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

Percentage Accurate: 97.4% → 97.8%

Time: 9.2s

Alternatives: 10

Speedup: 0.5×

• 25.4% 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: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t + \frac{x}{y} \cdot \left(z - t\right) \leq 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x \cdot \left(z - t\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ t (* (/ x y) (- z t))) 1e+302)
   (fma (/ x y) (- z t) t)
   (/ 1.0 (/ y (* x (- z t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t + ((x / y) * (z - t))) <= 1e+302) {
		tmp = fma((x / y), (z - t), t);
	} else {
		tmp = 1.0 / (y / (x * (z - t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(t + Float64(Float64(x / y) * Float64(z - t))) <= 1e+302)
		tmp = fma(Float64(x / y), Float64(z - t), t);
	else
		tmp = Float64(1.0 / Float64(y / Float64(x * Float64(z - t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(t + N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+302], N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision] + t), $MachinePrecision], N[(1.0 / N[(y / N[(x * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 x y) (-.f64 z t)) t) < 1.0000000000000001e302

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 98.8

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

   4. Applied rewrites 98.8%

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

   if 1.0000000000000001e302 < (+.f64 (*.f64 (/.f64 x y) (-.f64 z t)) t)

   1. Initial program 85.2%

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

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 86.4

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

   4. Applied rewrites 86.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 99.9

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 100.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t + \frac{x}{y} \cdot \left(z - t\right) \leq 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x \cdot \left(z - t\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{z - t}{y}\\ \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;t + \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ (- z t) y))))
   (if (<= (/ x y) -1e-22)
     t_1
     (if (<= (/ x y) 2e-10) (+ t (/ (* x z) y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((z - t) / y);
	double tmp;
	if ((x / y) <= -1e-22) {
		tmp = t_1;
	} else if ((x / y) <= 2e-10) {
		tmp = t + ((x * z) / 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 = x * ((z - t) / y)
    if ((x / y) <= (-1d-22)) then
        tmp = t_1
    else if ((x / y) <= 2d-10) then
        tmp = t + ((x * z) / 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 = x * ((z - t) / y);
	double tmp;
	if ((x / y) <= -1e-22) {
		tmp = t_1;
	} else if ((x / y) <= 2e-10) {
		tmp = t + ((x * z) / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * ((z - t) / y)
	tmp = 0
	if (x / y) <= -1e-22:
		tmp = t_1
	elif (x / y) <= 2e-10:
		tmp = t + ((x * z) / y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(z - t) / y))
	tmp = 0.0
	if (Float64(x / y) <= -1e-22)
		tmp = t_1;
	elseif (Float64(x / y) <= 2e-10)
		tmp = Float64(t + Float64(Float64(x * z) / y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * ((z - t) / y);
	tmp = 0.0;
	if ((x / y) <= -1e-22)
		tmp = t_1;
	elseif ((x / y) <= 2e-10)
		tmp = t + ((x * z) / y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(N[(z - t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -1e-22], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 2e-10], N[(t + N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -1e-22 or 2.00000000000000007e-10 < (/.f64 x y)

   1. Initial program 96.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 88.7

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

   7. Applied rewrites 88.7%

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

   9. Applied rewrites 91.2%

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

   if -1e-22 < (/.f64 x y) < 2.00000000000000007e-10

   1. Initial program 98.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{-22}:\\ \;\;\;\;x \cdot \frac{z - t}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;t + \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z - t}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -20:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z - t}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -20.0)
   (/ (* x (- z t)) y)
   (if (<= (/ x y) 1e-15) (fma (/ z y) x t) (* x (/ (- z t) y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -20.0) {
		tmp = (x * (z - t)) / y;
	} else if ((x / y) <= 1e-15) {
		tmp = fma((z / y), x, t);
	} else {
		tmp = x * ((z - t) / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -20.0)
		tmp = Float64(Float64(x * Float64(z - t)) / y);
	elseif (Float64(x / y) <= 1e-15)
		tmp = fma(Float64(z / y), x, t);
	else
		tmp = Float64(x * Float64(Float64(z - t) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -20.0], N[(N[(x * N[(z - t), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 1e-15], N[(N[(z / y), $MachinePrecision] * x + t), $MachinePrecision], N[(x * N[(N[(z - t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -20

   1. Initial program 96.7%

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

   3. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. 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. lower-*.f64 N/A

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

      5. lower--.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if -20 < (/.f64 x y) < 1.0000000000000001e-15

   1. Initial program 98.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 98.0

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

   4. Applied rewrites 98.0%

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

      1. lift-fma.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. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 93.8

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

   6. Applied rewrites 93.8%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 94.3

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

   9. Applied rewrites 94.3%

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

   if 1.0000000000000001e-15 < (/.f64 x y)

   1. Initial program 95.9%

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

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.3

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

   4. Applied rewrites 96.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 88.9

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

   7. Applied rewrites 88.9%

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

   9. Applied rewrites 91.2%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -20:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z - t}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \left(z - t\right)}{y}\\ \mathbf{if}\;\frac{x}{y} \leq -20:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x (- z t)) y)))
   (if (<= (/ x y) -20.0) t_1 (if (<= (/ x y) 5e-16) (fma (/ z y) x t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * (z - t)) / y;
	double tmp;
	if ((x / y) <= -20.0) {
		tmp = t_1;
	} else if ((x / y) <= 5e-16) {
		tmp = fma((z / y), x, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * Float64(z - t)) / y)
	tmp = 0.0
	if (Float64(x / y) <= -20.0)
		tmp = t_1;
	elseif (Float64(x / y) <= 5e-16)
		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[(x * N[(z - t), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -20.0], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 5e-16], N[(N[(z / y), $MachinePrecision] * x + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -20 or 5.0000000000000004e-16 < (/.f64 x y)

   1. Initial program 96.3%

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

   3. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. 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. lower-*.f64 N/A

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

      5. lower--.f64 90.5

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

   5. Applied rewrites 90.5%

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

   if -20 < (/.f64 x y) < 5.0000000000000004e-16

   1. Initial program 98.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 98.0

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

   4. Applied rewrites 98.0%

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

      1. lift-fma.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. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 94.5

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

   6. Applied rewrites 94.5%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 94.9

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

   9. Applied rewrites 94.9%

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

Alternative 5: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t + \frac{x}{y} \cdot \left(z - t\right) \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{y}, x \cdot \left(z - t\right), t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ t (* (/ x y) (- z t))) 2e+289)
   (fma (/ x y) (- z t) t)
   (fma (/ 1.0 y) (* x (- z t)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t + ((x / y) * (z - t))) <= 2e+289) {
		tmp = fma((x / y), (z - t), t);
	} else {
		tmp = fma((1.0 / y), (x * (z - t)), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(t + Float64(Float64(x / y) * Float64(z - t))) <= 2e+289)
		tmp = fma(Float64(x / y), Float64(z - t), t);
	else
		tmp = fma(Float64(1.0 / y), Float64(x * Float64(z - t)), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(t + N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+289], N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision] + t), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x * N[(z - t), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 x y) (-.f64 z t)) t) < 2.0000000000000001e289

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 98.8

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

   4. Applied rewrites 98.8%

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

   if 2.0000000000000001e289 < (+.f64 (*.f64 (/.f64 x y) (-.f64 z t)) t)

   1. Initial program 85.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t + \frac{x}{y} \cdot \left(z - t\right) \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{y}, x \cdot \left(z - t\right), t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{+98}:\\ \;\;\;\;t \cdot \frac{x}{-y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) 5e+27)
   (fma (/ z y) x t)
   (if (<= (/ x y) 2e+98) (* t (/ x (- y))) (* (/ x y) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= 5e+27) {
		tmp = fma((z / y), x, t);
	} else if ((x / y) <= 2e+98) {
		tmp = t * (x / -y);
	} else {
		tmp = (x / y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= 5e+27)
		tmp = fma(Float64(z / y), x, t);
	elseif (Float64(x / y) <= 2e+98)
		tmp = Float64(t * Float64(x / Float64(-y)));
	else
		tmp = Float64(Float64(x / y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], 5e+27], N[(N[(z / y), $MachinePrecision] * x + t), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 2e+98], N[(t * N[(x / (-y)), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < 4.99999999999999979e27

   1. Initial program 97.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 97.7

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

   4. Applied rewrites 97.7%

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

      1. lift-fma.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. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 92.1

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

   6. Applied rewrites 92.1%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 81.4

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

   9. Applied rewrites 81.4%

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

   if 4.99999999999999979e27 < (/.f64 x y) < 2e98

   1. Initial program 99.8%

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

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 85.6

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

   7. Applied rewrites 85.6%

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

   9. Applied rewrites 74.5%

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

   10. Taylor expanded in z around 0

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

   12. Applied rewrites 73.9%

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

   if 2e98 < (/.f64 x y)

   1. Initial program 93.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 65.0

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

   5. Applied rewrites 65.0%

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

   7. Applied rewrites 74.5%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq 5 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{+98}:\\ \;\;\;\;t \cdot \frac{x}{-y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t + \frac{x}{y} \cdot \left(z - t\right) \leq 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ t (* (/ x y) (- z t))) 1e+302)
   (fma (/ x y) (- z t) t)
   (/ (* x (- z t)) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t + ((x / y) * (z - t))) <= 1e+302) {
		tmp = fma((x / y), (z - t), t);
	} else {
		tmp = (x * (z - t)) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(t + Float64(Float64(x / y) * Float64(z - t))) <= 1e+302)
		tmp = fma(Float64(x / y), Float64(z - t), t);
	else
		tmp = Float64(Float64(x * Float64(z - t)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(t + N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+302], N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision] + t), $MachinePrecision], N[(N[(x * N[(z - t), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 x y) (-.f64 z t)) t) < 1.0000000000000001e302

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 98.8

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

   4. Applied rewrites 98.8%

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

   if 1.0000000000000001e302 < (+.f64 (*.f64 (/.f64 x y) (-.f64 z t)) t)

   1. Initial program 85.2%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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. lower-*.f64 N/A

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

      5. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t + \frac{x}{y} \cdot \left(z - t\right) \leq 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(z - t\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) 5e-16) (fma (/ z y) x t) (* (/ x y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= 5e-16) {
		tmp = fma((z / y), x, t);
	} else {
		tmp = (x / y) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= 5e-16)
		tmp = fma(Float64(z / y), x, t);
	else
		tmp = Float64(Float64(x / y) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], 5e-16], N[(N[(z / y), $MachinePrecision] * x + t), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < 5.0000000000000004e-16

   1. Initial program 97.6%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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. lower-fma.f64 97.6

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

   4. Applied rewrites 97.6%

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

      1. lift-fma.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. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 93.2

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

   6. Applied rewrites 93.2%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 82.4

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

   9. Applied rewrites 82.4%

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

   if 5.0000000000000004e-16 < (/.f64 x y)

   1. Initial program 96.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 56.9

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

   5. Applied rewrites 56.9%

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

   7. Applied rewrites 66.0%

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

Alternative 9: 81.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{x \cdot t}{y}\\ \mathbf{if}\;t \leq -560000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- t (/ (* x t) y))))
   (if (<= t -560000000000.0) t_1 (if (<= t 9e+36) (fma (/ z y) x t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = t - ((x * t) / y);
	double tmp;
	if (t <= -560000000000.0) {
		tmp = t_1;
	} else if (t <= 9e+36) {
		tmp = fma((z / y), x, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(t - Float64(Float64(x * t) / y))
	tmp = 0.0
	if (t <= -560000000000.0)
		tmp = t_1;
	elseif (t <= 9e+36)
		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[(t - N[(N[(x * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -560000000000.0], t$95$1, If[LessEqual[t, 9e+36], N[(N[(z / y), $MachinePrecision] * x + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.6e11 or 8.99999999999999994e36 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 88.5

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

   5. Applied rewrites 88.5%

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

   if -5.6e11 < t < 8.99999999999999994e36

   1. Initial program 95.3%

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
   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. lower-fma.f64 95.3

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

   4. Applied rewrites 95.3%

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

      1. lift-fma.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. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 94.2

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

   6. Applied rewrites 94.2%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 85.4

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

   9. Applied rewrites 85.4%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -560000000000:\\ \;\;\;\;t - \frac{x \cdot t}{y}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;t - \frac{x \cdot t}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 40.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.1%

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

3. Taylor expanded in z around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 39.9

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

5. Applied rewrites 39.9%

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

7. Applied rewrites 44.4%

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

Developer Target 1: 97.1% 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 2024238 
(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))