Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.3% → 99.6%

Time: 3.5s

Alternatives: 5

Speedup: 1.0×

• 23.6% 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: 99.3% 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 98.4%

   \[x \cdot y - z \cdot t \]
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 99.2

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

   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 99.2

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

4. Applied rewrites 99.2%

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

Alternative 2: 75.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{-19} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{-119}\right):\\ \;\;\;\;\left(-z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -5e-19) (not (<= (* z t) 2e-119)))
   (* (- z) t)
   (fma z t (* y x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e-19) || !((z * t) <= 2e-119)) {
		tmp = -z * t;
	} else {
		tmp = fma(z, t, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -5e-19) || !(Float64(z * t) <= 2e-119))
		tmp = Float64(Float64(-z) * t);
	else
		tmp = fma(z, t, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -5e-19], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e-119]], $MachinePrecision]], N[((-z) * t), $MachinePrecision], N[(z * t + N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5.0000000000000004e-19 or 2.00000000000000003e-119 < (*.f64 z t)

   1. Initial program 97.6%

      \[x \cdot y - z \cdot t \]
   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 72.6

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

   5. Applied rewrites 72.6%

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

   if -5.0000000000000004e-19 < (*.f64 z t) < 2.00000000000000003e-119

   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. +-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 100.0

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

         \[\leadsto {\left(\left(-z\right) \cdot t\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} + y \cdot x \]

      5. sqrt-pow1 N/A

         \[\leadsto \color{blue}{\sqrt{{\left(\left(-z\right) \cdot t\right)}^{2}}} + y \cdot x \]

      6. pow2 N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. neg-fabs N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. pow2 N/A

          \[\leadsto \sqrt{\color{blue}{{\left(z \cdot t\right)}^{2}}} + y \cdot x \]

      14. sqrt-pow1 N/A

          \[\leadsto \color{blue}{{\left(z \cdot t\right)}^{\left(\frac{2}{2}\right)}} + y \cdot x \]

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. lower-fma.f64 87.8

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

   6. Applied rewrites 87.8%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{-19} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{-119}\right):\\ \;\;\;\;\left(-z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t, y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% 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. Add Preprocessing
3. Add Preprocessing

Alternative 4: 52.5% accurate, 1.8× speedup

Math

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

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(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. 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 53.8

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

5. Applied rewrites 53.8%

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

Alternative 5: 3.9% 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. 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 53.8

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

5. Applied rewrites 53.8%

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

7. Applied rewrites 3.4%

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

Reproduce

herbie shell --seed 2024339 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))