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

Percentage Accurate: 97.8% → 97.8%

Time: 5.8s

Alternatives: 12

Speedup: 1.0×

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

Specification

Math

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

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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

3. Final simplification 98.3%

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

Alternative 2: 72.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+210}:\\ \;\;\;\;\frac{z \cdot x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -2 \cdot 10^{+80}:\\ \;\;\;\;\frac{-x}{y} \cdot t\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -1e+210)
   (/ (* z x) y)
   (if (<= (/ x y) -2e+80)
     (* (/ (- x) y) t)
     (if (<= (/ x y) 1e+156) (fma (/ z y) x t) (* (/ (- t) y) x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -1e+210) {
		tmp = (z * x) / y;
	} else if ((x / y) <= -2e+80) {
		tmp = (-x / y) * t;
	} else if ((x / y) <= 1e+156) {
		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) <= -1e+210)
		tmp = Float64(Float64(z * x) / y);
	elseif (Float64(x / y) <= -2e+80)
		tmp = Float64(Float64(Float64(-x) / y) * t);
	elseif (Float64(x / y) <= 1e+156)
		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], -1e+210], N[(N[(z * x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], -2e+80], N[(N[((-x) / y), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 1e+156], N[(N[(z / y), $MachinePrecision] * x + t), $MachinePrecision], N[(N[((-t) / y), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 x y) < -9.99999999999999927e209

   1. Initial program 89.5%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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 67.2

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

   5. Applied rewrites 67.2%

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

   7. Applied rewrites 67.5%

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

   if -9.99999999999999927e209 < (/.f64 x y) < -2e80

   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. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 70.3

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

   5. Applied rewrites 70.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 70.3%

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

   if -2e80 < (/.f64 x y) < 9.9999999999999998e155

   1. Initial program 99.6%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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 90.3

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

   4. Applied rewrites 90.3%

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

   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 81.7

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

   7. Applied rewrites 81.7%

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

   if 9.9999999999999998e155 < (/.f64 x y)

   1. Initial program 96.2%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 66.6

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 70.2%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+210}:\\ \;\;\;\;\frac{z \cdot x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -2 \cdot 10^{+80}:\\ \;\;\;\;\frac{-x}{y} \cdot t\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+17}:\\ \;\;\;\;\frac{z - t}{y} \cdot x\\ \mathbf{elif}\;\frac{x}{y} \leq 6:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -5e+17)
   (* (/ (- z t) y) x)
   (if (<= (/ x y) 6.0) (fma (/ z y) x t) (/ (* (- z t) x) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -5e+17) {
		tmp = ((z - t) / y) * x;
	} else if ((x / y) <= 6.0) {
		tmp = fma((z / y), x, t);
	} else {
		tmp = ((z - t) * x) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -5e+17)
		tmp = Float64(Float64(Float64(z - t) / y) * x);
	elseif (Float64(x / y) <= 6.0)
		tmp = fma(Float64(z / y), x, t);
	else
		tmp = Float64(Float64(Float64(z - t) * x) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -5e+17], N[(N[(N[(z - t), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 6.0], N[(N[(z / y), $MachinePrecision] * x + t), $MachinePrecision], N[(N[(N[(z - t), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -5e17

   1. Initial program 95.8%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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. 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 90.6

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

   5. Applied rewrites 90.6%

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

   7. Applied rewrites 93.2%

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

   if -5e17 < (/.f64 x y) < 6

   1. Initial program 99.6%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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 92.2

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

   4. Applied rewrites 92.2%

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

   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 93.7

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

   7. Applied rewrites 93.7%

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

   if 6 < (/.f64 x y)

   1. Initial program 98.2%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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. 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 91.0

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

   5. Applied rewrites 91.0%

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

Alternative 4: 92.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{y} \cdot x\\ \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (- z t) y) x)))
   (if (<= (/ x y) -5e+17) t_1 (if (<= (/ x y) 0.01) (fma (/ z y) x t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((z - t) / y) * x;
	double tmp;
	if ((x / y) <= -5e+17) {
		tmp = t_1;
	} else if ((x / y) <= 0.01) {
		tmp = fma((z / y), x, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(z - t) / y) * x)
	tmp = 0.0
	if (Float64(x / y) <= -5e+17)
		tmp = t_1;
	elseif (Float64(x / y) <= 0.01)
		tmp = fma(Float64(z / y), x, t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -5e+17], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 0.01], N[(N[(z / y), $MachinePrecision] * x + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -5e17 or 0.0100000000000000002 < (/.f64 x y)

   1. Initial program 96.9%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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. 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 90.1

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

   5. Applied rewrites 90.1%

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

   7. Applied rewrites 91.7%

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

   if -5e17 < (/.f64 x y) < 0.0100000000000000002

   1. Initial program 99.6%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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 92.2

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

   4. Applied rewrites 92.2%

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

   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 93.6

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

   7. Applied rewrites 93.6%

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

Alternative 5: 64.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{x}{y}\\ \mathbf{if}\;\frac{x}{y} \leq -200000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (/ x y))))
   (if (<= (/ x y) -200000000000.0)
     t_1
     (if (<= (/ x y) 2e-13) (* 1.0 t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (x / y);
	double tmp;
	if ((x / y) <= -200000000000.0) {
		tmp = t_1;
	} else if ((x / y) <= 2e-13) {
		tmp = 1.0 * 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 * (x / y)
    if ((x / y) <= (-200000000000.0d0)) then
        tmp = t_1
    else if ((x / y) <= 2d-13) then
        tmp = 1.0d0 * 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 * (x / y);
	double tmp;
	if ((x / y) <= -200000000000.0) {
		tmp = t_1;
	} else if ((x / y) <= 2e-13) {
		tmp = 1.0 * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * (x / y)
	tmp = 0
	if (x / y) <= -200000000000.0:
		tmp = t_1
	elif (x / y) <= 2e-13:
		tmp = 1.0 * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(x / y))
	tmp = 0.0
	if (Float64(x / y) <= -200000000000.0)
		tmp = t_1;
	elseif (Float64(x / y) <= 2e-13)
		tmp = Float64(1.0 * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = z * (x / y);
	tmp = 0.0;
	if ((x / y) <= -200000000000.0)
		tmp = t_1;
	elseif ((x / y) <= 2e-13)
		tmp = 1.0 * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -200000000000.0], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 2e-13], N[(1.0 * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -2e11 or 2.0000000000000001e-13 < (/.f64 x y)

   1. Initial program 97.0%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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 54.5

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

   5. Applied rewrites 54.5%

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

   if -2e11 < (/.f64 x y) < 2.0000000000000001e-13

   1. Initial program 99.6%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 79.6

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

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 78.8%

      \[\leadsto 1 \cdot t \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -200000000000:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{y} \cdot x\\ \mathbf{if}\;\frac{x}{y} \leq -200000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ z y) x)))
   (if (<= (/ x y) -200000000000.0)
     t_1
     (if (<= (/ x y) 2e-13) (* 1.0 t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z / y) * x;
	double tmp;
	if ((x / y) <= -200000000000.0) {
		tmp = t_1;
	} else if ((x / y) <= 2e-13) {
		tmp = 1.0 * 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) * x
    if ((x / y) <= (-200000000000.0d0)) then
        tmp = t_1
    else if ((x / y) <= 2d-13) then
        tmp = 1.0d0 * 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) * x;
	double tmp;
	if ((x / y) <= -200000000000.0) {
		tmp = t_1;
	} else if ((x / y) <= 2e-13) {
		tmp = 1.0 * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z / y) * x
	tmp = 0
	if (x / y) <= -200000000000.0:
		tmp = t_1
	elif (x / y) <= 2e-13:
		tmp = 1.0 * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z / y) * x)
	tmp = 0.0
	if (Float64(x / y) <= -200000000000.0)
		tmp = t_1;
	elseif (Float64(x / y) <= 2e-13)
		tmp = Float64(1.0 * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z / y) * x;
	tmp = 0.0;
	if ((x / y) <= -200000000000.0)
		tmp = t_1;
	elseif ((x / y) <= 2e-13)
		tmp = 1.0 * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -200000000000.0], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 2e-13], N[(1.0 * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -2e11 or 2.0000000000000001e-13 < (/.f64 x y)

   1. Initial program 97.0%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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 54.5

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

   5. Applied rewrites 54.5%

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

   7. Applied rewrites 51.2%

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

   if -2e11 < (/.f64 x y) < 2.0000000000000001e-13

   1. Initial program 99.6%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 79.6

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

   5. Applied rewrites 79.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 \cdot t \]
   7. Step-by-step derivation

   8. Applied rewrites 78.8%

      \[\leadsto 1 \cdot t \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -200000000000:\\ \;\;\;\;\frac{z}{y} \cdot x\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{-13}:\\ \;\;\;\;1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \mathbf{if}\;y \leq -9.6 \cdot 10^{-239}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-299}:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-187}:\\ \;\;\;\;\frac{-t}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ z y) x t)))
   (if (<= y -9.6e-239)
     t_1
     (if (<= y -2.3e-299)
       (* z (/ x y))
       (if (<= y 9e-187) (* (/ (- t) y) x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z / y), x, t);
	double tmp;
	if (y <= -9.6e-239) {
		tmp = t_1;
	} else if (y <= -2.3e-299) {
		tmp = z * (x / y);
	} else if (y <= 9e-187) {
		tmp = (-t / y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(z / y), x, t)
	tmp = 0.0
	if (y <= -9.6e-239)
		tmp = t_1;
	elseif (y <= -2.3e-299)
		tmp = Float64(z * Float64(x / y));
	elseif (y <= 9e-187)
		tmp = Float64(Float64(Float64(-t) / y) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / y), $MachinePrecision] * x + t), $MachinePrecision]}, If[LessEqual[y, -9.6e-239], t$95$1, If[LessEqual[y, -2.3e-299], N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e-187], N[(N[((-t) / y), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.59999999999999971e-239 or 8.9999999999999996e-187 < y

   1. Initial program 99.3%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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 79.0

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

   7. Applied rewrites 79.0%

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

   if -9.59999999999999971e-239 < y < -2.3000000000000001e-299

   1. Initial program 93.6%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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 62.9

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

   5. Applied rewrites 62.9%

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

   if -2.3000000000000001e-299 < y < 8.9999999999999996e-187

   1. Initial program 91.8%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 82.3

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 78.1%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{-239}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-299}:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-187}:\\ \;\;\;\;\frac{-t}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \mathbf{if}\;z \leq -2.85 \cdot 10^{+112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 70000000000000:\\ \;\;\;\;\left(1 - \frac{x}{y}\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ z y) x t)))
   (if (<= z -2.85e+112)
     t_1
     (if (<= z 70000000000000.0) (* (- 1.0 (/ x y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z / y), x, t);
	double tmp;
	if (z <= -2.85e+112) {
		tmp = t_1;
	} else if (z <= 70000000000000.0) {
		tmp = (1.0 - (x / y)) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(z / y), x, t)
	tmp = 0.0
	if (z <= -2.85e+112)
		tmp = t_1;
	elseif (z <= 70000000000000.0)
		tmp = Float64(Float64(1.0 - 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 / y), $MachinePrecision] * x + t), $MachinePrecision]}, If[LessEqual[z, -2.85e+112], t$95$1, If[LessEqual[z, 70000000000000.0], N[(N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.85000000000000016e112 or 7e13 < z

   1. Initial program 99.6%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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 90.4

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

   4. Applied rewrites 90.4%

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

   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 85.4

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

   7. Applied rewrites 85.4%

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

   if -2.85000000000000016e112 < z < 7e13

   1. Initial program 97.1%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 86.6

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

   5. Applied rewrites 86.6%

      \[\leadsto \color{blue}{\left(1 - \frac{x}{y}\right) \cdot t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 72.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{-169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-187}:\\ \;\;\;\;\frac{\left(-t\right) \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ z y) x t)))
   (if (<= y -3.1e-169) t_1 (if (<= y 9e-187) (/ (* (- t) x) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((z / y), x, t);
	double tmp;
	if (y <= -3.1e-169) {
		tmp = t_1;
	} else if (y <= 9e-187) {
		tmp = (-t * x) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(z / y), x, t)
	tmp = 0.0
	if (y <= -3.1e-169)
		tmp = t_1;
	elseif (y <= 9e-187)
		tmp = Float64(Float64(Float64(-t) * x) / y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / y), $MachinePrecision] * x + t), $MachinePrecision]}, If[LessEqual[y, -3.1e-169], t$95$1, If[LessEqual[y, 9e-187], N[(N[((-t) * x), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.1000000000000002e-169 or 8.9999999999999996e-187 < y

   1. Initial program 99.2%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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 97.0

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

   4. Applied rewrites 97.0%

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

   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 81.5

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

   7. Applied rewrites 81.5%

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

   if -3.1000000000000002e-169 < y < 8.9999999999999996e-187

   1. Initial program 94.8%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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. 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 85.5

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 58.9%

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

Alternative 10: 92.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.3%

   \[\frac{x}{y} \cdot \left(z - t\right) + t \]
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 92.1

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

4. Applied rewrites 92.1%

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

Alternative 11: 72.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.3%

   \[\frac{x}{y} \cdot \left(z - t\right) + t \]
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 92.1

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

4. Applied rewrites 92.1%

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

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 72.1

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

7. Applied rewrites 72.1%

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

Alternative 12: 38.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ 1 \cdot t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* 1.0 t))

C

double code(double x, double y, double z, double t) {
	return 1.0 * 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 = 1.0d0 * t
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 * t;
}

Python

def code(x, y, z, t):
	return 1.0 * t

Julia

function code(x, y, z, t)
	return Float64(1.0 * t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 * t;
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 * t), $MachinePrecision]

Derivation

1. Initial program 98.3%

   \[\frac{x}{y} \cdot \left(z - t\right) + t \]
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. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. associate-*l* N/A

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

   4. distribute-rgt1-in N/A

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

   5. +-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. mul-1-neg N/A

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

   8. unsub-neg N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 66.6

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

5. Applied rewrites 66.6%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 \cdot t \]
7. Step-by-step derivation

8. Applied rewrites 39.0%

   \[\leadsto 1 \cdot t \]
9. Add Preprocessing

Developer Target 1: 97.6% 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 2024298 
(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))