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

Percentage Accurate: 92.9% → 97.6%

Time: 8.0s

Alternatives: 13

Speedup: 1.1×

• 25.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 97.9

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

4. Applied rewrites 97.9%

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

Alternative 2: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z - x\right) \cdot y}{t} + x\\ t_2 := \frac{z - x}{t} \cdot y\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+281}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ (* (- z x) y) t) x)) (t_2 (* (/ (- z x) t) y)))
   (if (<= t_1 -1e+281) t_2 (if (<= t_1 2e+305) (+ (/ (* z y) t) x) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (((z - x) * y) / t) + x;
	double t_2 = ((z - x) / t) * y;
	double tmp;
	if (t_1 <= -1e+281) {
		tmp = t_2;
	} else if (t_1 <= 2e+305) {
		tmp = ((z * y) / t) + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (((z - x) * y) / t) + x
    t_2 = ((z - x) / t) * y
    if (t_1 <= (-1d+281)) then
        tmp = t_2
    else if (t_1 <= 2d+305) then
        tmp = ((z * y) / t) + 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 t_1 = (((z - x) * y) / t) + x;
	double t_2 = ((z - x) / t) * y;
	double tmp;
	if (t_1 <= -1e+281) {
		tmp = t_2;
	} else if (t_1 <= 2e+305) {
		tmp = ((z * y) / t) + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (((z - x) * y) / t) + x
	t_2 = ((z - x) / t) * y
	tmp = 0
	if t_1 <= -1e+281:
		tmp = t_2
	elif t_1 <= 2e+305:
		tmp = ((z * y) / t) + x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(z - x) * y) / t) + x)
	t_2 = Float64(Float64(Float64(z - x) / t) * y)
	tmp = 0.0
	if (t_1 <= -1e+281)
		tmp = t_2;
	elseif (t_1 <= 2e+305)
		tmp = Float64(Float64(Float64(z * y) / t) + x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (((z - x) * y) / t) + x;
	t_2 = ((z - x) / t) * y;
	tmp = 0.0;
	if (t_1 <= -1e+281)
		tmp = t_2;
	elseif (t_1 <= 2e+305)
		tmp = ((z * y) / t) + x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(z - x), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+281], t$95$2, If[LessEqual[t$95$1, 2e+305], N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < -1e281 or 1.9999999999999999e305 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t))

   1. Initial program 83.4%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 82.4

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

   5. Applied rewrites 82.4%

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

   7. Applied rewrites 95.6%

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

   if -1e281 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < 1.9999999999999999e305

   1. Initial program 98.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - x\right) \cdot y}{t} + x \leq -1 \cdot 10^{+281}:\\ \;\;\;\;\frac{z - x}{t} \cdot y\\ \mathbf{elif}\;\frac{\left(z - x\right) \cdot y}{t} + x \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{z - x}{t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z - x\right) \cdot y}{t} + x\\ t_2 := \frac{z - x}{t} \cdot y\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+281}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ (* (- z x) y) t) x)) (t_2 (* (/ (- z x) t) y)))
   (if (<= t_1 -1e+281) t_2 (if (<= t_1 2e+305) (fma (/ z t) y x) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (((z - x) * y) / t) + x;
	double t_2 = ((z - x) / t) * y;
	double tmp;
	if (t_1 <= -1e+281) {
		tmp = t_2;
	} else if (t_1 <= 2e+305) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(z - x) * y) / t) + x)
	t_2 = Float64(Float64(Float64(z - x) / t) * y)
	tmp = 0.0
	if (t_1 <= -1e+281)
		tmp = t_2;
	elseif (t_1 <= 2e+305)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(z - x), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+281], t$95$2, If[LessEqual[t$95$1, 2e+305], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < -1e281 or 1.9999999999999999e305 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t))

   1. Initial program 83.4%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 82.4

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

   5. Applied rewrites 82.4%

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

   7. Applied rewrites 95.6%

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

   if -1e281 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < 1.9999999999999999e305

   1. Initial program 98.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 84.6

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 74.5

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

   7. Applied rewrites 74.5%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - x\right) \cdot y}{t} + x \leq -1 \cdot 10^{+281}:\\ \;\;\;\;\frac{z - x}{t} \cdot y\\ \mathbf{elif}\;\frac{\left(z - x\right) \cdot y}{t} + x \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - x}{t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 39.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(z - x\right) \cdot y}{t} + x \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (/ (* (- z x) y) t) x) 2e+282) (/ (* z y) t) (* (/ z t) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((((z - x) * y) / t) + x) <= 2e+282) {
		tmp = (z * y) / t;
	} else {
		tmp = (z / t) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((((z - x) * y) / t) + x) <= 2d+282) then
        tmp = (z * y) / t
    else
        tmp = (z / t) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((((z - x) * y) / t) + x) <= 2e+282) {
		tmp = (z * y) / t;
	} else {
		tmp = (z / t) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((((z - x) * y) / t) + x) <= 2e+282:
		tmp = (z * y) / t
	else:
		tmp = (z / t) * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(z - x) * y) / t) + x) <= 2e+282)
		tmp = Float64(Float64(z * y) / t);
	else
		tmp = Float64(Float64(z / t) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((((z - x) * y) / t) + x) <= 2e+282)
		tmp = (z * y) / t;
	else
		tmp = (z / t) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[(N[(z - x), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], 2e+282], N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < 2.00000000000000007e282

   1. Initial program 96.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 37.5

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

   5. Applied rewrites 37.5%

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

   if 2.00000000000000007e282 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t))

   1. Initial program 80.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.4

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

   5. Applied rewrites 41.4%

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

   7. Applied rewrites 53.4%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - x\right) \cdot y}{t} + x \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{y}{t} + x\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-48}:\\ \;\;\;\;x - \frac{x \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* z (/ y t)) x)))
   (if (<= z -2.7e-22) t_1 (if (<= z 2.1e-48) (- x (/ (* x y) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * (y / t)) + x;
	double tmp;
	if (z <= -2.7e-22) {
		tmp = t_1;
	} else if (z <= 2.1e-48) {
		tmp = x - ((x * y) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * (y / t)) + x
    if (z <= (-2.7d-22)) then
        tmp = t_1
    else if (z <= 2.1d-48) then
        tmp = x - ((x * y) / t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * (y / t)) + x;
	double tmp;
	if (z <= -2.7e-22) {
		tmp = t_1;
	} else if (z <= 2.1e-48) {
		tmp = x - ((x * y) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * (y / t)) + x
	tmp = 0
	if z <= -2.7e-22:
		tmp = t_1
	elif z <= 2.1e-48:
		tmp = x - ((x * y) / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * Float64(y / t)) + x)
	tmp = 0.0
	if (z <= -2.7e-22)
		tmp = t_1;
	elseif (z <= 2.1e-48)
		tmp = Float64(x - Float64(Float64(x * y) / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * (y / t)) + x;
	tmp = 0.0;
	if (z <= -2.7e-22)
		tmp = t_1;
	elseif (z <= 2.1e-48)
		tmp = x - ((x * y) / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -2.7e-22], t$95$1, If[LessEqual[z, 2.1e-48], N[(x - N[(N[(x * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.7000000000000002e-22 or 2.09999999999999989e-48 < z

   1. Initial program 89.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 87.0

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

   4. Applied rewrites 87.0%

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

   5. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 87.3

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

   7. Applied rewrites 87.3%

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

   if -2.7000000000000002e-22 < z < 2.09999999999999989e-48

   1. Initial program 97.7%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 87.7

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

   5. Applied rewrites 87.7%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-22}:\\ \;\;\;\;z \cdot \frac{y}{t} + x\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-48}:\\ \;\;\;\;x - \frac{x \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{t} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{if}\;t \leq -6 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\left(z - x\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ z t) y x)))
   (if (<= t -6e-55) t_1 (if (<= t 3.5e-40) (/ (* (- z x) y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z / t), y, x);
	double tmp;
	if (t <= -6e-55) {
		tmp = t_1;
	} else if (t <= 3.5e-40) {
		tmp = ((z - x) * y) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(z / t), y, x)
	tmp = 0.0
	if (t <= -6e-55)
		tmp = t_1;
	elseif (t <= 3.5e-40)
		tmp = Float64(Float64(Float64(z - x) * y) / t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -6e-55], t$95$1, If[LessEqual[t, 3.5e-40], N[(N[(N[(z - x), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.00000000000000033e-55 or 3.5000000000000002e-40 < t

   1. Initial program 90.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 96.9

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 84.9

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

   7. Applied rewrites 84.9%

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

   if -6.00000000000000033e-55 < t < 3.5000000000000002e-40

   1. Initial program 97.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 86.0

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

   5. Applied rewrites 86.0%

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

Alternative 7: 82.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-48}:\\ \;\;\;\;x - \frac{x \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ z t) y x)))
   (if (<= z -2.7e-22) t_1 (if (<= z 2.1e-48) (- x (/ (* x y) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z / t), y, x);
	double tmp;
	if (z <= -2.7e-22) {
		tmp = t_1;
	} else if (z <= 2.1e-48) {
		tmp = x - ((x * y) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(z / t), y, x)
	tmp = 0.0
	if (z <= -2.7e-22)
		tmp = t_1;
	elseif (z <= 2.1e-48)
		tmp = Float64(x - Float64(Float64(x * y) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[z, -2.7e-22], t$95$1, If[LessEqual[z, 2.1e-48], N[(x - N[(N[(x * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.7000000000000002e-22 or 2.09999999999999989e-48 < z

   1. Initial program 89.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 87.0

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

   4. Applied rewrites 87.0%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 78.2

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

   7. Applied rewrites 78.2%

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

   if -2.7000000000000002e-22 < z < 2.09999999999999989e-48

   1. Initial program 97.7%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 87.7

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

   5. Applied rewrites 87.7%

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

Alternative 8: 70.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-290}:\\ \;\;\;\;\frac{\left(-x\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ z t) y x)))
   (if (<= t -2.7e-55) t_1 (if (<= t 3.5e-290) (/ (* (- x) y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z / t), y, x);
	double tmp;
	if (t <= -2.7e-55) {
		tmp = t_1;
	} else if (t <= 3.5e-290) {
		tmp = (-x * y) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(z / t), y, x)
	tmp = 0.0
	if (t <= -2.7e-55)
		tmp = t_1;
	elseif (t <= 3.5e-290)
		tmp = Float64(Float64(Float64(-x) * y) / t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -2.7e-55], t$95$1, If[LessEqual[t, 3.5e-290], N[(N[((-x) * y), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.70000000000000004e-55 or 3.49999999999999981e-290 < t

   1. Initial program 92.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 93.0

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

   4. Applied rewrites 93.0%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 78.7

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

   7. Applied rewrites 78.7%

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

   if -2.70000000000000004e-55 < t < 3.49999999999999981e-290

   1. Initial program 97.1%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 63.1%

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

Alternative 9: 70.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-290}:\\ \;\;\;\;\left(-x\right) \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ z t) y x)))
   (if (<= t -2.7e-55) t_1 (if (<= t 3.5e-290) (* (- x) (/ y t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z / t), y, x);
	double tmp;
	if (t <= -2.7e-55) {
		tmp = t_1;
	} else if (t <= 3.5e-290) {
		tmp = -x * (y / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(z / t), y, x)
	tmp = 0.0
	if (t <= -2.7e-55)
		tmp = t_1;
	elseif (t <= 3.5e-290)
		tmp = Float64(Float64(-x) * Float64(y / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -2.7e-55], t$95$1, If[LessEqual[t, 3.5e-290], N[((-x) * N[(y / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.70000000000000004e-55 or 3.49999999999999981e-290 < t

   1. Initial program 92.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 93.0

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

   4. Applied rewrites 93.0%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 78.7

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

   7. Applied rewrites 78.7%

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

   if -2.70000000000000004e-55 < t < 3.49999999999999981e-290

   1. Initial program 97.1%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 63.1%

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

   10. Applied rewrites 61.8%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-290}:\\ \;\;\;\;\left(-x\right) \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-290}:\\ \;\;\;\;\frac{-x}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ z t) y x)))
   (if (<= t -2.7e-55) t_1 (if (<= t 3.5e-290) (* (/ (- x) t) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z / t), y, x);
	double tmp;
	if (t <= -2.7e-55) {
		tmp = t_1;
	} else if (t <= 3.5e-290) {
		tmp = (-x / t) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(z / t), y, x)
	tmp = 0.0
	if (t <= -2.7e-55)
		tmp = t_1;
	elseif (t <= 3.5e-290)
		tmp = Float64(Float64(Float64(-x) / t) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t, -2.7e-55], t$95$1, If[LessEqual[t, 3.5e-290], N[(N[((-x) / t), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.70000000000000004e-55 or 3.49999999999999981e-290 < t

   1. Initial program 92.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 93.0

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

   4. Applied rewrites 93.0%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 78.7

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

   7. Applied rewrites 78.7%

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

   if -2.70000000000000004e-55 < t < 3.49999999999999981e-290

   1. Initial program 97.1%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 56.9%

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

Alternative 11: 72.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 89.3

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

4. Applied rewrites 89.3%

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

5. Taylor expanded in z around inf

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

   1. lower-/.f64 68.6

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

7. Applied rewrites 68.6%

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

Alternative 12: 40.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ z \cdot \frac{y}{t} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 89.3

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

4. Applied rewrites 89.3%

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

5. Taylor expanded in z around inf

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 40.8

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

7. Applied rewrites 40.8%

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

8. Final simplification 40.8%

   \[\leadsto z \cdot \frac{y}{t} \]
9. Add Preprocessing

Alternative 13: 37.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{z}{t} \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.4%

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

3. Taylor expanded in z around inf

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 38.2

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

5. Applied rewrites 38.2%

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

7. Applied rewrites 34.7%

   \[\leadsto \frac{z}{t} \cdot \color{blue}{y} \]
8. Add Preprocessing

Developer Target 1: 90.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024255 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (+ (* x (/ y t)) (* (- z) (/ y t)))))

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