Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.1% → 99.6%

Time: 5.0s

Alternatives: 6

Speedup: 1.0×

• 23.8% 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: 99.1% 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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

   \[x \cdot y - z \cdot t \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-neg.f64 99.6

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 75.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -0.002 \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+59}\right):\\ \;\;\;\;\left(-z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -0.002) (not (<= (* z t) 2e+59))) (* (- z) t) (* x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -0.002) || !((z * t) <= 2e+59)) {
		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 (((z * t) <= (-0.002d0)) .or. (.not. ((z * t) <= 2d+59))) 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 (((z * t) <= -0.002) || !((z * t) <= 2e+59)) {
		tmp = -z * t;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -0.002) or not ((z * t) <= 2e+59):
		tmp = -z * t
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -0.002) || !(Float64(z * t) <= 2e+59))
		tmp = Float64(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 (((z * t) <= -0.002) || ~(((z * t) <= 2e+59)))
		tmp = -z * t;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -0.002], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+59]], $MachinePrecision]], N[((-z) * t), $MachinePrecision], N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2e-3 or 1.99999999999999994e59 < (*.f64 z t)

   1. Initial program 97.5%

      \[x \cdot y - z \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \left(t \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 84.3

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

   5. Applied rewrites 84.3%

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

   if -2e-3 < (*.f64 z t) < 1.99999999999999994e59

   1. Initial program 100.0%

      \[x \cdot y - z \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 78.4%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{t \cdot z} \]
   8. Step-by-step derivation

      1. lower-*.f64 8.6

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

   9. Applied rewrites 8.6%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 79.2

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

   12. Applied rewrites 79.2%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -0.002 \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+59}\right):\\ \;\;\;\;\left(-z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

   \[x \cdot y - z \cdot t \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 99.2

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

4. Applied rewrites 99.2%

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

Alternative 4: 99.1% 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 98.8%

   \[x \cdot y - z \cdot t \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 5: 52.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

   \[x \cdot y - z \cdot t \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 99.2

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

4. Applied rewrites 99.2%

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

6. Applied rewrites 50.2%

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

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{t \cdot z} \]
8. Step-by-step derivation

   1. lower-*.f64 5.6

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

9. Applied rewrites 5.6%

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

10. Taylor expanded in x around inf

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

    1. lower-*.f64 51.4

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

12. Applied rewrites 51.4%

    \[\leadsto \color{blue}{x \cdot y} \]
13. Add Preprocessing

Alternative 6: 3.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

   \[x \cdot y - z \cdot t \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 99.2

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

4. Applied rewrites 99.2%

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

6. Applied rewrites 50.2%

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

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{t \cdot z} \]
8. Step-by-step derivation

   1. lower-*.f64 5.6

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

9. Applied rewrites 5.6%

   \[\leadsto \color{blue}{t \cdot z} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))