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

Percentage Accurate: 63.3% → 100.0%

Time: 11.1s

Alternatives: 4

Speedup: 3.0×

• 20.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 63.3% 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 64.8%

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

   1. associate-+l- 76.1%

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

   2. +-inverses 98.8%

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

   3. --rgt-identity 98.8%

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

   4. *-commutative 98.8%

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

   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. Add Preprocessing

Alternative 2: 77.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-61} \lor \neg \left(z \leq 6.5 \cdot 10^{+61}\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.2e-61) (not (<= z 6.5e+61))) (* y (- z)) (* y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.2e-61) || !(z <= 6.5e+61)) {
		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 <= (-2.2d-61)) .or. (.not. (z <= 6.5d+61))) 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 <= -2.2e-61) || !(z <= 6.5e+61)) {
		tmp = y * -z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.2e-61) or not (z <= 6.5e+61):
		tmp = y * -z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.2e-61) || !(z <= 6.5e+61))
		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 <= -2.2e-61) || ~((z <= 6.5e+61)))
		tmp = y * -z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.2e-61], N[Not[LessEqual[z, 6.5e+61]], $MachinePrecision]], N[(y * (-z)), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.20000000000000009e-61 or 6.4999999999999996e61 < z

   1. Initial program 72.9%

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

      1. associate-+l- 78.6%

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

      2. +-inverses 97.7%

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

      3. --rgt-identity 97.7%

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

      4. *-commutative 97.7%

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

      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 78.9%

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

      1. associate-*r* 78.9%

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

      2. neg-mul-1 78.9%

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

      3. *-commutative 78.9%

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

   7. Simplified 78.9%

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

   if -2.20000000000000009e-61 < z < 6.4999999999999996e61

   1. Initial program 55.6%

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

      1. associate-+l- 73.3%

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

      2. +-inverses 100.0%

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

      3. --rgt-identity 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-lft-out-- 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 76.6%

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

      1. *-commutative 76.6%

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

   7. Simplified 76.6%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-61} \lor \neg \left(z \leq 6.5 \cdot 10^{+61}\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
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 64.8%

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

   1. associate-+l- 76.1%

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

   2. +-inverses 98.8%

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

   3. --rgt-identity 98.8%

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

   4. *-commutative 98.8%

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

   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 51.3%

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

   1. *-commutative 51.3%

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

7. Simplified 51.3%

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

Alternative 4: 6.8% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 0.0)

C

double code(double x, double y, double z) {
	return 0.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 0.0;
}

Python

def code(x, y, z):
	return 0.0

Julia

function code(x, y, z)
	return 0.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_] := 0.0

Derivation

1. Initial program 64.8%

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

   1. associate-+l- 76.1%

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

   2. +-inverses 98.8%

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

   3. --rgt-identity 98.8%

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

   4. *-commutative 98.8%

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

   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 51.3%

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

   1. *-commutative 51.3%

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

7. Simplified 51.3%

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

   1. add-sqr-sqrt 25.5%

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

   2. sqrt-unprod 26.4%

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

   3. sqr-neg 26.4%

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

   4. sqrt-unprod 4.9%

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

   5. add-sqr-sqrt 8.0%

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

   6. distribute-lft-neg-in 8.0%

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

   7. neg-sub0 8.0%

      \[\leadsto \color{blue}{0 - y \cdot x} \]

9. Applied egg-rr 8.0%

   \[\leadsto \color{blue}{0 - y \cdot x} \]
10. Step-by-step derivation

    1. neg-sub0 8.0%

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

    2. distribute-lft-neg-in 8.0%

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

11. Simplified 8.0%

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

    1. add-log-exp 10.1%

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

    2. add-sqr-sqrt 10.1%

       \[\leadsto \log \color{blue}{\left(\sqrt{e^{\left(-y\right) \cdot x}} \cdot \sqrt{e^{\left(-y\right) \cdot x}}\right)} \]

    3. sqrt-unprod 10.1%

       \[\leadsto \log \color{blue}{\left(\sqrt{e^{\left(-y\right) \cdot x} \cdot e^{\left(-y\right) \cdot x}}\right)} \]

    4. distribute-lft-neg-out 10.1%

       \[\leadsto \log \left(\sqrt{e^{\left(-y\right) \cdot x} \cdot e^{\color{blue}{-y \cdot x}}}\right) \]

    5. distribute-rgt-neg-in 10.1%

       \[\leadsto \log \left(\sqrt{e^{\left(-y\right) \cdot x} \cdot e^{\color{blue}{y \cdot \left(-x\right)}}}\right) \]

    6. add-sqr-sqrt 3.8%

       \[\leadsto \log \left(\sqrt{e^{\left(-y\right) \cdot x} \cdot e^{\color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot \left(-x\right)}}\right) \]

    7. sqrt-unprod 8.1%

       \[\leadsto \log \left(\sqrt{e^{\left(-y\right) \cdot x} \cdot e^{\color{blue}{\sqrt{y \cdot y}} \cdot \left(-x\right)}}\right) \]

    8. sqr-neg 8.1%

       \[\leadsto \log \left(\sqrt{e^{\left(-y\right) \cdot x} \cdot e^{\sqrt{\color{blue}{\left(-y\right) \cdot \left(-y\right)}} \cdot \left(-x\right)}}\right) \]

    9. sqrt-unprod 4.4%

       \[\leadsto \log \left(\sqrt{e^{\left(-y\right) \cdot x} \cdot e^{\color{blue}{\left(\sqrt{-y} \cdot \sqrt{-y}\right)} \cdot \left(-x\right)}}\right) \]

    10. add-sqr-sqrt 6.7%

        \[\leadsto \log \left(\sqrt{e^{\left(-y\right) \cdot x} \cdot e^{\color{blue}{\left(-y\right)} \cdot \left(-x\right)}}\right) \]

    11. exp-prod 8.3%

        \[\leadsto \log \left(\sqrt{e^{\left(-y\right) \cdot x} \cdot \color{blue}{{\left(e^{-y}\right)}^{\left(-x\right)}}}\right) \]

    12. pow-flip 8.3%

        \[\leadsto \log \left(\sqrt{e^{\left(-y\right) \cdot x} \cdot \color{blue}{\frac{1}{{\left(e^{-y}\right)}^{x}}}}\right) \]

    13. exp-prod 6.7%

        \[\leadsto \log \left(\sqrt{e^{\left(-y\right) \cdot x} \cdot \frac{1}{\color{blue}{e^{\left(-y\right) \cdot x}}}}\right) \]

    14. rgt-mult-inverse 7.6%

        \[\leadsto \log \left(\sqrt{\color{blue}{1}}\right) \]

    15. metadata-eval 7.6%

        \[\leadsto \log \color{blue}{1} \]

    16. metadata-eval 7.6%

        \[\leadsto \color{blue}{0} \]

13. Applied egg-rr 7.6%

    \[\leadsto \color{blue}{0} \]
14. 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 2024165 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, B"
  :precision binary64

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

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