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

Percentage Accurate: 63.2% → 100.0%

Time: 7.6s

Alternatives: 3

Speedup: 3.3×

• 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: 63.2% 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, 3.3× 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]

Derivation

1. Initial program 61.5%

   \[\left(\left(x \cdot y + y \cdot y\right) - y \cdot z\right) - y \cdot y \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. distribute-lft-out-- N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 100.0

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

5. Applied rewrites 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 75.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \lor \neg \left(x \leq 3 \cdot 10^{-109}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.7) (not (<= x 3e-109))) (* x y) (* (- z) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.7) || !(x <= 3e-109)) {
		tmp = x * y;
	} 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 <= (-3.7d0)) .or. (.not. (x <= 3d-109))) then
        tmp = x * y
    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 <= -3.7) || !(x <= 3e-109)) {
		tmp = x * y;
	} else {
		tmp = -z * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.7) or not (x <= 3e-109):
		tmp = x * y
	else:
		tmp = -z * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.7) || !(x <= 3e-109))
		tmp = Float64(x * y);
	else
		tmp = Float64(Float64(-z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.7) || ~((x <= 3e-109)))
		tmp = x * y;
	else
		tmp = -z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.7], N[Not[LessEqual[x, 3e-109]], $MachinePrecision]], N[(x * y), $MachinePrecision], N[((-z) * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7000000000000002 or 3.00000000000000021e-109 < x

   1. Initial program 67.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-out-- N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 71.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 71.9

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

   4. Applied rewrites 71.9%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-out-- N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 77.3

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

   6. Applied rewrites 77.3%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 76.2

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

   9. Applied rewrites 76.2%

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

   if -3.7000000000000002 < x < 3.00000000000000021e-109

   1. Initial program 52.0%

      \[\left(\left(x \cdot y + y \cdot y\right) - y \cdot z\right) - y \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 86.7

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

   5. Applied rewrites 86.7%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \lor \neg \left(x \leq 3 \cdot 10^{-109}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-lft-out-- N/A

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

   8. lower-fma.f64 N/A

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

   9. lower--.f64 64.7

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 64.7

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

4. Applied rewrites 64.7%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate--l+ N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-lft-out-- N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 69.3

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

6. Applied rewrites 69.3%

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

7. Taylor expanded in x around inf

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

   1. lower-*.f64 54.2

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

9. Applied rewrites 54.2%

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

Developer Target 1: 100.0% accurate, 3.3× 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 2024337 
(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)))