Linear.V2:$cdot from linear-1.19.1.3, A

Percentage Accurate: 99.2% → 99.5%

Time: 1.8s

Alternatives: 4

Speedup: 1.0×

• 23.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.1× speedup

Math

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

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(x, y, Float64(z * t))
end

Wolfram

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

Derivation

1. Initial program 99.6%

   \[x \cdot y + z \cdot t \]
2. Step-by-step derivation

   1. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 67.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+154}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{+103} \lor \neg \left(x \leq -0.76\right) \land x \leq 1.6 \cdot 10^{-61}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4.5e+154)
   (* x y)
   (if (or (<= x -3.2e+103) (and (not (<= x -0.76)) (<= x 1.6e-61)))
     (* z t)
     (* x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.5e+154) {
		tmp = x * y;
	} else if ((x <= -3.2e+103) || (!(x <= -0.76) && (x <= 1.6e-61))) {
		tmp = z * t;
	} else {
		tmp = 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 (x <= (-4.5d+154)) then
        tmp = x * y
    else if ((x <= (-3.2d+103)) .or. (.not. (x <= (-0.76d0))) .and. (x <= 1.6d-61)) then
        tmp = z * t
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.5e+154) {
		tmp = x * y;
	} else if ((x <= -3.2e+103) || (!(x <= -0.76) && (x <= 1.6e-61))) {
		tmp = z * t;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -4.5e+154:
		tmp = x * y
	elif (x <= -3.2e+103) or (not (x <= -0.76) and (x <= 1.6e-61)):
		tmp = z * t
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4.5e+154)
		tmp = Float64(x * y);
	elseif ((x <= -3.2e+103) || (!(x <= -0.76) && (x <= 1.6e-61)))
		tmp = Float64(z * t);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -4.5e+154)
		tmp = x * y;
	elseif ((x <= -3.2e+103) || (~((x <= -0.76)) && (x <= 1.6e-61)))
		tmp = z * t;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4.5e+154], N[(x * y), $MachinePrecision], If[Or[LessEqual[x, -3.2e+103], And[N[Not[LessEqual[x, -0.76]], $MachinePrecision], LessEqual[x, 1.6e-61]]], N[(z * t), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.50000000000000009e154 or -3.19999999999999993e103 < x < -0.76000000000000001 or 1.6000000000000001e-61 < x

   1. Initial program 99.1%

      \[x \cdot y + z \cdot t \]

   2. Taylor expanded in x around inf 64.7%

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

   if -4.50000000000000009e154 < x < -3.19999999999999993e103 or -0.76000000000000001 < x < 1.6000000000000001e-61

   1. Initial program 100.0%

      \[x \cdot y + z \cdot t \]

   2. Taylor expanded in x around 0 77.9%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+154}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{+103} \lor \neg \left(x \leq -0.76\right) \land x \leq 1.6 \cdot 10^{-61}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[x \cdot y + z \cdot t \]

2. Final simplification 99.6%

   \[\leadsto z \cdot t + x \cdot y \]

Alternative 4: 53.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[x \cdot y + z \cdot t \]

2. Taylor expanded in x around 0 59.1%

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

3. Final simplification 59.1%

   \[\leadsto z \cdot t \]

Reproduce

herbie shell --seed 2023200 
(FPCore (x y z t)
  :name "Linear.V2:$cdot from linear-1.19.1.3, A"
  :precision binary64
  (+ (* x y) (* z t)))