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

Percentage Accurate: 93.3% → 95.4%

Time: 8.8s

Alternatives: 8

Speedup: 1.0×

• 24.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + ((y * (z - t)) / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + ((y * (z - t)) / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - t \leq -5 \cdot 10^{+143}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- z t) -5e+143) (+ x (* (- z t) (/ y a))) (+ x (/ (* (- z t) y) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z - t) <= -5e+143) {
		tmp = x + ((z - t) * (y / a));
	} else {
		tmp = x + (((z - t) * y) / a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z - t) <= (-5d+143)) then
        tmp = x + ((z - t) * (y / a))
    else
        tmp = x + (((z - t) * y) / a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z - t) <= -5e+143) {
		tmp = x + ((z - t) * (y / a));
	} else {
		tmp = x + (((z - t) * y) / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z - t) <= -5e+143:
		tmp = x + ((z - t) * (y / a))
	else:
		tmp = x + (((z - t) * y) / a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(z - t) <= -5e+143)
		tmp = Float64(x + Float64(Float64(z - t) * Float64(y / a)));
	else
		tmp = Float64(x + Float64(Float64(Float64(z - t) * y) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z - t) <= -5e+143)
		tmp = x + ((z - t) * (y / a));
	else
		tmp = x + (((z - t) * y) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(z - t), $MachinePrecision], -5e+143], N[(x + N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z t) < -5.00000000000000012e143

   1. Initial program 84.0%

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

      1. associate-*l/ 98.3%

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

   3. Simplified 98.3%

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

   if -5.00000000000000012e143 < (-.f64 z t)

   1. Initial program 99.2%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z - t \leq -5 \cdot 10^{+143}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(z - t\right) \cdot y}{a}\\ \end{array} \]

Alternative 2: 49.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+54}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -50000 \lor \neg \left(x \leq -2.8 \cdot 10^{-29}\right) \land x \leq 2.3 \cdot 10^{-99}:\\ \;\;\;\;t \cdot \frac{-y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -1.85e+54)
   x
   (if (or (<= x -50000.0) (and (not (<= x -2.8e-29)) (<= x 2.3e-99)))
     (* t (/ (- y) a))
     x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.85e+54) {
		tmp = x;
	} else if ((x <= -50000.0) || (!(x <= -2.8e-29) && (x <= 2.3e-99))) {
		tmp = t * (-y / a);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (x <= (-1.85d+54)) then
        tmp = x
    else if ((x <= (-50000.0d0)) .or. (.not. (x <= (-2.8d-29))) .and. (x <= 2.3d-99)) then
        tmp = t * (-y / a)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.85e+54) {
		tmp = x;
	} else if ((x <= -50000.0) || (!(x <= -2.8e-29) && (x <= 2.3e-99))) {
		tmp = t * (-y / a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -1.85e+54:
		tmp = x
	elif (x <= -50000.0) or (not (x <= -2.8e-29) and (x <= 2.3e-99)):
		tmp = t * (-y / a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -1.85e+54)
		tmp = x;
	elseif ((x <= -50000.0) || (!(x <= -2.8e-29) && (x <= 2.3e-99)))
		tmp = Float64(t * Float64(Float64(-y) / a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -1.85e+54)
		tmp = x;
	elseif ((x <= -50000.0) || (~((x <= -2.8e-29)) && (x <= 2.3e-99)))
		tmp = t * (-y / a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -1.85e+54], x, If[Or[LessEqual[x, -50000.0], And[N[Not[LessEqual[x, -2.8e-29]], $MachinePrecision], LessEqual[x, 2.3e-99]]], N[(t * N[((-y) / a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8500000000000001e54 or -5e4 < x < -2.8000000000000002e-29 or 2.2999999999999998e-99 < x

   1. Initial program 97.4%

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

      1. +-commutative 97.4%

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

      2. associate-*r/ 94.8%

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

      3. fma-def 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in y around 0 60.9%

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

   if -1.8500000000000001e54 < x < -5e4 or -2.8000000000000002e-29 < x < 2.2999999999999998e-99

   1. Initial program 93.5%

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

      1. +-commutative 93.5%

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

      2. associate-*r/ 91.1%

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

      3. fma-def 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in z around 0 66.0%

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

      1. +-commutative 66.0%

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

      2. mul-1-neg 66.0%

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

      3. unsub-neg 66.0%

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

      4. *-commutative 66.0%

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

      5. associate-*r/ 67.1%

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

   6. Simplified 67.1%

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

      1. clear-num 67.0%

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

      2. div-inv 66.3%

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

   8. Applied egg-rr 66.3%

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

   9. Taylor expanded in x around 0 54.1%

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

       1. mul-1-neg 54.1%

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

       2. associate-*r/ 53.3%

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

       3. distribute-lft-neg-in 53.3%

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

       4. *-commutative 53.3%

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

   11. Simplified 53.3%

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

   12. Taylor expanded in t around 0 54.1%

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

       1. associate-*l/ 55.5%

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

       2. *-commutative 55.5%

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

       3. neg-mul-1 55.5%

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

       4. distribute-rgt-neg-in 55.5%

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

       5. distribute-neg-frac 55.5%

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

   14. Simplified 55.5%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+54}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -50000 \lor \neg \left(x \leq -2.8 \cdot 10^{-29}\right) \land x \leq 2.3 \cdot 10^{-99}:\\ \;\;\;\;t \cdot \frac{-y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 49.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+52}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -0.118:\\ \;\;\;\;\frac{-t}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-29}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-99}:\\ \;\;\;\;t \cdot \frac{-y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -8.8e+52)
   x
   (if (<= x -0.118)
     (/ (- t) (/ a y))
     (if (<= x -2.3e-29) x (if (<= x 1.75e-99) (* t (/ (- y) a)) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -8.8e+52) {
		tmp = x;
	} else if (x <= -0.118) {
		tmp = -t / (a / y);
	} else if (x <= -2.3e-29) {
		tmp = x;
	} else if (x <= 1.75e-99) {
		tmp = t * (-y / a);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (x <= (-8.8d+52)) then
        tmp = x
    else if (x <= (-0.118d0)) then
        tmp = -t / (a / y)
    else if (x <= (-2.3d-29)) then
        tmp = x
    else if (x <= 1.75d-99) then
        tmp = t * (-y / a)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -8.8e+52) {
		tmp = x;
	} else if (x <= -0.118) {
		tmp = -t / (a / y);
	} else if (x <= -2.3e-29) {
		tmp = x;
	} else if (x <= 1.75e-99) {
		tmp = t * (-y / a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -8.8e+52:
		tmp = x
	elif x <= -0.118:
		tmp = -t / (a / y)
	elif x <= -2.3e-29:
		tmp = x
	elif x <= 1.75e-99:
		tmp = t * (-y / a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -8.8e+52)
		tmp = x;
	elseif (x <= -0.118)
		tmp = Float64(Float64(-t) / Float64(a / y));
	elseif (x <= -2.3e-29)
		tmp = x;
	elseif (x <= 1.75e-99)
		tmp = Float64(t * Float64(Float64(-y) / a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -8.8e+52)
		tmp = x;
	elseif (x <= -0.118)
		tmp = -t / (a / y);
	elseif (x <= -2.3e-29)
		tmp = x;
	elseif (x <= 1.75e-99)
		tmp = t * (-y / a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -8.8e+52], x, If[LessEqual[x, -0.118], N[((-t) / N[(a / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.3e-29], x, If[LessEqual[x, 1.75e-99], N[(t * N[((-y) / a), $MachinePrecision]), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.7999999999999999e52 or -0.11799999999999999 < x < -2.29999999999999991e-29 or 1.7499999999999999e-99 < x

   1. Initial program 97.4%

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

      1. +-commutative 97.4%

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

      2. associate-*r/ 94.8%

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

      3. fma-def 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in y around 0 60.9%

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

   if -8.7999999999999999e52 < x < -0.11799999999999999

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 89.2%

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

      3. fma-def 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in z around 0 79.8%

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

      1. +-commutative 79.8%

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

      2. mul-1-neg 79.8%

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

      3. unsub-neg 79.8%

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

      4. *-commutative 79.8%

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

      5. associate-*r/ 74.4%

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

   6. Simplified 74.4%

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

      1. clear-num 74.4%

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

      2. div-inv 74.5%

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

   8. Applied egg-rr 74.5%

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

   9. Taylor expanded in x around 0 57.6%

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

       1. mul-1-neg 57.6%

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

       2. associate-*r/ 50.6%

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

       3. distribute-lft-neg-in 50.6%

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

       4. *-commutative 50.6%

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

   11. Simplified 50.6%

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

       1. associate-/r/ 52.3%

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

       2. frac-2neg 52.3%

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

       3. distribute-frac-neg 52.3%

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

       4. frac-2neg 52.3%

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

   13. Applied egg-rr 52.3%

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

   if -2.29999999999999991e-29 < x < 1.7499999999999999e-99

   1. Initial program 92.2%

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

      1. +-commutative 92.2%

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

      2. associate-*r/ 91.5%

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

      3. fma-def 91.5%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in z around 0 63.3%

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

      1. +-commutative 63.3%

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

      2. mul-1-neg 63.3%

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

      3. unsub-neg 63.3%

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

      4. *-commutative 63.3%

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

      5. associate-*r/ 65.6%

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

   6. Simplified 65.6%

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

      1. clear-num 65.5%

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

      2. div-inv 64.6%

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

   8. Applied egg-rr 64.6%

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

   9. Taylor expanded in x around 0 53.5%

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

       1. mul-1-neg 53.5%

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

       2. associate-*r/ 53.8%

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

       3. distribute-lft-neg-in 53.8%

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

       4. *-commutative 53.8%

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

   11. Simplified 53.8%

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

   12. Taylor expanded in t around 0 53.5%

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

       1. associate-*l/ 56.2%

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

       2. *-commutative 56.2%

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

       3. neg-mul-1 56.2%

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

       4. distribute-rgt-neg-in 56.2%

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

       5. distribute-neg-frac 56.2%

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

   14. Simplified 56.2%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+52}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -0.118:\\ \;\;\;\;\frac{-t}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-29}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-99}:\\ \;\;\;\;t \cdot \frac{-y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 49.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+54}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -128000:\\ \;\;\;\;\frac{t \cdot \left(-y\right)}{a}\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-29}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-99}:\\ \;\;\;\;t \cdot \frac{-y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -6.8e+54)
   x
   (if (<= x -128000.0)
     (/ (* t (- y)) a)
     (if (<= x -2.5e-29) x (if (<= x 2.3e-99) (* t (/ (- y) a)) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -6.8e+54) {
		tmp = x;
	} else if (x <= -128000.0) {
		tmp = (t * -y) / a;
	} else if (x <= -2.5e-29) {
		tmp = x;
	} else if (x <= 2.3e-99) {
		tmp = t * (-y / a);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (x <= (-6.8d+54)) then
        tmp = x
    else if (x <= (-128000.0d0)) then
        tmp = (t * -y) / a
    else if (x <= (-2.5d-29)) then
        tmp = x
    else if (x <= 2.3d-99) then
        tmp = t * (-y / a)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -6.8e+54) {
		tmp = x;
	} else if (x <= -128000.0) {
		tmp = (t * -y) / a;
	} else if (x <= -2.5e-29) {
		tmp = x;
	} else if (x <= 2.3e-99) {
		tmp = t * (-y / a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -6.8e+54:
		tmp = x
	elif x <= -128000.0:
		tmp = (t * -y) / a
	elif x <= -2.5e-29:
		tmp = x
	elif x <= 2.3e-99:
		tmp = t * (-y / a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -6.8e+54)
		tmp = x;
	elseif (x <= -128000.0)
		tmp = Float64(Float64(t * Float64(-y)) / a);
	elseif (x <= -2.5e-29)
		tmp = x;
	elseif (x <= 2.3e-99)
		tmp = Float64(t * Float64(Float64(-y) / a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -6.8e+54)
		tmp = x;
	elseif (x <= -128000.0)
		tmp = (t * -y) / a;
	elseif (x <= -2.5e-29)
		tmp = x;
	elseif (x <= 2.3e-99)
		tmp = t * (-y / a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -6.8e+54], x, If[LessEqual[x, -128000.0], N[(N[(t * (-y)), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[x, -2.5e-29], x, If[LessEqual[x, 2.3e-99], N[(t * N[((-y) / a), $MachinePrecision]), $MachinePrecision], x]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.8000000000000001e54 or -128000 < x < -2.49999999999999993e-29 or 2.2999999999999998e-99 < x

   1. Initial program 97.4%

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

      1. +-commutative 97.4%

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

      2. associate-*r/ 94.8%

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

      3. fma-def 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in y around 0 60.9%

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

   if -6.8000000000000001e54 < x < -128000

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 89.2%

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

      3. fma-def 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in z around 0 79.8%

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

      1. +-commutative 79.8%

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

      2. mul-1-neg 79.8%

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

      3. unsub-neg 79.8%

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

      4. *-commutative 79.8%

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

      5. associate-*r/ 74.4%

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

   6. Simplified 74.4%

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

      1. clear-num 74.4%

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

      2. div-inv 74.5%

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

   8. Applied egg-rr 74.5%

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

   9. Taylor expanded in x around 0 57.6%

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

       1. associate-*r/ 57.6%

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

       2. *-commutative 57.6%

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

       3. neg-mul-1 57.6%

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

       4. distribute-rgt-neg-in 57.6%

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

   11. Simplified 57.6%

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

   if -2.49999999999999993e-29 < x < 2.2999999999999998e-99

   1. Initial program 92.2%

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

      1. +-commutative 92.2%

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

      2. associate-*r/ 91.5%

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

      3. fma-def 91.5%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in z around 0 63.3%

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

      1. +-commutative 63.3%

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

      2. mul-1-neg 63.3%

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

      3. unsub-neg 63.3%

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

      4. *-commutative 63.3%

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

      5. associate-*r/ 65.6%

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

   6. Simplified 65.6%

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

      1. clear-num 65.5%

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

      2. div-inv 64.6%

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

   8. Applied egg-rr 64.6%

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

   9. Taylor expanded in x around 0 53.5%

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

       1. mul-1-neg 53.5%

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

       2. associate-*r/ 53.8%

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

       3. distribute-lft-neg-in 53.8%

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

       4. *-commutative 53.8%

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

   11. Simplified 53.8%

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

   12. Taylor expanded in t around 0 53.5%

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

       1. associate-*l/ 56.2%

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

       2. *-commutative 56.2%

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

       3. neg-mul-1 56.2%

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

       4. distribute-rgt-neg-in 56.2%

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

       5. distribute-neg-frac 56.2%

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

   14. Simplified 56.2%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+54}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -128000:\\ \;\;\;\;\frac{t \cdot \left(-y\right)}{a}\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-29}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-99}:\\ \;\;\;\;t \cdot \frac{-y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 73.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.86 \cdot 10^{+136}:\\ \;\;\;\;\frac{-t}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq 1.18 \cdot 10^{+119} \lor \neg \left(t \leq 5.4 \cdot 10^{+178}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.86e+136)
   (/ (- t) (/ a y))
   (if (or (<= t 1.18e+119) (not (<= t 5.4e+178)))
     (+ x (* z (/ y a)))
     (* y (/ (- t) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.86e+136) {
		tmp = -t / (a / y);
	} else if ((t <= 1.18e+119) || !(t <= 5.4e+178)) {
		tmp = x + (z * (y / a));
	} else {
		tmp = y * (-t / a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-1.86d+136)) then
        tmp = -t / (a / y)
    else if ((t <= 1.18d+119) .or. (.not. (t <= 5.4d+178))) then
        tmp = x + (z * (y / a))
    else
        tmp = y * (-t / a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.86e+136) {
		tmp = -t / (a / y);
	} else if ((t <= 1.18e+119) || !(t <= 5.4e+178)) {
		tmp = x + (z * (y / a));
	} else {
		tmp = y * (-t / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -1.86e+136:
		tmp = -t / (a / y)
	elif (t <= 1.18e+119) or not (t <= 5.4e+178):
		tmp = x + (z * (y / a))
	else:
		tmp = y * (-t / a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.86e+136)
		tmp = Float64(Float64(-t) / Float64(a / y));
	elseif ((t <= 1.18e+119) || !(t <= 5.4e+178))
		tmp = Float64(x + Float64(z * Float64(y / a)));
	else
		tmp = Float64(y * Float64(Float64(-t) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -1.86e+136)
		tmp = -t / (a / y);
	elseif ((t <= 1.18e+119) || ~((t <= 5.4e+178)))
		tmp = x + (z * (y / a));
	else
		tmp = y * (-t / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.86e+136], N[((-t) / N[(a / y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 1.18e+119], N[Not[LessEqual[t, 5.4e+178]], $MachinePrecision]], N[(x + N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[((-t) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.86e136

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-*r/ 85.9%

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

      3. fma-def 85.9%

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

   3. Simplified 85.9%

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

   4. Taylor expanded in z around 0 93.0%

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

      1. +-commutative 93.0%

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

      2. mul-1-neg 93.0%

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

      3. unsub-neg 93.0%

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

      4. *-commutative 93.0%

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

      5. associate-*r/ 93.0%

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

   6. Simplified 93.0%

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

      1. clear-num 93.0%

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

      2. div-inv 93.1%

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

   8. Applied egg-rr 93.1%

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

   9. Taylor expanded in x around 0 76.1%

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

       1. mul-1-neg 76.1%

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

       2. associate-*r/ 63.8%

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

       3. distribute-lft-neg-in 63.8%

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

       4. *-commutative 63.8%

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

   11. Simplified 63.8%

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

       1. associate-/r/ 76.1%

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

       2. frac-2neg 76.1%

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

       3. distribute-frac-neg 76.1%

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

       4. frac-2neg 76.1%

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

   13. Applied egg-rr 76.1%

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

   if -1.86e136 < t < 1.1799999999999999e119 or 5.40000000000000036e178 < t

   1. Initial program 95.6%

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

      1. +-commutative 95.6%

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

      2. associate-*r/ 94.3%

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

      3. fma-def 94.3%

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

   3. Simplified 94.3%

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

   4. Taylor expanded in t around 0 78.4%

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

      1. associate-*l/ 79.9%

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

      2. *-commutative 79.9%

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

   6. Simplified 79.9%

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

   if 1.1799999999999999e119 < t < 5.40000000000000036e178

   1. Initial program 83.9%

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

      1. +-commutative 83.9%

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

      2. associate-*r/ 99.7%

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

      3. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 83.9%

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

      1. +-commutative 83.9%

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

      2. mul-1-neg 83.9%

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

      3. unsub-neg 83.9%

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

      4. *-commutative 83.9%

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

      5. associate-*r/ 92.0%

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

   6. Simplified 92.0%

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

      1. clear-num 91.8%

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

      2. div-inv 92.1%

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

   8. Applied egg-rr 92.1%

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

   9. Taylor expanded in x around 0 59.7%

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

       1. mul-1-neg 59.7%

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

       2. associate-*r/ 75.5%

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

       3. distribute-lft-neg-in 75.5%

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

       4. *-commutative 75.5%

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

   11. Simplified 75.5%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.86 \cdot 10^{+136}:\\ \;\;\;\;\frac{-t}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq 1.18 \cdot 10^{+119} \lor \neg \left(t \leq 5.4 \cdot 10^{+178}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-t}{a}\\ \end{array} \]

Alternative 6: 86.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+23}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{+68}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.8e+23)
   (+ x (* z (/ y a)))
   (if (<= z 7.4e+68) (- x (* t (/ y a))) (+ x (/ z (/ a y))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.8e+23) {
		tmp = x + (z * (y / a));
	} else if (z <= 7.4e+68) {
		tmp = x - (t * (y / a));
	} else {
		tmp = x + (z / (a / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4.8d+23)) then
        tmp = x + (z * (y / a))
    else if (z <= 7.4d+68) then
        tmp = x - (t * (y / a))
    else
        tmp = x + (z / (a / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.8e+23) {
		tmp = x + (z * (y / a));
	} else if (z <= 7.4e+68) {
		tmp = x - (t * (y / a));
	} else {
		tmp = x + (z / (a / y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.8e+23:
		tmp = x + (z * (y / a))
	elif z <= 7.4e+68:
		tmp = x - (t * (y / a))
	else:
		tmp = x + (z / (a / y))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.8e+23)
		tmp = Float64(x + Float64(z * Float64(y / a)));
	elseif (z <= 7.4e+68)
		tmp = Float64(x - Float64(t * Float64(y / a)));
	else
		tmp = Float64(x + Float64(z / Float64(a / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -4.8e+23)
		tmp = x + (z * (y / a));
	elseif (z <= 7.4e+68)
		tmp = x - (t * (y / a));
	else
		tmp = x + (z / (a / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.8e+23], N[(x + N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.4e+68], N[(x - N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(z / N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.8e23

   1. Initial program 93.4%

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

      1. +-commutative 93.4%

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

      2. associate-*r/ 91.7%

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

      3. fma-def 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in t around 0 84.6%

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

      1. associate-*l/ 91.2%

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

      2. *-commutative 91.2%

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

   6. Simplified 91.2%

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

   if -4.8e23 < z < 7.39999999999999996e68

   1. Initial program 96.5%

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

      1. +-commutative 96.5%

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

      2. associate-*r/ 94.8%

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

      3. fma-def 94.8%

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

   3. Simplified 94.8%

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

   4. Taylor expanded in z around 0 88.0%

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

      1. +-commutative 88.0%

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

      2. mul-1-neg 88.0%

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

      3. unsub-neg 88.0%

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

      4. *-commutative 88.0%

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

      5. associate-*r/ 90.0%

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

   6. Simplified 90.0%

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

   if 7.39999999999999996e68 < z

   1. Initial program 96.0%

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

      1. +-commutative 96.0%

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

      2. associate-*r/ 90.0%

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

      3. fma-def 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in t around 0 88.5%

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

      1. associate-*l/ 90.5%

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

      2. *-commutative 90.5%

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

   6. Simplified 90.5%

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

      1. clear-num 90.5%

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

      2. div-inv 90.5%

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

   8. Applied egg-rr 90.5%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+23}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{+68}:\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \end{array} \]

Alternative 7: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + ((z - t) * (y / a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

   1. associate-*l/ 97.3%

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

3. Simplified 97.3%

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

4. Final simplification 97.3%

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

Alternative 8: 40.2% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

   1. +-commutative 95.7%

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

   2. associate-*r/ 93.2%

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

   3. fma-def 93.2%

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

3. Simplified 93.2%

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

4. Taylor expanded in y around 0 41.1%

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

5. Final simplification 41.1%

   \[\leadsto x \]

Developer target: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{z - t}\\ \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{t_1}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ a (- z t))))
   (if (< y -1.0761266216389975e-10)
     (+ x (/ 1.0 (/ t_1 y)))
     (if (< y 2.894426862792089e-49)
       (+ x (/ (* y (- z t)) a))
       (+ x (/ y t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x + (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) / a);
	} else {
		tmp = x + (y / t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a / (z - t)
    if (y < (-1.0761266216389975d-10)) then
        tmp = x + (1.0d0 / (t_1 / y))
    else if (y < 2.894426862792089d-49) then
        tmp = x + ((y * (z - t)) / a)
    else
        tmp = x + (y / t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a / (z - t);
	double tmp;
	if (y < -1.0761266216389975e-10) {
		tmp = x + (1.0 / (t_1 / y));
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) / a);
	} else {
		tmp = x + (y / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a / (z - t)
	tmp = 0
	if y < -1.0761266216389975e-10:
		tmp = x + (1.0 / (t_1 / y))
	elif y < 2.894426862792089e-49:
		tmp = x + ((y * (z - t)) / a)
	else:
		tmp = x + (y / t_1)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a / Float64(z - t))
	tmp = 0.0
	if (y < -1.0761266216389975e-10)
		tmp = Float64(x + Float64(1.0 / Float64(t_1 / y)));
	elseif (y < 2.894426862792089e-49)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / a));
	else
		tmp = Float64(x + Float64(y / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a / (z - t);
	tmp = 0.0;
	if (y < -1.0761266216389975e-10)
		tmp = x + (1.0 / (t_1 / y));
	elseif (y < 2.894426862792089e-49)
		tmp = x + ((y * (z - t)) / a);
	else
		tmp = x + (y / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -1.0761266216389975e-10], N[(x + N[(1.0 / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[y, 2.894426862792089e-49], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2023224 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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