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

Percentage Accurate: 98.9% → 99.5%

Time: 2.2s

Alternatives: 5

Speedup: 1.0×

• 24.2% 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: 98.9% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. +-commutative 98.4%

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

   2. fma-def 99.2%

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

3. Applied egg-rr 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 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 98.4%

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

   1. fma-def 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 3: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{-84}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{+86}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -6.4e-84) (* z t) (if (<= t 1.12e+86) (* x y) (* z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -6.4e-84) {
		tmp = z * t;
	} else if (t <= 1.12e+86) {
		tmp = x * y;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-6.4d-84)) then
        tmp = z * t
    else if (t <= 1.12d+86) then
        tmp = x * y
    else
        tmp = z * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -6.4e-84) {
		tmp = z * t;
	} else if (t <= 1.12e+86) {
		tmp = x * y;
	} else {
		tmp = z * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -6.4e-84:
		tmp = z * t
	elif t <= 1.12e+86:
		tmp = x * y
	else:
		tmp = z * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -6.4e-84)
		tmp = Float64(z * t);
	elseif (t <= 1.12e+86)
		tmp = Float64(x * y);
	else
		tmp = Float64(z * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -6.4e-84)
		tmp = z * t;
	elseif (t <= 1.12e+86)
		tmp = x * y;
	else
		tmp = z * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -6.4e-84], N[(z * t), $MachinePrecision], If[LessEqual[t, 1.12e+86], N[(x * y), $MachinePrecision], N[(z * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.3999999999999999e-84 or 1.12e86 < t

   1. Initial program 96.7%

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

   2. Taylor expanded in x around 0 66.3%

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

   if -6.3999999999999999e-84 < t < 1.12e86

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 73.6%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{-84}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{+86}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot t\\ \end{array} \]

Alternative 4: 98.9% 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]

Derivation

1. Initial program 98.4%

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

2. Final simplification 98.4%

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

Alternative 5: 51.5% 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 98.4%

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

2. Taylor expanded in x around 0 48.9%

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

3. Final simplification 48.9%

   \[\leadsto z \cdot t \]

Reproduce

herbie shell --seed 2023208 
(FPCore (x y z t)
  :name "Linear.V2:$cdot from linear-1.19.1.3, A"
  :precision binary64
  (+ (* x y) (* z t)))