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

Percentage Accurate: 63.1% → 100.0%

Time: 2.9s

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: 63.1% 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 55.9%

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

   1. +-commutative 55.9%

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

   2. associate--l+ 55.9%

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

   3. +-commutative 55.9%

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

   4. associate--l+ 69.1%

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

   5. +-inverses 97.6%

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

   6. +-rgt-identity 97.6%

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

   7. *-commutative 97.6%

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

   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 2: 76.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+160}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -1.36 \cdot 10^{+128} \lor \neg \left(x \leq -1.86 \cdot 10^{-72}\right) \land x \leq 1.55 \cdot 10^{-14}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.1e+160)
   (* y x)
   (if (or (<= x -1.36e+128) (and (not (<= x -1.86e-72)) (<= x 1.55e-14)))
     (* y (- z))
     (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.1e+160) {
		tmp = y * x;
	} else if ((x <= -1.36e+128) || (!(x <= -1.86e-72) && (x <= 1.55e-14))) {
		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 (x <= (-3.1d+160)) then
        tmp = y * x
    else if ((x <= (-1.36d+128)) .or. (.not. (x <= (-1.86d-72))) .and. (x <= 1.55d-14)) 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 (x <= -3.1e+160) {
		tmp = y * x;
	} else if ((x <= -1.36e+128) || (!(x <= -1.86e-72) && (x <= 1.55e-14))) {
		tmp = y * -z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.1e+160:
		tmp = y * x
	elif (x <= -1.36e+128) or (not (x <= -1.86e-72) and (x <= 1.55e-14)):
		tmp = y * -z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.1e+160)
		tmp = Float64(y * x);
	elseif ((x <= -1.36e+128) || (!(x <= -1.86e-72) && (x <= 1.55e-14)))
		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 (x <= -3.1e+160)
		tmp = y * x;
	elseif ((x <= -1.36e+128) || (~((x <= -1.86e-72)) && (x <= 1.55e-14)))
		tmp = y * -z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.1e+160], N[(y * x), $MachinePrecision], If[Or[LessEqual[x, -1.36e+128], And[N[Not[LessEqual[x, -1.86e-72]], $MachinePrecision], LessEqual[x, 1.55e-14]]], N[(y * (-z)), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.0999999999999998e160 or -1.35999999999999991e128 < x < -1.85999999999999994e-72 or 1.55000000000000002e-14 < x

   1. Initial program 63.7%

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

      1. +-commutative 63.7%

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

      2. associate--l+ 63.7%

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

      3. +-commutative 63.7%

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

      4. associate--l+ 69.1%

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

      5. +-inverses 95.7%

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

      6. +-rgt-identity 95.7%

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

      7. *-commutative 95.7%

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

      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 80.3%

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

   if -3.0999999999999998e160 < x < -1.35999999999999991e128 or -1.85999999999999994e-72 < x < 1.55000000000000002e-14

   1. Initial program 46.6%

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

      1. +-commutative 46.6%

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

      2. associate--l+ 46.6%

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

      3. +-commutative 46.6%

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

      4. associate--l+ 69.2%

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

      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 0 87.7%

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

      1. mul-1-neg 87.7%

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

      2. distribute-rgt-neg-out 87.7%

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

   6. Simplified 87.7%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+160}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq -1.36 \cdot 10^{+128} \lor \neg \left(x \leq -1.86 \cdot 10^{-72}\right) \land x \leq 1.55 \cdot 10^{-14}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]

Alternative 3: 52.8% 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 55.9%

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

   1. +-commutative 55.9%

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

   2. associate--l+ 55.9%

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

   3. +-commutative 55.9%

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

   4. associate--l+ 69.1%

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

   5. +-inverses 97.6%

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

   6. +-rgt-identity 97.6%

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

   7. *-commutative 97.6%

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

   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 52.3%

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

5. Final simplification 52.3%

   \[\leadsto y \cdot x \]

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 2023173 
(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)))