Linear.Quaternion:$c/ from linear-1.19.1.3, B

Percentage Accurate: 63.2% → 100.0%

Time: 4.3s

Alternatives: 3

Speedup: 3.0×

• 21.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 63.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.4%

   \[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y \]
2. Step-by-step derivation

   1. associate-+l- 75.0%

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

   2. +-inverses 97.6%

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

   3. --rgt-identity 97.6%

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

   4. *-commutative 97.6%

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

   5. distribute-lft-out-- 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{y \cdot \left(x - z\right)} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto y \cdot \left(x - z\right) \]
6. Add Preprocessing

Alternative 2: 74.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.28 \cdot 10^{-17} \lor \neg \left(x \leq -5.5 \cdot 10^{-59} \lor \neg \left(x \leq -5.5 \cdot 10^{-134}\right) \land x \leq 1.7 \cdot 10^{+115}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.28e-17)
         (not
          (or (<= x -5.5e-59) (and (not (<= x -5.5e-134)) (<= x 1.7e+115)))))
   (* y x)
   (* z (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.28e-17) || !((x <= -5.5e-59) || (!(x <= -5.5e-134) && (x <= 1.7e+115)))) {
		tmp = y * x;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.28d-17)) .or. (.not. (x <= (-5.5d-59)) .or. (.not. (x <= (-5.5d-134))) .and. (x <= 1.7d+115))) then
        tmp = y * x
    else
        tmp = z * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.28e-17) || !((x <= -5.5e-59) || (!(x <= -5.5e-134) && (x <= 1.7e+115)))) {
		tmp = y * x;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.28e-17) or not ((x <= -5.5e-59) or (not (x <= -5.5e-134) and (x <= 1.7e+115))):
		tmp = y * x
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.28e-17) || !((x <= -5.5e-59) || (!(x <= -5.5e-134) && (x <= 1.7e+115))))
		tmp = Float64(y * x);
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.28e-17) || ~(((x <= -5.5e-59) || (~((x <= -5.5e-134)) && (x <= 1.7e+115)))))
		tmp = y * x;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.28e-17], N[Not[Or[LessEqual[x, -5.5e-59], And[N[Not[LessEqual[x, -5.5e-134]], $MachinePrecision], LessEqual[x, 1.7e+115]]]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.28e-17 or -5.50000000000000014e-59 < x < -5.5000000000000002e-134 or 1.7e115 < x

   1. Initial program 66.9%

      \[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y \]
   2. Step-by-step derivation

      1. associate-+l- 74.1%

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

      2. +-inverses 96.6%

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

      3. --rgt-identity 96.6%

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

      4. *-commutative 96.6%

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

      5. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{y \cdot \left(x - z\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 86.2%

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

      1. *-commutative 86.2%

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

   7. Simplified 86.2%

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

   if -1.28e-17 < x < -5.50000000000000014e-59 or -5.5000000000000002e-134 < x < 1.7e115

   1. Initial program 56.6%

      \[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y \]
   2. Step-by-step derivation

      1. associate-+l- 75.7%

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

      2. +-inverses 98.5%

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

      3. --rgt-identity 98.5%

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

      4. *-commutative 98.5%

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

      5. distribute-lft-out-- 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{y \cdot \left(x - z\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 83.3%

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

      1. associate-*r* 83.3%

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

      2. *-commutative 83.3%

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

      3. mul-1-neg 83.3%

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

   7. Simplified 83.3%

      \[\leadsto \color{blue}{z \cdot \left(-y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.28 \cdot 10^{-17} \lor \neg \left(x \leq -5.5 \cdot 10^{-59} \lor \neg \left(x \leq -5.5 \cdot 10^{-134}\right) \land x \leq 1.7 \cdot 10^{+115}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.4%

   \[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y \]
2. Step-by-step derivation

   1. associate-+l- 75.0%

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

   2. +-inverses 97.6%

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

   3. --rgt-identity 97.6%

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

   4. *-commutative 97.6%

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

   5. distribute-lft-out-- 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{y \cdot \left(x - z\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 53.4%

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

   1. *-commutative 53.4%

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

7. Simplified 53.4%

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

8. Final simplification 53.4%

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

Developer target: 100.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024041 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (* (- x z) y)

  (+ (- (- (* x y) (* y z)) (* y y)) (* y y)))