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

Percentage Accurate: 92.3% → 98.0%

Time: 8.9s

Alternatives: 9

Speedup: 1.1×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.3% 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: 98.0% 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 92.4%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. remove-double-neg N/A

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

   4. remove-double-neg N/A

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

   5. lift-/.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 98.1

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

4. Applied rewrites 98.1%

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

Alternative 2: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{-38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{x}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma z (/ y t) x)))
   (if (<= z -4.2e-38) t_1 (if (<= z 1.7e-5) (fma (- (/ x t)) y x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(z, (y / t), x);
	double tmp;
	if (z <= -4.2e-38) {
		tmp = t_1;
	} else if (z <= 1.7e-5) {
		tmp = fma(-(x / t), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(z, Float64(y / t), x)
	tmp = 0.0
	if (z <= -4.2e-38)
		tmp = t_1;
	elseif (z <= 1.7e-5)
		tmp = fma(Float64(-Float64(x / t)), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -4.2e-38], t$95$1, If[LessEqual[z, 1.7e-5], N[((-N[(x / t), $MachinePrecision]) * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.20000000000000026e-38 or 1.7e-5 < z

   1. Initial program 94.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. lower-fma.f64 92.6

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

   7. Applied rewrites 92.6%

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

   if -4.20000000000000026e-38 < z < 1.7e-5

   1. Initial program 90.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.4

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

   7. Applied rewrites 90.4%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{x}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{-37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma z (/ y t) x)))
   (if (<= z -1.35e-37) t_1 (if (<= z 2.5e-5) (fma (/ y t) (- x) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(z, (y / t), x);
	double tmp;
	if (z <= -1.35e-37) {
		tmp = t_1;
	} else if (z <= 2.5e-5) {
		tmp = fma((y / t), -x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(z, Float64(y / t), x)
	tmp = 0.0
	if (z <= -1.35e-37)
		tmp = t_1;
	elseif (z <= 2.5e-5)
		tmp = fma(Float64(y / t), Float64(-x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.35e-37], t$95$1, If[LessEqual[z, 2.5e-5], N[(N[(y / t), $MachinePrecision] * (-x) + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.35000000000000008e-37 or 2.50000000000000012e-5 < z

   1. Initial program 94.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. lower-fma.f64 92.6

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

   7. Applied rewrites 92.6%

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

   if -1.35000000000000008e-37 < z < 2.50000000000000012e-5

   1. Initial program 90.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 96.1

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.4

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

   7. Applied rewrites 90.4%

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

Alternative 4: 81.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{-38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{t} \cdot \left(t - y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma z (/ y t) x)))
   (if (<= z -4.2e-38) t_1 (if (<= z 1.7e-5) (* (/ x t) (- t y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(z, (y / t), x);
	double tmp;
	if (z <= -4.2e-38) {
		tmp = t_1;
	} else if (z <= 1.7e-5) {
		tmp = (x / t) * (t - y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(z, Float64(y / t), x)
	tmp = 0.0
	if (z <= -4.2e-38)
		tmp = t_1;
	elseif (z <= 1.7e-5)
		tmp = Float64(Float64(x / t) * Float64(t - y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -4.2e-38], t$95$1, If[LessEqual[z, 1.7e-5], N[(N[(x / t), $MachinePrecision] * N[(t - y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.20000000000000026e-38 or 1.7e-5 < z

   1. Initial program 94.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. lower-fma.f64 92.6

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

   7. Applied rewrites 92.6%

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

   if -4.20000000000000026e-38 < z < 1.7e-5

   1. Initial program 90.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.4

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

   7. Applied rewrites 90.4%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out-- N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 72.7

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

   10. Applied rewrites 72.7%

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

       1. lift--.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. clear-num N/A

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

       4. associate-/r/ N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. associate-*l/ N/A

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

       8. *-lft-identity N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 84.2

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

   12. Applied rewrites 84.2%

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

Alternative 5: 84.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - x}{t}\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (/ (- z x) t))))
   (if (<= y -5.5e+34) t_1 (if (<= y 2.65e+74) (fma z (/ y t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * ((z - x) / t);
	double tmp;
	if (y <= -5.5e+34) {
		tmp = t_1;
	} else if (y <= 2.65e+74) {
		tmp = fma(z, (y / t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(Float64(z - x) / t))
	tmp = 0.0
	if (y <= -5.5e+34)
		tmp = t_1;
	elseif (y <= 2.65e+74)
		tmp = fma(z, Float64(y / t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.5e+34], t$95$1, If[LessEqual[y, 2.65e+74], N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.4999999999999996e34 or 2.6499999999999999e74 < y

   1. Initial program 86.6%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 86.7

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

   5. Applied rewrites 86.7%

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

   if -5.4999999999999996e34 < y < 2.6499999999999999e74

   1. Initial program 97.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 84.2

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

   5. Applied rewrites 84.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. lower-fma.f64 86.3

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

   7. Applied rewrites 86.3%

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

Alternative 6: 73.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \mathbf{if}\;z \leq 9 \cdot 10^{-280}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-109}:\\ \;\;\;\;-y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma z (/ y t) x)))
   (if (<= z 9e-280) t_1 (if (<= z 4e-109) (- (* y (/ x t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(z, (y / t), x);
	double tmp;
	if (z <= 9e-280) {
		tmp = t_1;
	} else if (z <= 4e-109) {
		tmp = -(y * (x / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(z, Float64(y / t), x)
	tmp = 0.0
	if (z <= 9e-280)
		tmp = t_1;
	elseif (z <= 4e-109)
		tmp = Float64(-Float64(y * Float64(x / t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, 9e-280], t$95$1, If[LessEqual[z, 4e-109], (-N[(y * N[(x / t), $MachinePrecision]), $MachinePrecision]), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < 8.9999999999999991e-280 or 4e-109 < z

   1. Initial program 92.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 79.5

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

   5. Applied rewrites 79.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. lower-fma.f64 86.4

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

   7. Applied rewrites 86.4%

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

   if 8.9999999999999991e-280 < z < 4e-109

   1. Initial program 94.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 90.3

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 68.1

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

   7. Applied rewrites 68.1%

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

   8. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 57.8

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

   10. Applied rewrites 57.8%

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

       1. lift-neg.f64 N/A

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

       2. associate-/l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 61.6

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

   12. Applied rewrites 61.6%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9 \cdot 10^{-280}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-109}:\\ \;\;\;\;-y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t \cdot x}{t}\\ \mathbf{if}\;t \leq -1.95 \cdot 10^{+71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+59}:\\ \;\;\;\;\frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* t x) t)))
   (if (<= t -1.95e+71) t_1 (if (<= t 1.05e+59) (* (/ y t) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t * x) / t;
	double tmp;
	if (t <= -1.95e+71) {
		tmp = t_1;
	} else if (t <= 1.05e+59) {
		tmp = (y / t) * z;
	} 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 = (t * x) / t
    if (t <= (-1.95d+71)) then
        tmp = t_1
    else if (t <= 1.05d+59) then
        tmp = (y / t) * z
    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 = (t * x) / t;
	double tmp;
	if (t <= -1.95e+71) {
		tmp = t_1;
	} else if (t <= 1.05e+59) {
		tmp = (y / t) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t * x) / t
	tmp = 0
	if t <= -1.95e+71:
		tmp = t_1
	elif t <= 1.05e+59:
		tmp = (y / t) * z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t * x) / t)
	tmp = 0.0
	if (t <= -1.95e+71)
		tmp = t_1;
	elseif (t <= 1.05e+59)
		tmp = Float64(Float64(y / t) * z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t * x) / t;
	tmp = 0.0;
	if (t <= -1.95e+71)
		tmp = t_1;
	elseif (t <= 1.05e+59)
		tmp = (y / t) * z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * x), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[t, -1.95e+71], t$95$1, If[LessEqual[t, 1.05e+59], N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.9500000000000001e71 or 1.04999999999999992e59 < t

   1. Initial program 83.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-neg N/A

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

      4. remove-double-neg N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 85.2

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

   7. Applied rewrites 85.2%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-out-- N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 48.5

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

   10. Applied rewrites 48.5%

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

   11. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 46.2

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

   13. Applied rewrites 46.2%

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

   if -1.9500000000000001e71 < t < 1.04999999999999992e59

   1. Initial program 97.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 49.8

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

   5. Applied rewrites 49.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lift-/.f64 N/A

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

      4. lower-*.f64 56.1

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

   7. Applied rewrites 56.1%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.95 \cdot 10^{+71}:\\ \;\;\;\;\frac{t \cdot x}{t}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+59}:\\ \;\;\;\;\frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

3. Taylor expanded in z around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 72.5

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

5. Applied rewrites 72.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. lift-/.f64 N/A

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

   9. lower-fma.f64 78.1

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

7. Applied rewrites 78.1%

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

Alternative 9: 41.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.4%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 37.1

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

5. Applied rewrites 37.1%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. lift-/.f64 N/A

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

   4. lower-*.f64 42.2

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

7. Applied rewrites 42.2%

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

8. Final simplification 42.2%

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

Developer Target 1: 91.4% 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 2024219 
(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)))