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

Percentage Accurate: 93.2% → 96.5%

Time: 11.1s

Alternatives: 12

Speedup: 1.1×

• 25.2% 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: 93.2% 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: 96.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-38}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y 1.35e-38) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= 1.35e-38) {
		tmp = x + ((y * (z - t)) / a);
	} else {
		tmp = x + (y / (a / (z - t)));
	}
	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 (y <= 1.35d-38) then
        tmp = x + ((y * (z - t)) / a)
    else
        tmp = x + (y / (a / (z - t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= 1.35e-38) {
		tmp = x + ((y * (z - t)) / a);
	} else {
		tmp = x + (y / (a / (z - t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= 1.35e-38:
		tmp = x + ((y * (z - t)) / a)
	else:
		tmp = x + (y / (a / (z - t)))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= 1.35e-38)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / a));
	else
		tmp = Float64(x + Float64(y / Float64(a / Float64(z - t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= 1.35e-38)
		tmp = x + ((y * (z - t)) / a);
	else
		tmp = x + (y / (a / (z - t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, 1.35e-38], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35000000000000003e-38

   1. Initial program 97.2%

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

   if 1.35000000000000003e-38 < y

   1. Initial program 87.5%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

Alternative 2: 82.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+146}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+20}:\\ \;\;\;\;x - \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)))
   (if (<= t_1 -1e+146) t_1 (if (<= t_1 2e+20) (- x (/ (* y t) a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double tmp;
	if (t_1 <= -1e+146) {
		tmp = t_1;
	} else if (t_1 <= 2e+20) {
		tmp = x - ((y * t) / a);
	} else {
		tmp = 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 = (y * (z - t)) / a
    if (t_1 <= (-1d+146)) then
        tmp = t_1
    else if (t_1 <= 2d+20) then
        tmp = x - ((y * t) / a)
    else
        tmp = 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 = (y * (z - t)) / a;
	double tmp;
	if (t_1 <= -1e+146) {
		tmp = t_1;
	} else if (t_1 <= 2e+20) {
		tmp = x - ((y * t) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	tmp = 0
	if t_1 <= -1e+146:
		tmp = t_1
	elif t_1 <= 2e+20:
		tmp = x - ((y * t) / a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	tmp = 0.0
	if (t_1 <= -1e+146)
		tmp = t_1;
	elseif (t_1 <= 2e+20)
		tmp = Float64(x - Float64(Float64(y * t) / a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (z - t)) / a;
	tmp = 0.0;
	if (t_1 <= -1e+146)
		tmp = t_1;
	elseif (t_1 <= 2e+20)
		tmp = x - ((y * t) / a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+146], t$95$1, If[LessEqual[t$95$1, 2e+20], N[(x - N[(N[(y * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -9.99999999999999934e145 or 2e20 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 91.3%

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

   3. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 86.5

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

   5. Simplified 86.5%

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

   if -9.99999999999999934e145 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2e20

   1. Initial program 98.2%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 88.0

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

   5. Simplified 88.0%

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

Alternative 3: 59.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ t_2 := z \cdot \frac{y}{a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+130}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+20}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- z t)) a)) (t_2 (* z (/ y a))))
   (if (<= t_1 -2e+130) t_2 (if (<= t_1 2e+20) x t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = z * (y / a);
	double tmp;
	if (t_1 <= -2e+130) {
		tmp = t_2;
	} else if (t_1 <= 2e+20) {
		tmp = x;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (y * (z - t)) / a
    t_2 = z * (y / a)
    if (t_1 <= (-2d+130)) then
        tmp = t_2
    else if (t_1 <= 2d+20) then
        tmp = x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (z - t)) / a;
	double t_2 = z * (y / a);
	double tmp;
	if (t_1 <= -2e+130) {
		tmp = t_2;
	} else if (t_1 <= 2e+20) {
		tmp = x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (z - t)) / a
	t_2 = z * (y / a)
	tmp = 0
	if t_1 <= -2e+130:
		tmp = t_2
	elif t_1 <= 2e+20:
		tmp = x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / a)
	t_2 = Float64(z * Float64(y / a))
	tmp = 0.0
	if (t_1 <= -2e+130)
		tmp = t_2;
	elseif (t_1 <= 2e+20)
		tmp = x;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (z - t)) / a;
	t_2 = z * (y / a);
	tmp = 0.0;
	if (t_1 <= -2e+130)
		tmp = t_2;
	elseif (t_1 <= 2e+20)
		tmp = x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+130], t$95$2, If[LessEqual[t$95$1, 2e+20], x, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -2.0000000000000001e130 or 2e20 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 91.3%

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

   3. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 46.9

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

   5. Simplified 46.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 50.7

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

   7. Applied egg-rr 50.7%

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

   if -2.0000000000000001e130 < (/.f64 (*.f64 y (-.f64 z t)) a) < 2e20

   1. Initial program 98.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 74.6%

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

Alternative 4: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{if}\;z \leq -6.6 \cdot 10^{+83}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, -t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y a) z x)))
   (if (<= z -6.6e+83) t_1 (if (<= z 7.5e+64) (fma (/ y a) (- t) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / a), z, x);
	double tmp;
	if (z <= -6.6e+83) {
		tmp = t_1;
	} else if (z <= 7.5e+64) {
		tmp = fma((y / a), -t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y / a), z, x)
	tmp = 0.0
	if (z <= -6.6e+83)
		tmp = t_1;
	elseif (z <= 7.5e+64)
		tmp = fma(Float64(y / a), Float64(-t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[z, -6.6e+83], t$95$1, If[LessEqual[z, 7.5e+64], N[(N[(y / a), $MachinePrecision] * (-t) + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.59999999999999969e83 or 7.5000000000000005e64 < z

   1. Initial program 93.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 98.8

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{z}, x\right) \]
   6. Step-by-step derivation

   7. Simplified 90.7%

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

   if -6.59999999999999969e83 < z < 7.5000000000000005e64

   1. Initial program 95.0%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 94.2

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

   4. Applied egg-rr 94.2%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 85.0

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

   7. Simplified 85.0%

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

Alternative 5: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+64}:\\ \;\;\;\;x - \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y a) z x)))
   (if (<= z -3.2e+72) t_1 (if (<= z 1.85e+64) (- x (/ (* y t) a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / a), z, x);
	double tmp;
	if (z <= -3.2e+72) {
		tmp = t_1;
	} else if (z <= 1.85e+64) {
		tmp = x - ((y * t) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y / a), z, x)
	tmp = 0.0
	if (z <= -3.2e+72)
		tmp = t_1;
	elseif (z <= 1.85e+64)
		tmp = Float64(x - Float64(Float64(y * t) / a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[z, -3.2e+72], t$95$1, If[LessEqual[z, 1.85e+64], N[(x - N[(N[(y * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.2000000000000001e72 or 1.84999999999999992e64 < z

   1. Initial program 93.1%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 98.8

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{z}, x\right) \]
   6. Step-by-step derivation

   7. Simplified 90.7%

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

   if -3.2000000000000001e72 < z < 1.84999999999999992e64

   1. Initial program 95.0%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 84.7

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

   5. Simplified 84.7%

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

Alternative 6: 77.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{+200}:\\ \;\;\;\;\frac{y \cdot t}{-a}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.2e+200)
   (/ (* y t) (- a))
   (if (<= t 7.2e+189) (fma (/ y a) z x) (* t (/ y (- a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.2e+200) {
		tmp = (y * t) / -a;
	} else if (t <= 7.2e+189) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = t * (y / -a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.2e+200)
		tmp = Float64(Float64(y * t) / Float64(-a));
	elseif (t <= 7.2e+189)
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = Float64(t * Float64(y / Float64(-a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.2e+200], N[(N[(y * t), $MachinePrecision] / (-a)), $MachinePrecision], If[LessEqual[t, 7.2e+189], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], N[(t * N[(y / (-a)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.2e200

   1. Initial program 95.6%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. /-lowering-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 87.3

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

   5. Simplified 87.3%

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-out N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. *-lowering-*.f64 N/A

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

      7. neg-lowering-neg.f64 91.3

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

   7. Applied egg-rr 91.3%

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

   if -1.2e200 < t < 7.20000000000000017e189

   1. Initial program 94.4%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 96.5

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

   4. Applied egg-rr 96.5%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{z}, x\right) \]
   6. Step-by-step derivation

   7. Simplified 79.7%

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

   if 7.20000000000000017e189 < t

   1. Initial program 92.3%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. /-lowering-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 73.0

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

   5. Simplified 73.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{+200}:\\ \;\;\;\;\frac{y \cdot t}{-a}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{y}{-a}\\ \mathbf{if}\;t \leq -1.2 \cdot 10^{+203}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (/ y (- a)))))
   (if (<= t -1.2e+203) t_1 (if (<= t 7.5e+189) (fma (/ y a) z x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (y / -a);
	double tmp;
	if (t <= -1.2e+203) {
		tmp = t_1;
	} else if (t <= 7.5e+189) {
		tmp = fma((y / a), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(y / Float64(-a)))
	tmp = 0.0
	if (t <= -1.2e+203)
		tmp = t_1;
	elseif (t <= 7.5e+189)
		tmp = fma(Float64(y / a), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(y / (-a)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.2e+203], t$95$1, If[LessEqual[t, 7.5e+189], N[(N[(y / a), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.2000000000000001e203 or 7.49999999999999955e189 < t

   1. Initial program 93.9%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. /-lowering-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. neg-lowering-neg.f64 79.8

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

   5. Simplified 79.8%

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

   if -1.2000000000000001e203 < t < 7.49999999999999955e189

   1. Initial program 94.4%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 96.5

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

   4. Applied egg-rr 96.5%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{z}, x\right) \]
   6. Step-by-step derivation

   7. Simplified 79.7%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{+203}:\\ \;\;\;\;t \cdot \frac{y}{-a}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+50}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y 2e+50) (+ x (/ (* y (- z t)) a)) (fma (/ (- z t) a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= 2e+50) {
		tmp = x + ((y * (z - t)) / a);
	} else {
		tmp = fma(((z - t) / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= 2e+50)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / a));
	else
		tmp = fma(Float64(Float64(z - t) / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, 2e+50], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.0000000000000002e50

   1. Initial program 97.5%

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

   if 2.0000000000000002e50 < y

   1. Initial program 82.7%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

Alternative 9: 97.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. --lowering--.f64 96.0

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

4. Applied egg-rr 96.0%

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

Alternative 10: 71.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. --lowering--.f64 96.0

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

4. Applied egg-rr 96.0%

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

5. Taylor expanded in z around inf

   \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{z}, x\right) \]
6. Step-by-step derivation

7. Simplified 68.6%

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

Alternative 11: 68.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

3. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. /-lowering-/.f64 66.1

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

5. Simplified 66.1%

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

Alternative 12: 39.5% accurate, 23.0× 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 94.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 36.5%

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

Developer Target 1: 99.3% accurate, 0.5× 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 2024205 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -430450648655599/4000000000000000000000000) (+ x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2894426862792089/10000000000000000000000000000000000000000000000000000000000000000) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t)))))))

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