Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1

Percentage Accurate: 97.8% → 97.8%

Time: 7.6s

Alternatives: 10

Speedup: 1.1×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. div-sub N/A

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

   4. lower--.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 97.7

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

4. Applied rewrites 97.7%

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

Alternative 2: 94.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-15} \lor \neg \left(\frac{x}{y} \leq 2 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ x y) -2e-15) (not (<= (/ x y) 2e-24)))
   (* (/ x y) (- z t))
   (fma (/ z y) x t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x / y) <= -2e-15) || !((x / y) <= 2e-24)) {
		tmp = (x / y) * (z - t);
	} else {
		tmp = fma((z / y), x, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x / y) <= -2e-15) || !(Float64(x / y) <= 2e-24))
		tmp = Float64(Float64(x / y) * Float64(z - t));
	else
		tmp = fma(Float64(z / y), x, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x / y), $MachinePrecision], -2e-15], N[Not[LessEqual[N[(x / y), $MachinePrecision], 2e-24]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision], N[(N[(z / y), $MachinePrecision] * x + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -2.0000000000000002e-15 or 1.99999999999999985e-24 < (/.f64 x y)

   1. Initial program 96.9%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 64.7

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

   5. Applied rewrites 64.7%

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

   6. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 94.6

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

   8. Applied rewrites 94.6%

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

   10. Applied rewrites 96.0%

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

   if -2.0000000000000002e-15 < (/.f64 x y) < 1.99999999999999985e-24

   1. Initial program 98.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 94.7

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

   4. Applied rewrites 94.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 96.2

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

   7. Applied rewrites 96.2%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-15} \lor \neg \left(\frac{x}{y} \leq 2 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\frac{z}{y} \cdot x + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -2e-15)
   (/ (* (- z t) x) y)
   (if (<= (/ x y) 2e-24) (+ (* (/ z y) x) t) (* (/ x y) (- z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -2e-15) {
		tmp = ((z - t) * x) / y;
	} else if ((x / y) <= 2e-24) {
		tmp = ((z / y) * x) + t;
	} else {
		tmp = (x / y) * (z - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x / y) <= (-2d-15)) then
        tmp = ((z - t) * x) / y
    else if ((x / y) <= 2d-24) then
        tmp = ((z / y) * x) + t
    else
        tmp = (x / y) * (z - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -2e-15) {
		tmp = ((z - t) * x) / y;
	} else if ((x / y) <= 2e-24) {
		tmp = ((z / y) * x) + t;
	} else {
		tmp = (x / y) * (z - t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= -2e-15:
		tmp = ((z - t) * x) / y
	elif (x / y) <= 2e-24:
		tmp = ((z / y) * x) + t
	else:
		tmp = (x / y) * (z - t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -2e-15)
		tmp = Float64(Float64(Float64(z - t) * x) / y);
	elseif (Float64(x / y) <= 2e-24)
		tmp = Float64(Float64(Float64(z / y) * x) + t);
	else
		tmp = Float64(Float64(x / y) * Float64(z - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x / y) <= -2e-15)
		tmp = ((z - t) * x) / y;
	elseif ((x / y) <= 2e-24)
		tmp = ((z / y) * x) + t;
	else
		tmp = (x / y) * (z - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -2e-15], N[(N[(N[(z - t), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 2e-24], N[(N[(N[(z / y), $MachinePrecision] * x), $MachinePrecision] + t), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -2.0000000000000002e-15

   1. Initial program 95.2%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 96.7

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

   8. Applied rewrites 96.7%

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

   if -2.0000000000000002e-15 < (/.f64 x y) < 1.99999999999999985e-24

   1. Initial program 98.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 94.7

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

   4. Applied rewrites 94.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 96.2

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

   7. Applied rewrites 96.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-+.f64 96.2

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

   9. Applied rewrites 96.2%

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

   if 1.99999999999999985e-24 < (/.f64 x y)

   1. Initial program 98.5%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 92.7

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

   8. Applied rewrites 92.7%

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

   10. Applied rewrites 96.7%

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

Alternative 4: 93.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -2e-15)
   (/ (* (- z t) x) y)
   (if (<= (/ x y) 2e-24) (fma (/ z y) x t) (* (/ x y) (- z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -2e-15) {
		tmp = ((z - t) * x) / y;
	} else if ((x / y) <= 2e-24) {
		tmp = fma((z / y), x, t);
	} else {
		tmp = (x / y) * (z - t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -2e-15)
		tmp = Float64(Float64(Float64(z - t) * x) / y);
	elseif (Float64(x / y) <= 2e-24)
		tmp = fma(Float64(z / y), x, t);
	else
		tmp = Float64(Float64(x / y) * Float64(z - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -2e-15], N[(N[(N[(z - t), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 2e-24], N[(N[(z / y), $MachinePrecision] * x + t), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -2.0000000000000002e-15

   1. Initial program 95.2%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 96.7

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

   8. Applied rewrites 96.7%

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

   if -2.0000000000000002e-15 < (/.f64 x y) < 1.99999999999999985e-24

   1. Initial program 98.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 94.7

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

   4. Applied rewrites 94.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 96.2

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

   7. Applied rewrites 96.2%

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

   if 1.99999999999999985e-24 < (/.f64 x y)

   1. Initial program 98.5%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 56.0

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 92.7

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

   8. Applied rewrites 92.7%

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

   10. Applied rewrites 96.7%

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

Alternative 5: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{y} \cdot z\\ \mathbf{elif}\;\frac{x}{y} \leq 4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -2e-15)
   (* (/ x y) z)
   (if (<= (/ x y) 4e+154) (fma (/ z y) x t) (* (/ (- t) y) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -2e-15) {
		tmp = (x / y) * z;
	} else if ((x / y) <= 4e+154) {
		tmp = fma((z / y), x, t);
	} else {
		tmp = (-t / y) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -2e-15)
		tmp = Float64(Float64(x / y) * z);
	elseif (Float64(x / y) <= 4e+154)
		tmp = fma(Float64(z / y), x, t);
	else
		tmp = Float64(Float64(Float64(-t) / y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -2e-15], N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 4e+154], N[(N[(z / y), $MachinePrecision] * x + t), $MachinePrecision], N[(N[((-t) / y), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -2.0000000000000002e-15

   1. Initial program 95.2%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 74.3

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

   5. Applied rewrites 74.3%

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

   if -2.0000000000000002e-15 < (/.f64 x y) < 4.00000000000000015e154

   1. Initial program 98.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 93.9

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

   4. Applied rewrites 93.9%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 88.8

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

   7. Applied rewrites 88.8%

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

   if 4.00000000000000015e154 < (/.f64 x y)

   1. Initial program 97.3%

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

   3. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 62.0%

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

Alternative 6: 73.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -2e-15) (* (/ x y) z) (fma (/ z y) x t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -2e-15) {
		tmp = (x / y) * z;
	} else {
		tmp = fma((z / y), x, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -2e-15)
		tmp = Float64(Float64(x / y) * z);
	else
		tmp = fma(Float64(z / y), x, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -2e-15], N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision], N[(N[(z / y), $MachinePrecision] * x + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -2.0000000000000002e-15

   1. Initial program 95.2%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 74.3

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

   5. Applied rewrites 74.3%

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

   if -2.0000000000000002e-15 < (/.f64 x y)

   1. Initial program 98.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 95.0

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 80.9

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

   7. Applied rewrites 80.9%

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

Alternative 7: 83.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+127} \lor \neg \left(t \leq 3.5 \cdot 10^{+97}\right):\\ \;\;\;\;\left(1 - \frac{x}{y}\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1e+127) (not (<= t 3.5e+97)))
   (* (- 1.0 (/ x y)) t)
   (fma (/ z y) x t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1e+127) || !(t <= 3.5e+97)) {
		tmp = (1.0 - (x / y)) * t;
	} else {
		tmp = fma((z / y), x, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1e+127) || !(t <= 3.5e+97))
		tmp = Float64(Float64(1.0 - Float64(x / y)) * t);
	else
		tmp = fma(Float64(z / y), x, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1e+127], N[Not[LessEqual[t, 3.5e+97]], $MachinePrecision]], N[(N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[(z / y), $MachinePrecision] * x + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.99999999999999955e126 or 3.5000000000000001e97 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 95.1

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

   5. Applied rewrites 95.1%

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

   if -9.99999999999999955e126 < t < 3.5000000000000001e97

   1. Initial program 96.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 93.9

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

   4. Applied rewrites 93.9%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 82.5

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

   7. Applied rewrites 82.5%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+127} \lor \neg \left(t \leq 3.5 \cdot 10^{+97}\right):\\ \;\;\;\;\left(1 - \frac{x}{y}\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 97.7

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

4. Applied rewrites 97.7%

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

Alternative 9: 40.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in z around inf

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 41.9

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

5. Applied rewrites 41.9%

   \[\leadsto \color{blue}{\frac{x}{y} \cdot z} \]
6. Add Preprocessing

Alternative 10: 37.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in x around inf

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

   1. div-sub N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 55.2

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

5. Applied rewrites 55.2%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 38.5%

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

Developer Target 1: 97.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{if}\;z < 2.759456554562692 \cdot 10^{-282}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* (/ x y) (- z t)) t)))
   (if (< z 2.759456554562692e-282)
     t_1
     (if (< z 2.326994450874436e-110) (+ (* x (/ (- z t) y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x / y) * (z - t)) + t;
	double tmp;
	if (z < 2.759456554562692e-282) {
		tmp = t_1;
	} else if (z < 2.326994450874436e-110) {
		tmp = (x * ((z - t) / 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 = ((x / y) * (z - t)) + t
    if (z < 2.759456554562692d-282) then
        tmp = t_1
    else if (z < 2.326994450874436d-110) then
        tmp = (x * ((z - t) / 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 = ((x / y) * (z - t)) + t;
	double tmp;
	if (z < 2.759456554562692e-282) {
		tmp = t_1;
	} else if (z < 2.326994450874436e-110) {
		tmp = (x * ((z - t) / y)) + t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((x / y) * (z - t)) + t
	tmp = 0
	if z < 2.759456554562692e-282:
		tmp = t_1
	elif z < 2.326994450874436e-110:
		tmp = (x * ((z - t) / y)) + t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x / y) * Float64(z - t)) + t)
	tmp = 0.0
	if (z < 2.759456554562692e-282)
		tmp = t_1;
	elseif (z < 2.326994450874436e-110)
		tmp = Float64(Float64(x * Float64(Float64(z - t) / y)) + t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((x / y) * (z - t)) + t;
	tmp = 0.0;
	if (z < 2.759456554562692e-282)
		tmp = t_1;
	elseif (z < 2.326994450874436e-110)
		tmp = (x * ((z - t) / y)) + t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, If[Less[z, 2.759456554562692e-282], t$95$1, If[Less[z, 2.326994450874436e-110], N[(N[(x * N[(N[(z - t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024339 
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z 689864138640673/250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (* (/ x y) (- z t)) t) (if (< z 581748612718609/25000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (* x (/ (- z t) y)) t) (+ (* (/ x y) (- z t)) t))))

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