Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, B

Percentage Accurate: 99.8% → 99.8%

Time: 5.4s

Alternatives: 5

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot 3\right) \cdot y - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* (* x 3.0) y) z))

C

double code(double x, double y, double z) {
	return ((x * 3.0) * y) - 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 = ((x * 3.0d0) * y) - z
end function

Java

public static double code(double x, double y, double z) {
	return ((x * 3.0) * y) - z;
}

Python

def code(x, y, z):
	return ((x * 3.0) * y) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * 3.0) * y) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * 3.0) * y) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 3\right) \cdot y - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* (* x 3.0) y) z))

C

double code(double x, double y, double z) {
	return ((x * 3.0) * y) - 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 = ((x * 3.0d0) * y) - z
end function

Java

public static double code(double x, double y, double z) {
	return ((x * 3.0) * y) - z;
}

Python

def code(x, y, z):
	return ((x * 3.0) * y) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * 3.0) * y) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * 3.0) * y) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(3, y \cdot x, -z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma 3.0 (* y x) (- z)))

C

double code(double x, double y, double z) {
	return fma(3.0, (y * x), -z);
}

Julia

function code(x, y, z)
	return fma(3.0, Float64(y * x), Float64(-z))
end

Wolfram

code[x_, y_, z_] := N[(3.0 * N[(y * x), $MachinePrecision] + (-z)), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. associate-*l* 99.8%

      \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right)} - z \]

   2. fma-neg 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3 \cdot y, -z\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3 \cdot y, -z\right)} \]
4. Step-by-step derivation

   1. fma-udef 99.8%

      \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right) + \left(-z\right)} \]

   2. add-sqr-sqrt 54.8%

      \[\leadsto x \cdot \left(3 \cdot y\right) + \color{blue}{\sqrt{-z} \cdot \sqrt{-z}} \]

   3. sqrt-unprod 61.3%

      \[\leadsto x \cdot \left(3 \cdot y\right) + \color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}} \]

   4. sqr-neg 61.3%

      \[\leadsto x \cdot \left(3 \cdot y\right) + \sqrt{\color{blue}{z \cdot z}} \]

   5. sqrt-unprod 22.0%

      \[\leadsto x \cdot \left(3 \cdot y\right) + \color{blue}{\sqrt{z} \cdot \sqrt{z}} \]

   6. add-sqr-sqrt 47.4%

      \[\leadsto x \cdot \left(3 \cdot y\right) + \color{blue}{z} \]

5. Applied egg-rr 47.4%

   \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right) + z} \]
6. Step-by-step derivation

   1. *-commutative 47.4%

      \[\leadsto \color{blue}{\left(3 \cdot y\right) \cdot x} + z \]

   2. associate-*r* 47.4%

      \[\leadsto \color{blue}{3 \cdot \left(y \cdot x\right)} + z \]

   3. *-commutative 47.4%

      \[\leadsto 3 \cdot \color{blue}{\left(x \cdot y\right)} + z \]

   4. fma-def 47.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3, x \cdot y, z\right)} \]

   5. *-commutative 47.4%

      \[\leadsto \mathsf{fma}\left(3, \color{blue}{y \cdot x}, z\right) \]

7. Simplified 47.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(3, y \cdot x, z\right)} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 22.0%

      \[\leadsto \mathsf{fma}\left(3, y \cdot x, \color{blue}{\sqrt{z} \cdot \sqrt{z}}\right) \]

   2. sqrt-unprod 61.3%

      \[\leadsto \mathsf{fma}\left(3, y \cdot x, \color{blue}{\sqrt{z \cdot z}}\right) \]

   3. add-sqr-sqrt 20.7%

      \[\leadsto \mathsf{fma}\left(3, y \cdot x, \sqrt{\color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)} \cdot z}\right) \]

   4. add-sqr-sqrt 20.7%

      \[\leadsto \mathsf{fma}\left(3, y \cdot x, \sqrt{\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)}}\right) \]

   5. sqr-neg 20.7%

      \[\leadsto \mathsf{fma}\left(3, y \cdot x, \sqrt{\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \color{blue}{\left(\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)\right)}}\right) \]

   6. neg-mul-1 20.7%

      \[\leadsto \mathsf{fma}\left(3, y \cdot x, \sqrt{\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \left(\left(-\sqrt{z}\right) \cdot \color{blue}{\left(-1 \cdot \sqrt{z}\right)}\right)}\right) \]

   7. neg-mul-1 20.7%

      \[\leadsto \mathsf{fma}\left(3, y \cdot x, \sqrt{\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \left(\color{blue}{\left(-1 \cdot \sqrt{z}\right)} \cdot \left(-1 \cdot \sqrt{z}\right)\right)}\right) \]

   8. swap-sqr 20.7%

      \[\leadsto \mathsf{fma}\left(3, y \cdot x, \sqrt{\color{blue}{\left(\sqrt{z} \cdot \left(-1 \cdot \sqrt{z}\right)\right) \cdot \left(\sqrt{z} \cdot \left(-1 \cdot \sqrt{z}\right)\right)}}\right) \]

   9. sqrt-unprod 0.0%

      \[\leadsto \mathsf{fma}\left(3, y \cdot x, \color{blue}{\sqrt{\sqrt{z} \cdot \left(-1 \cdot \sqrt{z}\right)} \cdot \sqrt{\sqrt{z} \cdot \left(-1 \cdot \sqrt{z}\right)}}\right) \]

   10. add-sqr-sqrt 44.7%

       \[\leadsto \mathsf{fma}\left(3, y \cdot x, \color{blue}{\sqrt{z} \cdot \left(-1 \cdot \sqrt{z}\right)}\right) \]

   11. neg-mul-1 44.7%

       \[\leadsto \mathsf{fma}\left(3, y \cdot x, \sqrt{z} \cdot \color{blue}{\left(-\sqrt{z}\right)}\right) \]

   12. distribute-rgt-neg-out 44.7%

       \[\leadsto \mathsf{fma}\left(3, y \cdot x, \color{blue}{-\sqrt{z} \cdot \sqrt{z}}\right) \]

   13. add-sqr-sqrt 99.9%

       \[\leadsto \mathsf{fma}\left(3, y \cdot x, -\color{blue}{z}\right) \]

   14. fma-neg 99.9%

       \[\leadsto \color{blue}{3 \cdot \left(y \cdot x\right) - z} \]

   15. add-sqr-sqrt 44.7%

       \[\leadsto 3 \cdot \left(y \cdot x\right) - \color{blue}{\sqrt{z} \cdot \sqrt{z}} \]

   16. sqrt-unprod 56.8%

       \[\leadsto 3 \cdot \left(y \cdot x\right) - \color{blue}{\sqrt{z \cdot z}} \]

   17. add-sqr-sqrt 32.5%

       \[\leadsto 3 \cdot \left(y \cdot x\right) - \sqrt{\color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)} \cdot z} \]

   18. add-sqr-sqrt 32.4%

       \[\leadsto 3 \cdot \left(y \cdot x\right) - \sqrt{\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)}} \]

   19. sqr-neg 32.4%

       \[\leadsto 3 \cdot \left(y \cdot x\right) - \sqrt{\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \color{blue}{\left(\left(-\sqrt{z}\right) \cdot \left(-\sqrt{z}\right)\right)}} \]

   20. neg-mul-1 32.4%

       \[\leadsto 3 \cdot \left(y \cdot x\right) - \sqrt{\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \left(\left(-\sqrt{z}\right) \cdot \color{blue}{\left(-1 \cdot \sqrt{z}\right)}\right)} \]

   21. neg-mul-1 32.4%

       \[\leadsto 3 \cdot \left(y \cdot x\right) - \sqrt{\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \left(\color{blue}{\left(-1 \cdot \sqrt{z}\right)} \cdot \left(-1 \cdot \sqrt{z}\right)\right)} \]

9. Applied egg-rr 44.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(3, y \cdot x, -\left({z}^{0.25} \cdot \sqrt{z}\right) \cdot {z}^{0.25}\right) + \mathsf{fma}\left(-{z}^{0.25} \cdot \sqrt{z}, {z}^{0.25}, \left({z}^{0.25} \cdot \sqrt{z}\right) \cdot {z}^{0.25}\right)} \]
10. Step-by-step derivation

    1. associate-*r* 44.7%

       \[\leadsto \mathsf{fma}\left(3, y \cdot x, -\color{blue}{{z}^{0.25} \cdot \left(\sqrt{z} \cdot {z}^{0.25}\right)}\right) + \mathsf{fma}\left(-{z}^{0.25} \cdot \sqrt{z}, {z}^{0.25}, \left({z}^{0.25} \cdot \sqrt{z}\right) \cdot {z}^{0.25}\right) \]

    2. fma-udef 44.7%

       \[\leadsto \mathsf{fma}\left(3, y \cdot x, -{z}^{0.25} \cdot \left(\sqrt{z} \cdot {z}^{0.25}\right)\right) + \color{blue}{\left(\left(-{z}^{0.25} \cdot \sqrt{z}\right) \cdot {z}^{0.25} + \left({z}^{0.25} \cdot \sqrt{z}\right) \cdot {z}^{0.25}\right)} \]

    3. distribute-rgt-out 44.7%

       \[\leadsto \mathsf{fma}\left(3, y \cdot x, -{z}^{0.25} \cdot \left(\sqrt{z} \cdot {z}^{0.25}\right)\right) + \color{blue}{{z}^{0.25} \cdot \left(\left(-{z}^{0.25} \cdot \sqrt{z}\right) + {z}^{0.25} \cdot \sqrt{z}\right)} \]

    4. distribute-lft-out 44.7%

       \[\leadsto \mathsf{fma}\left(3, y \cdot x, -{z}^{0.25} \cdot \left(\sqrt{z} \cdot {z}^{0.25}\right)\right) + \color{blue}{\left({z}^{0.25} \cdot \left(-{z}^{0.25} \cdot \sqrt{z}\right) + {z}^{0.25} \cdot \left({z}^{0.25} \cdot \sqrt{z}\right)\right)} \]

11. Simplified 99.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(3, y \cdot x, -z\right)} \]

12. Final simplification 99.9%

    \[\leadsto \mathsf{fma}\left(3, y \cdot x, -z\right) \]

Alternative 2: 67.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+116} \lor \neg \left(x \leq 8 \cdot 10^{-93}\right):\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -7.2e+116) (not (<= x 8e-93))) (* 3.0 (* y x)) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.2e+116) || !(x <= 8e-93)) {
		tmp = 3.0 * (y * x);
	} else {
		tmp = -z;
	}
	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 <= (-7.2d+116)) .or. (.not. (x <= 8d-93))) then
        tmp = 3.0d0 * (y * x)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.2e+116) || !(x <= 8e-93)) {
		tmp = 3.0 * (y * x);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -7.2e+116) or not (x <= 8e-93):
		tmp = 3.0 * (y * x)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -7.2e+116) || !(x <= 8e-93))
		tmp = Float64(3.0 * Float64(y * x));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -7.2e+116) || ~((x <= 8e-93)))
		tmp = 3.0 * (y * x);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -7.2e+116], N[Not[LessEqual[x, 8e-93]], $MachinePrecision]], N[(3.0 * N[(y * x), $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -7.19999999999999941e116 or 7.9999999999999992e-93 < x

   1. Initial program 99.8%

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

      1. associate-*l* 99.7%

         \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right)} - z \]

      2. fma-neg 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3 \cdot y, -z\right)} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3 \cdot y, -z\right)} \]
   4. Step-by-step derivation

      1. fma-udef 99.7%

         \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right) + \left(-z\right)} \]

      2. add-sqr-sqrt 53.3%

         \[\leadsto x \cdot \left(3 \cdot y\right) + \color{blue}{\sqrt{-z} \cdot \sqrt{-z}} \]

      3. sqrt-unprod 80.1%

         \[\leadsto x \cdot \left(3 \cdot y\right) + \color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}} \]

      4. sqr-neg 80.1%

         \[\leadsto x \cdot \left(3 \cdot y\right) + \sqrt{\color{blue}{z \cdot z}} \]

      5. sqrt-unprod 39.3%

         \[\leadsto x \cdot \left(3 \cdot y\right) + \color{blue}{\sqrt{z} \cdot \sqrt{z}} \]

      6. add-sqr-sqrt 74.0%

         \[\leadsto x \cdot \left(3 \cdot y\right) + \color{blue}{z} \]

   5. Applied egg-rr 74.0%

      \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right) + z} \]
   6. Step-by-step derivation

      1. *-commutative 74.0%

         \[\leadsto \color{blue}{\left(3 \cdot y\right) \cdot x} + z \]

      2. associate-*r* 74.1%

         \[\leadsto \color{blue}{3 \cdot \left(y \cdot x\right)} + z \]

      3. *-commutative 74.1%

         \[\leadsto 3 \cdot \color{blue}{\left(x \cdot y\right)} + z \]

      4. fma-def 74.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(3, x \cdot y, z\right)} \]

      5. *-commutative 74.0%

         \[\leadsto \mathsf{fma}\left(3, \color{blue}{y \cdot x}, z\right) \]

   7. Simplified 74.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3, y \cdot x, z\right)} \]
   8. Step-by-step derivation

      1. fma-udef 74.1%

         \[\leadsto \color{blue}{3 \cdot \left(y \cdot x\right) + z} \]

      2. flip-+ 32.0%

         \[\leadsto \color{blue}{\frac{\left(3 \cdot \left(y \cdot x\right)\right) \cdot \left(3 \cdot \left(y \cdot x\right)\right) - z \cdot z}{3 \cdot \left(y \cdot x\right) - z}} \]

      3. *-commutative 32.0%

         \[\leadsto \frac{\left(3 \cdot \left(y \cdot x\right)\right) \cdot \color{blue}{\left(\left(y \cdot x\right) \cdot 3\right)} - z \cdot z}{3 \cdot \left(y \cdot x\right) - z} \]

      4. *-commutative 32.0%

         \[\leadsto \frac{\color{blue}{\left(\left(y \cdot x\right) \cdot 3\right)} \cdot \left(\left(y \cdot x\right) \cdot 3\right) - z \cdot z}{3 \cdot \left(y \cdot x\right) - z} \]

      5. swap-sqr 31.9%

         \[\leadsto \frac{\color{blue}{\left(\left(y \cdot x\right) \cdot \left(y \cdot x\right)\right) \cdot \left(3 \cdot 3\right)} - z \cdot z}{3 \cdot \left(y \cdot x\right) - z} \]

      6. pow2 31.9%

         \[\leadsto \frac{\color{blue}{{\left(y \cdot x\right)}^{2}} \cdot \left(3 \cdot 3\right) - z \cdot z}{3 \cdot \left(y \cdot x\right) - z} \]

      7. metadata-eval 31.9%

         \[\leadsto \frac{{\left(y \cdot x\right)}^{2} \cdot \color{blue}{9} - z \cdot z}{3 \cdot \left(y \cdot x\right) - z} \]

      8. *-commutative 31.9%

         \[\leadsto \frac{{\left(y \cdot x\right)}^{2} \cdot 9 - z \cdot z}{\color{blue}{\left(y \cdot x\right) \cdot 3} - z} \]

      9. associate-*l* 31.8%

         \[\leadsto \frac{{\left(y \cdot x\right)}^{2} \cdot 9 - z \cdot z}{\color{blue}{y \cdot \left(x \cdot 3\right)} - z} \]

      10. fma-neg 31.8%

          \[\leadsto \frac{{\left(y \cdot x\right)}^{2} \cdot 9 - z \cdot z}{\color{blue}{\mathsf{fma}\left(y, x \cdot 3, -z\right)}} \]

      11. add-sqr-sqrt 13.5%

          \[\leadsto \frac{{\left(y \cdot x\right)}^{2} \cdot 9 - z \cdot z}{\mathsf{fma}\left(y, x \cdot 3, -\color{blue}{\sqrt{z} \cdot \sqrt{z}}\right)} \]

      12. distribute-rgt-neg-out 13.5%

          \[\leadsto \frac{{\left(y \cdot x\right)}^{2} \cdot 9 - z \cdot z}{\mathsf{fma}\left(y, x \cdot 3, \color{blue}{\sqrt{z} \cdot \left(-\sqrt{z}\right)}\right)} \]

      13. neg-mul-1 13.5%

          \[\leadsto \frac{{\left(y \cdot x\right)}^{2} \cdot 9 - z \cdot z}{\mathsf{fma}\left(y, x \cdot 3, \sqrt{z} \cdot \color{blue}{\left(-1 \cdot \sqrt{z}\right)}\right)} \]

      14. add-sqr-sqrt 0.0%

          \[\leadsto \frac{{\left(y \cdot x\right)}^{2} \cdot 9 - z \cdot z}{\mathsf{fma}\left(y, x \cdot 3, \color{blue}{\sqrt{\sqrt{z} \cdot \left(-1 \cdot \sqrt{z}\right)} \cdot \sqrt{\sqrt{z} \cdot \left(-1 \cdot \sqrt{z}\right)}}\right)} \]

      15. sqrt-unprod 17.0%

          \[\leadsto \frac{{\left(y \cdot x\right)}^{2} \cdot 9 - z \cdot z}{\mathsf{fma}\left(y, x \cdot 3, \color{blue}{\sqrt{\left(\sqrt{z} \cdot \left(-1 \cdot \sqrt{z}\right)\right) \cdot \left(\sqrt{z} \cdot \left(-1 \cdot \sqrt{z}\right)\right)}}\right)} \]

      16. swap-sqr 17.0%

          \[\leadsto \frac{{\left(y \cdot x\right)}^{2} \cdot 9 - z \cdot z}{\mathsf{fma}\left(y, x \cdot 3, \sqrt{\color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \left(\left(-1 \cdot \sqrt{z}\right) \cdot \left(-1 \cdot \sqrt{z}\right)\right)}}\right)} \]

      17. add-sqr-sqrt 17.0%

          \[\leadsto \frac{{\left(y \cdot x\right)}^{2} \cdot 9 - z \cdot z}{\mathsf{fma}\left(y, x \cdot 3, \sqrt{\color{blue}{z} \cdot \left(\left(-1 \cdot \sqrt{z}\right) \cdot \left(-1 \cdot \sqrt{z}\right)\right)}\right)} \]

      18. neg-mul-1 17.0%

          \[\leadsto \frac{{\left(y \cdot x\right)}^{2} \cdot 9 - z \cdot z}{\mathsf{fma}\left(y, x \cdot 3, \sqrt{z \cdot \left(\color{blue}{\left(-\sqrt{z}\right)} \cdot \left(-1 \cdot \sqrt{z}\right)\right)}\right)} \]

   9. Applied egg-rr 45.0%

      \[\leadsto \color{blue}{\frac{{\left(y \cdot x\right)}^{2} \cdot 9 - z \cdot z}{\mathsf{fma}\left(y, 3 \cdot x, z\right)}} \]

   10. Taylor expanded in y around inf 74.7%

       \[\leadsto \color{blue}{3 \cdot \left(y \cdot x\right)} \]

   if -7.19999999999999941e116 < x < 7.9999999999999992e-93

   1. Initial program 99.9%

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

      1. associate-*l* 99.9%

         \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right)} - z \]

      2. fma-neg 99.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3 \cdot y, -z\right)} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3 \cdot y, -z\right)} \]

   4. Taylor expanded in x around 0 73.6%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   5. Step-by-step derivation

      1. mul-1-neg 73.6%

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

   6. Simplified 73.6%

      \[\leadsto \color{blue}{-z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+116} \lor \neg \left(x \leq 8 \cdot 10^{-93}\right):\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(y \cdot x\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* 3.0 (* y x)) z))

C

double code(double x, double y, double z) {
	return (3.0 * (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 = (3.0d0 * (y * x)) - z
end function

Java

public static double code(double x, double y, double z) {
	return (3.0 * (y * x)) - z;
}

Python

def code(x, y, z):
	return (3.0 * (y * x)) - z

Julia

function code(x, y, z)
	return Float64(Float64(3.0 * Float64(y * x)) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (3.0 * (y * x)) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(3.0 * N[(y * x), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(x \cdot 3\right) \cdot y - z \]

2. Taylor expanded in x around 0 99.9%

   \[\leadsto \color{blue}{3 \cdot \left(y \cdot x\right)} - z \]

3. Final simplification 99.9%

   \[\leadsto 3 \cdot \left(y \cdot x\right) - z \]

Alternative 4: 50.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 99.9%

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

   1. associate-*l* 99.8%

      \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right)} - z \]

   2. fma-neg 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3 \cdot y, -z\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3 \cdot y, -z\right)} \]

4. Taylor expanded in x around 0 53.2%

   \[\leadsto \color{blue}{-1 \cdot z} \]
5. Step-by-step derivation

   1. mul-1-neg 53.2%

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

6. Simplified 53.2%

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

7. Final simplification 53.2%

   \[\leadsto -z \]

Alternative 5: 2.3% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 99.9%

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

   1. associate-*l* 99.8%

      \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right)} - z \]

   2. fma-neg 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3 \cdot y, -z\right)} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3 \cdot y, -z\right)} \]
4. Step-by-step derivation

   1. fma-udef 99.8%

      \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right) + \left(-z\right)} \]

   2. add-sqr-sqrt 54.8%

      \[\leadsto x \cdot \left(3 \cdot y\right) + \color{blue}{\sqrt{-z} \cdot \sqrt{-z}} \]

   3. sqrt-unprod 61.3%

      \[\leadsto x \cdot \left(3 \cdot y\right) + \color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}} \]

   4. sqr-neg 61.3%

      \[\leadsto x \cdot \left(3 \cdot y\right) + \sqrt{\color{blue}{z \cdot z}} \]

   5. sqrt-unprod 22.0%

      \[\leadsto x \cdot \left(3 \cdot y\right) + \color{blue}{\sqrt{z} \cdot \sqrt{z}} \]

   6. add-sqr-sqrt 47.4%

      \[\leadsto x \cdot \left(3 \cdot y\right) + \color{blue}{z} \]

5. Applied egg-rr 47.4%

   \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right) + z} \]
6. Step-by-step derivation

   1. *-commutative 47.4%

      \[\leadsto \color{blue}{\left(3 \cdot y\right) \cdot x} + z \]

   2. associate-*r* 47.4%

      \[\leadsto \color{blue}{3 \cdot \left(y \cdot x\right)} + z \]

   3. *-commutative 47.4%

      \[\leadsto 3 \cdot \color{blue}{\left(x \cdot y\right)} + z \]

   4. fma-def 47.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3, x \cdot y, z\right)} \]

   5. *-commutative 47.4%

      \[\leadsto \mathsf{fma}\left(3, \color{blue}{y \cdot x}, z\right) \]

7. Simplified 47.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(3, y \cdot x, z\right)} \]

8. Taylor expanded in y around 0 2.2%

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

9. Final simplification 2.2%

   \[\leadsto z \]

Developer target: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(3 \cdot y\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* x (* 3.0 y)) z))

C

double code(double x, double y, double z) {
	return (x * (3.0 * y)) - 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 = (x * (3.0d0 * y)) - z
end function

Java

public static double code(double x, double y, double z) {
	return (x * (3.0 * y)) - z;
}

Python

def code(x, y, z):
	return (x * (3.0 * y)) - z

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(3.0 * y)) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (3.0 * y)) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(3.0 * y), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Reproduce

herbie shell --seed 2023257 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (- (* x (* 3.0 y)) z)

  (- (* (* x 3.0) y) z))