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

Percentage Accurate: 63.0% → 99.0%

Time: 4.7s

Alternatives: 4

Speedup: 3.0×

• 21.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 63.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

   1. +-commutative 64.0%

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

   2. associate--l+ 64.0%

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

   3. +-commutative 64.0%

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

   4. associate--l+ 75.8%

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

   5. +-inverses 99.6%

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

   6. +-rgt-identity 99.6%

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

   7. *-commutative 99.6%

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

   8. 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. Step-by-step derivation

   1. flip-- 64.2%

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

   2. associate-*r/ 56.2%

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

5. Applied egg-rr 56.2%

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

   1. associate-/l* 64.0%

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

   2. difference-of-squares 65.0%

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

   3. associate-/r* 99.6%

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

   4. *-inverses 99.6%

      \[\leadsto \frac{y}{\frac{\color{blue}{1}}{x - z}} \]

7. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{y}{\frac{1}{x - z}}} \]
8. Step-by-step derivation

   1. associate-/r/ 100.0%

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

   2. /-rgt-identity 100.0%

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

   3. sub-neg 100.0%

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

   4. distribute-lft-in 99.6%

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

   5. *-commutative 99.6%

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

   6. distribute-lft-neg-out 99.6%

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

   7. unsub-neg 99.6%

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

   8. *-commutative 99.6%

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

9. Applied egg-rr 99.6%

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

    1. fma-neg 100.0%

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

    2. *-commutative 100.0%

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

    3. distribute-rgt-neg-in 100.0%

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

11. Applied egg-rr 100.0%

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

12. Final simplification 100.0%

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

Alternative 2: 78.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-23} \lor \neg \left(z \leq 4.4 \cdot 10^{-23}\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.35e-23) (not (<= z 4.4e-23))) (* y (- z)) (* y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.35e-23) || !(z <= 4.4e-23)) {
		tmp = y * -z;
	} else {
		tmp = y * x;
	}
	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 ((z <= (-1.35d-23)) .or. (.not. (z <= 4.4d-23))) then
        tmp = y * -z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.35e-23) || !(z <= 4.4e-23)) {
		tmp = y * -z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.35e-23) or not (z <= 4.4e-23):
		tmp = y * -z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.35e-23) || !(z <= 4.4e-23))
		tmp = Float64(y * Float64(-z));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.35e-23) || ~((z <= 4.4e-23)))
		tmp = y * -z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.35e-23], N[Not[LessEqual[z, 4.4e-23]], $MachinePrecision]], N[(y * (-z)), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.34999999999999992e-23 or 4.3999999999999999e-23 < z

   1. Initial program 73.3%

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

      1. +-commutative 73.3%

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

      2. associate--l+ 73.3%

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

      3. +-commutative 73.3%

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

      4. associate--l+ 80.0%

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

      5. +-inverses 99.2%

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

      6. +-rgt-identity 99.2%

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

      7. *-commutative 99.2%

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

      8. 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. Taylor expanded in x around 0 82.1%

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

      1. mul-1-neg 82.1%

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

      2. distribute-rgt-neg-out 82.1%

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

   6. Simplified 82.1%

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

   if -1.34999999999999992e-23 < z < 4.3999999999999999e-23

   1. Initial program 53.5%

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

      1. +-commutative 53.5%

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

      2. associate--l+ 53.5%

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

      3. +-commutative 53.5%

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

      4. associate--l+ 71.1%

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

      5. +-inverses 100.0%

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

      6. +-rgt-identity 100.0%

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

      7. *-commutative 100.0%

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

      8. 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. Taylor expanded in x around inf 83.1%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-23} \lor \neg \left(z \leq 4.4 \cdot 10^{-23}\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3: 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 64.0%

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

   1. +-commutative 64.0%

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

   2. associate--l+ 64.0%

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

   3. +-commutative 64.0%

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

   4. associate--l+ 75.8%

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

   5. +-inverses 99.6%

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

   6. +-rgt-identity 99.6%

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

   7. *-commutative 99.6%

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

   8. 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. Final simplification 100.0%

   \[\leadsto y \cdot \left(x - z\right) \]

Alternative 4: 53.6% 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 64.0%

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

   1. +-commutative 64.0%

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

   2. associate--l+ 64.0%

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

   3. +-commutative 64.0%

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

   4. associate--l+ 75.8%

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

   5. +-inverses 99.6%

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

   6. +-rgt-identity 99.6%

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

   7. *-commutative 99.6%

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

   8. 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. Taylor expanded in x around inf 50.0%

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

5. Final simplification 50.0%

   \[\leadsto y \cdot x \]

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 2023195 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, C"
  :precision binary64

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

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