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

Percentage Accurate: 97.6% → 97.6%

Time: 9.7s

Alternatives: 8

Speedup: 1.0×

• 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: 97.6% 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: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ t + \frac{z - t}{\frac{y}{x}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(t + N[(N[(z - t), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $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 0 91.5%

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

   1. associate-/l* 94.0%

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

   2. *-commutative 94.0%

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

   3. associate-/r/ 97.9%

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

5. Simplified 97.9%

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

6. Final simplification 97.9%

   \[\leadsto t + \frac{z - t}{\frac{y}{x}} \]
7. Add Preprocessing

Alternative 2: 83.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-16} \lor \neg \left(z \leq 1.2 \cdot 10^{-70}\right):\\ \;\;\;\;t + x \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.5e-16) (not (<= z 1.2e-70)))
   (+ t (* x (/ z y)))
   (* t (- 1.0 (/ x y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.5e-16) || !(z <= 1.2e-70)) {
		tmp = t + (x * (z / y));
	} else {
		tmp = t * (1.0 - (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-2.5d-16)) .or. (.not. (z <= 1.2d-70))) then
        tmp = t + (x * (z / y))
    else
        tmp = t * (1.0d0 - (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.5e-16) || !(z <= 1.2e-70)) {
		tmp = t + (x * (z / y));
	} else {
		tmp = t * (1.0 - (x / y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.5e-16) or not (z <= 1.2e-70):
		tmp = t + (x * (z / y))
	else:
		tmp = t * (1.0 - (x / y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.5e-16) || !(z <= 1.2e-70))
		tmp = Float64(t + Float64(x * Float64(z / y)));
	else
		tmp = Float64(t * Float64(1.0 - Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2.5e-16) || ~((z <= 1.2e-70)))
		tmp = t + (x * (z / y));
	else
		tmp = t * (1.0 - (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.5e-16], N[Not[LessEqual[z, 1.2e-70]], $MachinePrecision]], N[(t + N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.5000000000000002e-16 or 1.2000000000000001e-70 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 80.0%

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

      1. associate-/l* 88.1%

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

   5. Simplified 88.1%

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

   if -2.5000000000000002e-16 < z < 1.2000000000000001e-70

   1. Initial program 95.3%

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

      1. associate-*l/ 95.9%

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

      2. associate-/l* 92.9%

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

      3. fma-define 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in z around 0 84.7%

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

      1. mul-1-neg 84.7%

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

      2. *-rgt-identity 84.7%

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

      3. associate-/l* 87.2%

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

      4. distribute-rgt-neg-in 87.2%

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

      5. mul-1-neg 87.2%

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

      6. distribute-lft-in 87.2%

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

      7. mul-1-neg 87.2%

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

      8. unsub-neg 87.2%

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

   7. Simplified 87.2%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-16} \lor \neg \left(z \leq 1.2 \cdot 10^{-70}\right):\\ \;\;\;\;t + x \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-16} \lor \neg \left(z \leq 1.05 \cdot 10^{-73}\right):\\ \;\;\;\;t + z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.2e-16) (not (<= z 1.05e-73)))
   (+ t (* z (/ x y)))
   (* t (- 1.0 (/ x y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.2e-16) || !(z <= 1.05e-73)) {
		tmp = t + (z * (x / y));
	} else {
		tmp = t * (1.0 - (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-3.2d-16)) .or. (.not. (z <= 1.05d-73))) then
        tmp = t + (z * (x / y))
    else
        tmp = t * (1.0d0 - (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.2e-16) || !(z <= 1.05e-73)) {
		tmp = t + (z * (x / y));
	} else {
		tmp = t * (1.0 - (x / y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -3.2e-16) or not (z <= 1.05e-73):
		tmp = t + (z * (x / y))
	else:
		tmp = t * (1.0 - (x / y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -3.2e-16) || !(z <= 1.05e-73))
		tmp = Float64(t + Float64(z * Float64(x / y)));
	else
		tmp = Float64(t * Float64(1.0 - Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -3.2e-16) || ~((z <= 1.05e-73)))
		tmp = t + (z * (x / y));
	else
		tmp = t * (1.0 - (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -3.2e-16], N[Not[LessEqual[z, 1.05e-73]], $MachinePrecision]], N[(t + N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.20000000000000023e-16 or 1.0499999999999999e-73 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 80.0%

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

      1. associate-/l* 88.1%

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

   5. Simplified 88.1%

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

      1. clear-num 88.1%

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

      2. un-div-inv 88.2%

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

   7. Applied egg-rr 88.2%

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

      1. associate-/r/ 91.6%

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

   9. Applied egg-rr 91.6%

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

   if -3.20000000000000023e-16 < z < 1.0499999999999999e-73

   1. Initial program 95.3%

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

      1. associate-*l/ 95.9%

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

      2. associate-/l* 92.9%

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

      3. fma-define 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in z around 0 84.7%

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

      1. mul-1-neg 84.7%

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

      2. *-rgt-identity 84.7%

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

      3. associate-/l* 87.2%

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

      4. distribute-rgt-neg-in 87.2%

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

      5. mul-1-neg 87.2%

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

      6. distribute-lft-in 87.2%

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

      7. mul-1-neg 87.2%

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

      8. unsub-neg 87.2%

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

   7. Simplified 87.2%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-16} \lor \neg \left(z \leq 1.05 \cdot 10^{-73}\right):\\ \;\;\;\;t + z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-16} \lor \neg \left(z \leq 1.9 \cdot 10^{-69}\right):\\ \;\;\;\;t + z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{t}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.65e-16) (not (<= z 1.9e-69)))
   (+ t (* z (/ x y)))
   (- t (/ t (/ y x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.65e-16) || !(z <= 1.9e-69)) {
		tmp = t + (z * (x / y));
	} else {
		tmp = t - (t / (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.65d-16)) .or. (.not. (z <= 1.9d-69))) then
        tmp = t + (z * (x / y))
    else
        tmp = t - (t / (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.65e-16) || !(z <= 1.9e-69)) {
		tmp = t + (z * (x / y));
	} else {
		tmp = t - (t / (y / x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.65e-16) or not (z <= 1.9e-69):
		tmp = t + (z * (x / y))
	else:
		tmp = t - (t / (y / x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.65e-16) || !(z <= 1.9e-69))
		tmp = Float64(t + Float64(z * Float64(x / y)));
	else
		tmp = Float64(t - Float64(t / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.65e-16) || ~((z <= 1.9e-69)))
		tmp = t + (z * (x / y));
	else
		tmp = t - (t / (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.65e-16], N[Not[LessEqual[z, 1.9e-69]], $MachinePrecision]], N[(t + N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t - N[(t / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.64999999999999994e-16 or 1.8999999999999999e-69 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 80.0%

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

      1. associate-/l* 88.1%

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

   5. Simplified 88.1%

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

      1. clear-num 88.1%

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

      2. un-div-inv 88.2%

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

   7. Applied egg-rr 88.2%

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

      1. associate-/r/ 91.6%

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

   9. Applied egg-rr 91.6%

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

   if -1.64999999999999994e-16 < z < 1.8999999999999999e-69

   1. Initial program 95.3%

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

   3. Taylor expanded in z around 0 84.7%

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

      1. associate-*r/ 84.7%

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

      2. mul-1-neg 84.7%

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

      3. *-commutative 84.7%

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

      4. distribute-rgt-neg-out 84.7%

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

      5. associate-/l* 83.3%

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

      6. distribute-neg-frac 83.3%

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

      7. distribute-neg-frac2 83.3%

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

   5. Simplified 83.3%

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

   6. Taylor expanded in x around 0 84.7%

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

      1. neg-mul-1 84.7%

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

      2. sub-neg 84.7%

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

      3. associate-/l* 87.2%

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

   8. Simplified 87.2%

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

      1. clear-num 87.1%

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

      2. div-inv 87.2%

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

   10. Applied egg-rr 87.2%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-16} \lor \neg \left(z \leq 1.9 \cdot 10^{-69}\right):\\ \;\;\;\;t + z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{t}{\frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-68} \lor \neg \left(x \leq 9.4 \cdot 10^{+70}\right):\\ \;\;\;\;x \cdot \frac{t}{-y}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.7e-68) (not (<= x 9.4e+70))) (* x (/ t (- y))) t))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.7e-68) || !(x <= 9.4e+70)) {
		tmp = x * (t / -y);
	} else {
		tmp = 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 <= (-2.7d-68)) .or. (.not. (x <= 9.4d+70))) then
        tmp = x * (t / -y)
    else
        tmp = t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.7e-68) || !(x <= 9.4e+70)) {
		tmp = x * (t / -y);
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.7e-68) or not (x <= 9.4e+70):
		tmp = x * (t / -y)
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.7e-68) || !(x <= 9.4e+70))
		tmp = Float64(x * Float64(t / Float64(-y)));
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.7e-68) || ~((x <= 9.4e+70)))
		tmp = x * (t / -y);
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.7e-68], N[Not[LessEqual[x, 9.4e+70]], $MachinePrecision]], N[(x * N[(t / (-y)), $MachinePrecision]), $MachinePrecision], t]

Derivation

1. Split input into 2 regimes

2. if x < -2.7000000000000002e-68 or 9.3999999999999996e70 < x

   1. Initial program 96.0%

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

   3. Taylor expanded in z around 0 53.7%

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

      1. associate-*r/ 53.7%

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

      2. mul-1-neg 53.7%

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

      3. *-commutative 53.7%

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

      4. distribute-rgt-neg-out 53.7%

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

      5. associate-/l* 57.7%

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

      6. distribute-neg-frac 57.7%

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

      7. distribute-neg-frac2 57.7%

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

   5. Simplified 57.7%

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

   6. Taylor expanded in x around 0 53.7%

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

      1. neg-mul-1 53.7%

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

      2. sub-neg 53.7%

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

      3. associate-/l* 62.4%

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

   8. Simplified 62.4%

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

      1. clear-num 62.3%

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

      2. div-inv 62.4%

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

   10. Applied egg-rr 62.4%

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

   11. Taylor expanded in y around 0 41.8%

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

       1. associate-*l/ 43.4%

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

       2. associate-*r* 43.4%

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

       3. *-commutative 43.4%

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

       4. neg-mul-1 43.4%

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

       5. distribute-frac-neg2 43.4%

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

   13. Simplified 43.4%

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

   if -2.7000000000000002e-68 < x < 9.3999999999999996e70

   1. Initial program 99.9%

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

      1. associate-*l/ 99.1%

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

      2. associate-/l* 89.3%

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

      3. fma-define 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in x around 0 70.1%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-68} \lor \neg \left(x \leq 9.4 \cdot 10^{+70}\right):\\ \;\;\;\;x \cdot \frac{t}{-y}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Final simplification 97.7%

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

Alternative 7: 65.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ t \cdot \left(1 - \frac{x}{y}\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

   1. associate-*l/ 91.5%

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

   2. associate-/l* 94.0%

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

   3. fma-define 94.0%

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

3. Simplified 94.0%

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

5. Taylor expanded in z around 0 66.2%

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

   1. mul-1-neg 66.2%

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

   2. *-rgt-identity 66.2%

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

   3. associate-/l* 71.0%

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

   4. distribute-rgt-neg-in 71.0%

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

   5. mul-1-neg 71.0%

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

   6. distribute-lft-in 71.0%

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

   7. mul-1-neg 71.0%

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

   8. unsub-neg 71.0%

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

7. Simplified 71.0%

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

8. Final simplification 71.0%

   \[\leadsto t \cdot \left(1 - \frac{x}{y}\right) \]
9. Add Preprocessing

Alternative 8: 38.1% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x, y, z, t)
	return t
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := t

Derivation

1. Initial program 97.7%

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

   1. associate-*l/ 91.5%

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

   2. associate-/l* 94.0%

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

   3. fma-define 94.0%

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

3. Simplified 94.0%

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

5. Taylor expanded in x around 0 40.1%

   \[\leadsto \color{blue}{t} \]

6. Final simplification 40.1%

   \[\leadsto t \]
7. Add Preprocessing

Developer target: 97.4% accurate, 0.5× 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 2024071 
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"
  :precision binary64

  :alt
  (if (< z 2.759456554562692e-282) (+ (* (/ x y) (- z t)) t) (if (< z 2.326994450874436e-110) (+ (* x (/ (- z t) y)) t) (+ (* (/ x y) (- z t)) t)))

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