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

Percentage Accurate: 63.0% → 100.0%

Time: 6.5s

Alternatives: 3

Speedup: 3.0×

• 20.5% 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: 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 64.7%

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

   1. cancel-sign-sub-inv 64.7%

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

   2. *-commutative 64.7%

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

   3. associate-+l+ 64.7%

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

   4. +-commutative 64.7%

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

   5. associate-+r+ 64.7%

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

   6. associate--l+ 79.3%

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

   7. +-inverses 99.2%

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

   8. +-rgt-identity 99.2%

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

   9. cancel-sign-sub-inv 99.2%

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

   10. 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. Add Preprocessing

Alternative 2: 78.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-15} \lor \neg \left(x \leq 2 \cdot 10^{-32}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.7e-15) (not (<= x 2e-32))) (* y x) (* z (- y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.7e-15) || !(x <= 2e-32)) {
		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 <= (-1.7d-15)) .or. (.not. (x <= 2d-32))) 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 <= -1.7e-15) || !(x <= 2e-32)) {
		tmp = y * x;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.7e-15) or not (x <= 2e-32):
		tmp = y * x
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.7e-15) || !(x <= 2e-32))
		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 <= -1.7e-15) || ~((x <= 2e-32)))
		tmp = y * x;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.7e-15], N[Not[LessEqual[x, 2e-32]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7e-15 or 2.00000000000000011e-32 < x

   1. Initial program 66.3%

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

      1. cancel-sign-sub-inv 66.3%

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

      2. *-commutative 66.3%

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

      3. associate-+l+ 66.3%

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

      4. +-commutative 66.3%

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

      5. associate-+r+ 66.3%

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

      6. associate--l+ 74.8%

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

      7. +-inverses 98.6%

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

      8. +-rgt-identity 98.6%

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

      9. cancel-sign-sub-inv 98.6%

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

      10. 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 79.7%

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

      1. *-commutative 79.7%

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

   7. Simplified 79.7%

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

   if -1.7e-15 < x < 2.00000000000000011e-32

   1. Initial program 62.8%

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

      1. cancel-sign-sub-inv 62.8%

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

      2. *-commutative 62.8%

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

      3. associate-+l+ 62.8%

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

      4. +-commutative 62.8%

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

      5. associate-+r+ 62.8%

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

      6. associate--l+ 85.0%

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

      7. +-inverses 100.0%

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

      8. +-rgt-identity 100.0%

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

      9. cancel-sign-sub-inv 100.0%

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

      10. 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 88.6%

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

      1. mul-1-neg 88.6%

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

      2. *-commutative 88.6%

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

      3. distribute-rgt-neg-in 88.6%

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

   7. Simplified 88.6%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-15} \lor \neg \left(x \leq 2 \cdot 10^{-32}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.0% 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 64.7%

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

   1. cancel-sign-sub-inv 64.7%

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

   2. *-commutative 64.7%

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

   3. associate-+l+ 64.7%

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

   4. +-commutative 64.7%

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

   5. associate-+r+ 64.7%

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

   6. associate--l+ 79.3%

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

   7. +-inverses 99.2%

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

   8. +-rgt-identity 99.2%

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

   9. cancel-sign-sub-inv 99.2%

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

   10. 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 55.0%

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

   1. *-commutative 55.0%

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

7. Simplified 55.0%

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

Developer Target 1: 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 2024170 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, C"
  :precision binary64

  :alt
  (! :herbie-platform default (* (- x z) y))

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