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

Percentage Accurate: 97.8% → 97.8%

Time: 10.1s

Alternatives: 9

Speedup: 1.1×

• 25.5% 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, 0.8× 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 99.1%

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

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

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

   5. --lowering--.f64 N/A

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

   6. /-lowering-/.f64 99.2

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

4. Applied egg-rr 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 94.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z - t\right) \cdot x}{y}\\ \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 10:\\ \;\;\;\;t + z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* (- z t) x) y)))
   (if (<= (/ x y) -2e+27)
     t_1
     (if (<= (/ x y) 10.0) (+ t (* z (/ x y))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((z - t) * x) / y;
	double tmp;
	if ((x / y) <= -2e+27) {
		tmp = t_1;
	} else if ((x / y) <= 10.0) {
		tmp = t + (z * (x / y));
	} 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 - t) * x) / y
    if ((x / y) <= (-2d+27)) then
        tmp = t_1
    else if ((x / y) <= 10.0d0) then
        tmp = t + (z * (x / y))
    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 - t) * x) / y;
	double tmp;
	if ((x / y) <= -2e+27) {
		tmp = t_1;
	} else if ((x / y) <= 10.0) {
		tmp = t + (z * (x / y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((z - t) * x) / y
	tmp = 0
	if (x / y) <= -2e+27:
		tmp = t_1
	elif (x / y) <= 10.0:
		tmp = t + (z * (x / y))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(z - t) * x) / y)
	tmp = 0.0
	if (Float64(x / y) <= -2e+27)
		tmp = t_1;
	elseif (Float64(x / y) <= 10.0)
		tmp = Float64(t + Float64(z * Float64(x / y)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((z - t) * x) / y;
	tmp = 0.0;
	if ((x / y) <= -2e+27)
		tmp = t_1;
	elseif ((x / y) <= 10.0)
		tmp = t + (z * (x / y));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -2e+27], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 10.0], N[(t + N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -2e27 or 10 < (/.f64 x y)

   1. Initial program 98.9%

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. --lowering--.f64 98.9

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

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{z}{y} - \frac{t}{y}\right)} \]
   6. 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. /-lowering-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. *-lowering-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. --lowering--.f64 89.8

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

   7. Simplified 89.8%

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

   if -2e27 < (/.f64 x y) < 10

   1. Initial program 99.3%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 99.1%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{+27}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 10:\\ \;\;\;\;t + z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{x}{y}\\ \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (/ x y))))
   (if (<= (/ x y) -5e-6) t_1 (if (<= (/ x y) 5e-11) 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) <= -5e-6) {
		tmp = t_1;
	} else if ((x / y) <= 5e-11) {
		tmp = 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) <= (-5d-6)) then
        tmp = t_1
    else if ((x / y) <= 5d-11) then
        tmp = 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) <= -5e-6) {
		tmp = t_1;
	} else if ((x / y) <= 5e-11) {
		tmp = t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * (x / y)
	tmp = 0
	if (x / y) <= -5e-6:
		tmp = t_1
	elif (x / y) <= 5e-11:
		tmp = 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) <= -5e-6)
		tmp = t_1;
	elseif (Float64(x / y) <= 5e-11)
		tmp = 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) <= -5e-6)
		tmp = t_1;
	elseif ((x / y) <= 5e-11)
		tmp = 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], -5e-6], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 5e-11], t, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -5.00000000000000041e-6 or 5.00000000000000018e-11 < (/.f64 x y)

   1. Initial program 99.0%

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

   3. Taylor expanded in z around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{x \cdot z}{y} + 0} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{z}{y}} + 0 \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, 0\right)} \]

      4. /-lowering-/.f64 53.4

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{z}{y}}, 0\right) \]

   5. Simplified 53.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z}{y}, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 60.0

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

   7. Applied egg-rr 60.0%

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

   if -5.00000000000000041e-6 < (/.f64 x y) < 5.00000000000000018e-11

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 78.4%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{-6}:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3300000000:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t + z \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3300000000.0) (* t (- 1.0 (/ x y))) (+ t (* z (/ x y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3300000000.0) {
		tmp = t * (1.0 - (x / y));
	} else {
		tmp = t + (z * (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 (t <= (-3300000000.0d0)) then
        tmp = t * (1.0d0 - (x / y))
    else
        tmp = t + (z * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3300000000.0) {
		tmp = t * (1.0 - (x / y));
	} else {
		tmp = t + (z * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3300000000.0:
		tmp = t * (1.0 - (x / y))
	else:
		tmp = t + (z * (x / y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3300000000.0)
		tmp = Float64(t * Float64(1.0 - Float64(x / y)));
	else
		tmp = Float64(t + Float64(z * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -3300000000.0)
		tmp = t * (1.0 - (x / y));
	else
		tmp = t + (z * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3300000000.0], N[(t * N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t + N[(z * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.3e9

   1. Initial program 99.9%

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-in N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

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

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

      13. *-lowering-*.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. unsub-neg N/A

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

      20. --lowering--.f64 N/A

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

      21. /-lowering-/.f64 91.6

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

   7. Simplified 91.6%

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

   if -3.3e9 < t

   1. Initial program 98.9%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 86.6%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3300000000:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t + z \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -105000000:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -105000000.0) (* t (- 1.0 (/ x y))) (fma (/ x y) z t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -105000000.0) {
		tmp = t * (1.0 - (x / y));
	} else {
		tmp = fma((x / y), z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -105000000.0)
		tmp = Float64(t * Float64(1.0 - Float64(x / y)));
	else
		tmp = fma(Float64(x / y), z, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -105000000.0], N[(t * N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * z + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.05e8

   1. Initial program 99.9%

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-in N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

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

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

      13. *-lowering-*.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

      18. metadata-eval N/A

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

      19. unsub-neg N/A

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

      20. --lowering--.f64 N/A

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

      21. /-lowering-/.f64 91.6

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

   7. Simplified 91.6%

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

   if -1.05e8 < t

   1. Initial program 98.9%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 86.6%

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. /-lowering-/.f64 86.6

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

   7. Applied egg-rr 86.6%

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

Alternative 6: 97.8% 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 99.1%

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

3. Final simplification 99.1%

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

Alternative 7: 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 99.1%

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

   1. accelerator-lowering-fma.f64 N/A

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

   2. /-lowering-/.f64 N/A

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

   3. --lowering--.f64 99.1

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

4. Applied egg-rr 99.1%

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

Alternative 8: 76.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in z around inf

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

5. Simplified 81.5%

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

   1. accelerator-lowering-fma.f64 N/A

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

   2. /-lowering-/.f64 81.5

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

7. Applied egg-rr 81.5%

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

Alternative 9: 38.5% accurate, 23.0× 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 99.1%

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

3. Taylor expanded in x around 0

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

5. Simplified 44.2%

   \[\leadsto \color{blue}{t} \]
6. 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 2024195 
(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))