t <= -9.8e-88) {
		tmp = y * (z / t);
	} else if (t <= -1.2e-126) {
		tmp = z * (x / z);
	} else if (t <= 6.5e-26) {
		tmp = z * (y / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.65e+115:
		tmp = x
	elif t <= -9.8e-88:
		tmp = y * (z / t)
	elif t <= -1.2e-126:
		tmp = z * (x / z)
	elif t <= 6.5e-26:
		tmp = z * (y / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.65e+115)
		tmp = x;
	elseif (t <= -9.8e-88)
		tmp = Float64(y * Float64(z / t));
	elseif (t <= -1.2e-126)
		tmp = Float64(z * Float64(x / z));
	elseif (t <= 6.5e-26)
		tmp = Float64(z * Float64(y / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.65e+115)
		tmp = x;
	elseif (t <= -9.8e-88)
		tmp = y * (z / t);
	elseif (t <= -1.2e-126)
		tmp = z * (x / z);
	elseif (t <= 6.5e-26)
		tmp = z * (y / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.65e+115], x, If[LessEqual[t, -9.8e-88], N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.2e-126], N[(z * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.5e-26], N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision], x]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.65000000000000003e115 or 6.5e-26 < t

   1. Initial program 89.0%

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

   3. Taylor expanded in y around 0 72.3%

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

   if -1.65000000000000003e115 < t < -9.80000000000000055e-88

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/l* 99.6%

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

      4. fma-define 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in z around inf 60.9%

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

      1. associate-*r/ 60.8%

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

   7. Simplified 60.8%

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

   if -9.80000000000000055e-88 < t < -1.20000000000000003e-126

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 75.1%

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

   4. Taylor expanded in z around inf 74.8%

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

   5. Taylor expanded in x around inf 75.6%

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

   if -1.20000000000000003e-126 < t < 6.5e-26

   1. Initial program 98.9%

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

   3. Taylor expanded in z around inf 66.0%

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

   4. Taylor expanded in z around inf 66.8%

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

   5. Taylor expanded in x around 0 56.8%

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

Alternative 7: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-58}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-38}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{y}{t} + \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.6e-58)
   (+ x (/ (* z y) t))
   (if (<= z 4e-38) (* x (- 1.0 (/ y t))) (* z (+ (/ y t) (/ x z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.6e-58) {
		tmp = x + ((z * y) / t);
	} else if (z <= 4e-38) {
		tmp = x * (1.0 - (y / t));
	} else {
		tmp = z * ((y / t) + (x / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-2.6d-58)) then
        tmp = x + ((z * y) / t)
    else if (z <= 4d-38) then
        tmp = x * (1.0d0 - (y / t))
    else
        tmp = z * ((y / t) + (x / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.6e-58) {
		tmp = x + ((z * y) / t);
	} else if (z <= 4e-38) {
		tmp = x * (1.0 - (y / t));
	} else {
		tmp = z * ((y / t) + (x / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.6e-58:
		tmp = x + ((z * y) / t)
	elif z <= 4e-38:
		tmp = x * (1.0 - (y / t))
	else:
		tmp = z * ((y / t) + (x / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.6e-58)
		tmp = Float64(x + Float64(Float64(z * y) / t));
	elseif (z <= 4e-38)
		tmp = Float64(x * Float64(1.0 - Float64(y / t)));
	else
		tmp = Float64(z * Float64(Float64(y / t) + Float64(x / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2.6e-58)
		tmp = x + ((z * y) / t);
	elseif (z <= 4e-38)
		tmp = x * (1.0 - (y / t));
	else
		tmp = z * ((y / t) + (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.6e-58], N[(x + N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e-38], N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(y / t), $MachinePrecision] + N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.60000000000000007e-58

   1. Initial program 97.5%

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

   3. Taylor expanded in z around inf 89.9%

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

   if -2.60000000000000007e-58 < z < 3.9999999999999998e-38

   1. Initial program 94.5%

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

   3. Taylor expanded in x around inf 91.5%

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

      1. mul-1-neg 91.5%

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

      2. unsub-neg 91.5%

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

   5. Simplified 91.5%

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

   if 3.9999999999999998e-38 < z

   1. Initial program 91.8%

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

   3. Taylor expanded in z around inf 87.0%

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

   4. Taylor expanded in z around inf 92.4%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-58}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-38}:\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{y}{t} + \frac{x}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-100} \lor \neg \left(t \leq 5.4 \cdot 10^{-98}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - x\right) \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.15e-100) (not (<= t 5.4e-98)))
   (+ x (* y (/ z t)))
   (/ (* (- z x) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.15e-100) || !(t <= 5.4e-98)) {
		tmp = x + (y * (z / t));
	} else {
		tmp = ((z - x) * y) / 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 ((t <= (-1.15d-100)) .or. (.not. (t <= 5.4d-98))) then
        tmp = x + (y * (z / t))
    else
        tmp = ((z - x) * y) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.15e-100) || !(t <= 5.4e-98)) {
		tmp = x + (y * (z / t));
	} else {
		tmp = ((z - x) * y) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.15e-100) or not (t <= 5.4e-98):
		tmp = x + (y * (z / t))
	else:
		tmp = ((z - x) * y) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.15e-100) || !(t <= 5.4e-98))
		tmp = Float64(x + Float64(y * Float64(z / t)));
	else
		tmp = Float64(Float64(Float64(z - x) * y) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.15e-100) || ~((t <= 5.4e-98)))
		tmp = x + (y * (z / t));
	else
		tmp = ((z - x) * y) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.15e-100], N[Not[LessEqual[t, 5.4e-98]], $MachinePrecision]], N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z - x), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.14999999999999997e-100 or 5.3999999999999997e-98 < t

   1. Initial program 92.5%

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

   3. Taylor expanded in z around inf 88.0%

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

      1. associate-/l* 90.7%

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

   5. Simplified 90.7%

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

   if -1.14999999999999997e-100 < t < 5.3999999999999997e-98

   1. Initial program 98.8%

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

   3. Taylor expanded in y around -inf 89.9%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-100} \lor \neg \left(t \leq 5.4 \cdot 10^{-98}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - x\right) \cdot y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-104} \lor \neg \left(t \leq 7 \cdot 10^{-96}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(z - x\right) \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -5.5e-104) (not (<= t 7e-96)))
   (+ x (* y (/ z t)))
   (* (- z x) (/ y t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -5.5e-104) || !(t <= 7e-96)) {
		tmp = x + (y * (z / t));
	} else {
		tmp = (z - x) * (y / 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 ((t <= (-5.5d-104)) .or. (.not. (t <= 7d-96))) then
        tmp = x + (y * (z / t))
    else
        tmp = (z - x) * (y / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -5.5e-104) || !(t <= 7e-96)) {
		tmp = x + (y * (z / t));
	} else {
		tmp = (z - x) * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -5.5e-104) or not (t <= 7e-96):
		tmp = x + (y * (z / t))
	else:
		tmp = (z - x) * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -5.5e-104) || !(t <= 7e-96))
		tmp = Float64(x + Float64(y * Float64(z / t)));
	else
		tmp = Float64(Float64(z - x) * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -5.5e-104) || ~((t <= 7e-96)))
		tmp = x + (y * (z / t));
	else
		tmp = (z - x) * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -5.5e-104], N[Not[LessEqual[t, 7e-96]], $MachinePrecision]], N[(x + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z - x), $MachinePrecision] * N[(y / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.4999999999999998e-104 or 6.9999999999999998e-96 < t

   1. Initial program 92.5%

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

   3. Taylor expanded in z around inf 88.0%

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

      1. associate-/l* 90.7%

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

   5. Simplified 90.7%

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

   if -5.4999999999999998e-104 < t < 6.9999999999999998e-96

   1. Initial program 98.8%

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

      1. +-commutative 98.8%

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

      2. *-commutative 98.8%

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

      3. associate-/l* 97.8%

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

      4. fma-define 97.8%

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

   4. Applied egg-rr 97.8%

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

   5. Taylor expanded in y around inf 73.8%

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

      1. div-sub 78.2%

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

      2. associate-/l* 89.9%

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

      3. associate-*l/ 88.8%

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

   7. Simplified 88.8%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-104} \lor \neg \left(t \leq 7 \cdot 10^{-96}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(z - x\right) \cdot \frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{+71} \lor \neg \left(t \leq 6.4 \cdot 10^{-26}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z - x\right) \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.55e+71) (not (<= t 6.4e-26)))
   (* x (- 1.0 (/ y t)))
   (* (- z x) (/ y t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.55e+71) || !(t <= 6.4e-26)) {
		tmp = x * (1.0 - (y / t));
	} else {
		tmp = (z - x) * (y / 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 ((t <= (-1.55d+71)) .or. (.not. (t <= 6.4d-26))) then
        tmp = x * (1.0d0 - (y / t))
    else
        tmp = (z - x) * (y / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.55e+71) || !(t <= 6.4e-26)) {
		tmp = x * (1.0 - (y / t));
	} else {
		tmp = (z - x) * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.55e+71) or not (t <= 6.4e-26):
		tmp = x * (1.0 - (y / t))
	else:
		tmp = (z - x) * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.55e+71) || !(t <= 6.4e-26))
		tmp = Float64(x * Float64(1.0 - Float64(y / t)));
	else
		tmp = Float64(Float64(z - x) * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.55e+71) || ~((t <= 6.4e-26)))
		tmp = x * (1.0 - (y / t));
	else
		tmp = (z - x) * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.55e+71], N[Not[LessEqual[t, 6.4e-26]], $MachinePrecision]], N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z - x), $MachinePrecision] * N[(y / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.55000000000000009e71 or 6.4000000000000002e-26 < t

   1. Initial program 89.6%

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

   3. Taylor expanded in x around inf 78.6%

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

      1. mul-1-neg 78.6%

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

      2. unsub-neg 78.6%

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

   5. Simplified 78.6%

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

   if -1.55000000000000009e71 < t < 6.4000000000000002e-26

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-/l* 98.5%

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

      4. fma-define 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in y around inf 71.1%

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

      1. div-sub 74.1%

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

      2. associate-/l* 81.7%

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

      3. associate-*l/ 80.9%

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

   7. Simplified 80.9%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{+71} \lor \neg \left(t \leq 6.4 \cdot 10^{-26}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z - x\right) \cdot \frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-83} \lor \neg \left(x \leq 3.1 \cdot 10^{-25}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -7e-83) (not (<= x 3.1e-25)))
   (* x (- 1.0 (/ y t)))
   (* y (/ (- z x) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -7e-83) || !(x <= 3.1e-25)) {
		tmp = x * (1.0 - (y / t));
	} else {
		tmp = y * ((z - x) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-7d-83)) .or. (.not. (x <= 3.1d-25))) then
        tmp = x * (1.0d0 - (y / t))
    else
        tmp = y * ((z - x) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -7e-83) || !(x <= 3.1e-25)) {
		tmp = x * (1.0 - (y / t));
	} else {
		tmp = y * ((z - x) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -7e-83) or not (x <= 3.1e-25):
		tmp = x * (1.0 - (y / t))
	else:
		tmp = y * ((z - x) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -7e-83) || !(x <= 3.1e-25))
		tmp = Float64(x * Float64(1.0 - Float64(y / t)));
	else
		tmp = Float64(y * Float64(Float64(z - x) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -7e-83) || ~((x <= 3.1e-25)))
		tmp = x * (1.0 - (y / t));
	else
		tmp = y * ((z - x) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -7e-83], N[Not[LessEqual[x, 3.1e-25]], $MachinePrecision]], N[(x * N[(1.0 - N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.00000000000000061e-83 or 3.09999999999999995e-25 < x

   1. Initial program 94.1%

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

   3. Taylor expanded in x around inf 84.1%

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

      1. mul-1-neg 84.1%

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

      2. unsub-neg 84.1%

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

   5. Simplified 84.1%

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

   if -7.00000000000000061e-83 < x < 3.09999999999999995e-25

   1. Initial program 95.8%

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

      1. +-commutative 95.8%

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

      2. *-commutative 95.8%

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

      3. associate-/l* 96.8%

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

      4. fma-define 96.8%

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

   4. Applied egg-rr 96.8%

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

   5. Taylor expanded in y around inf 69.6%

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

      1. div-sub 69.6%

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

   7. Simplified 69.6%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-83} \lor \neg \left(x \leq 3.1 \cdot 10^{-25}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+46} \lor \neg \left(z \leq 3.7 \cdot 10^{-53}\right):\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.02e+46) (not (<= z 3.7e-53))) (* y (/ z t)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.02e+46) || !(z <= 3.7e-53)) {
		tmp = y * (z / t);
	} else {
		tmp = 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 <= (-1.02d+46)) .or. (.not. (z <= 3.7d-53))) then
        tmp = y * (z / t)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.02e+46) || !(z <= 3.7e-53)) {
		tmp = y * (z / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.02e+46) or not (z <= 3.7e-53):
		tmp = y * (z / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.02e+46) || !(z <= 3.7e-53))
		tmp = Float64(y * Float64(z / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.02e+46) || ~((z <= 3.7e-53)))
		tmp = y * (z / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.02e+46], N[Not[LessEqual[z, 3.7e-53]], $MachinePrecision]], N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.0199999999999999e46 or 3.69999999999999982e-53 < z

   1. Initial program 94.2%

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

      1. +-commutative 94.2%

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

      2. *-commutative 94.2%

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

      3. associate-/l* 98.5%

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

      4. fma-define 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in z around inf 61.7%

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

      1. associate-*r/ 60.1%

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

   7. Simplified 60.1%

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

   if -1.0199999999999999e46 < z < 3.69999999999999982e-53

   1. Initial program 95.2%

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

   3. Taylor expanded in y around 0 59.6%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+46} \lor \neg \left(z \leq 3.7 \cdot 10^{-53}\right):\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.7%

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

3. Taylor expanded in z around 0 90.4%

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

   1. +-commutative 90.4%

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

   2. *-commutative 90.4%

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

   3. associate-*r/ 89.6%

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

   4. mul-1-neg 89.6%

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

   5. associate-/l* 92.1%

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

   6. distribute-lft-neg-in 92.1%

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

   7. distribute-rgt-in 98.8%

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

   8. sub-neg 98.8%

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

5. Simplified 98.8%

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

6. Final simplification 98.8%

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

Alternative 14: 38.5% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.7%

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

3. Taylor expanded in y around 0 43.0%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Developer target: 91.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024108 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :alt
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

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