Linear.V3:cross from linear-1.19.1.3

Percentage Accurate: 99.3% → 99.6%

Time: 5.7s

Alternatives: 4

Speedup: 1.0×

• 22.8% 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: 99.3% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0 - z, t, x \cdot y\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma (- 0.0 z) t (* x y)))

C

double code(double x, double y, double z, double t) {
	return fma((0.0 - z), t, (x * y));
}

Julia

function code(x, y, z, t)
	return fma(Float64(0.0 - z), t, Float64(x * y))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(0.0 - z), $MachinePrecision] * t + N[(x * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{0 - z}, t, x \cdot y\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{0 - z}, t, x \cdot y\right) \]

   7. *-lowering-*.f64 99.2

      \[\leadsto \mathsf{fma}\left(0 - z, t, \color{blue}{x \cdot y}\right) \]

4. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0 - z, t, x \cdot y\right)} \]
5. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 99.2

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

6. Applied egg-rr 99.2%

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

7. Final simplification 99.2%

   \[\leadsto \mathsf{fma}\left(0 - z, t, x \cdot y\right) \]
8. Add Preprocessing

Alternative 2: 75.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(0 - t\right)\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{-23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{-72}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (- 0.0 t))))
   (if (<= (* z t) -2e-23) t_1 (if (<= (* z t) 4e-72) (* x y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (0.0 - t);
	double tmp;
	if ((z * t) <= -2e-23) {
		tmp = t_1;
	} else if ((z * t) <= 4e-72) {
		tmp = x * 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 = z * (0.0d0 - t)
    if ((z * t) <= (-2d-23)) then
        tmp = t_1
    else if ((z * t) <= 4d-72) then
        tmp = x * 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 = z * (0.0 - t);
	double tmp;
	if ((z * t) <= -2e-23) {
		tmp = t_1;
	} else if ((z * t) <= 4e-72) {
		tmp = x * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * (0.0 - t)
	tmp = 0
	if (z * t) <= -2e-23:
		tmp = t_1
	elif (z * t) <= 4e-72:
		tmp = x * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(0.0 - t))
	tmp = 0.0
	if (Float64(z * t) <= -2e-23)
		tmp = t_1;
	elseif (Float64(z * t) <= 4e-72)
		tmp = Float64(x * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = z * (0.0 - t);
	tmp = 0.0;
	if ((z * t) <= -2e-23)
		tmp = t_1;
	elseif ((z * t) <= 4e-72)
		tmp = x * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(0.0 - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e-23], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 4e-72], N[(x * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.99999999999999992e-23 or 3.9999999999999999e-72 < (*.f64 z t)

   1. Initial program 97.8%

      \[x \cdot y - z \cdot t \]
   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. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - t \cdot z} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - t \cdot z} \]

      4. +-rgt-identity N/A

         \[\leadsto 0 - \color{blue}{\left(t \cdot z + 0\right)} \]

      5. accelerator-lowering-fma.f64 77.9

         \[\leadsto 0 - \color{blue}{\mathsf{fma}\left(t, z, 0\right)} \]

   5. Simplified 77.9%

      \[\leadsto \color{blue}{0 - \mathsf{fma}\left(t, z, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto 0 - \color{blue}{t \cdot z} \]

      2. *-commutative N/A

         \[\leadsto 0 - \color{blue}{z \cdot t} \]

      3. *-lowering-*.f64 77.9

         \[\leadsto 0 - \color{blue}{z \cdot t} \]

   7. Applied egg-rr 77.9%

      \[\leadsto 0 - \color{blue}{z \cdot t} \]

   if -1.99999999999999992e-23 < (*.f64 z t) < 3.9999999999999999e-72

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot y + 0} \]

      2. accelerator-lowering-fma.f64 81.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]

   5. Simplified 81.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 81.0

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

   7. Applied egg-rr 81.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{-23}:\\ \;\;\;\;z \cdot \left(0 - t\right)\\ \mathbf{elif}\;z \cdot t \leq 4 \cdot 10^{-72}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(0 - t\right)\\ \end{array} \]
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 98.8%

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

Alternative 4: 52.3% 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. Taylor expanded in x around inf

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot y + 0} \]

   2. accelerator-lowering-fma.f64 51.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]

5. Simplified 51.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 51.3

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

7. Applied egg-rr 51.3%

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

8. Final simplification 51.3%

   \[\leadsto x \cdot y \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024195 
(FPCore (x y z t)
  :name "Linear.V3:cross from linear-1.19.1.3"
  :precision binary64
  (- (* x y) (* z t)))