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

Percentage Accurate: 99.8% → 99.8%

Time: 5.0s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\left(x \cdot 3\right) \cdot y - z \]
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. Final simplification 99.9%

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

Alternative 2: 69.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+105}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-76}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.25e+105) (- z) (if (<= z 2e-76) (* 3.0 (* x y)) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.25e+105) {
		tmp = -z;
	} else if (z <= 2e-76) {
		tmp = 3.0 * (x * y);
	} 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 (z <= (-1.25d+105)) then
        tmp = -z
    else if (z <= 2d-76) then
        tmp = 3.0d0 * (x * y)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.25e+105) {
		tmp = -z;
	} else if (z <= 2e-76) {
		tmp = 3.0 * (x * y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.25e+105:
		tmp = -z
	elif z <= 2e-76:
		tmp = 3.0 * (x * y)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.25e+105)
		tmp = Float64(-z);
	elseif (z <= 2e-76)
		tmp = Float64(3.0 * Float64(x * y));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.25e+105)
		tmp = -z;
	elseif (z <= 2e-76)
		tmp = 3.0 * (x * y);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.25e+105], (-z), If[LessEqual[z, 2e-76], N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25000000000000011e105 or 1.99999999999999985e-76 < z

   1. Initial program 99.1%

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

      1. associate-*l* 100.0%

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

      2. fma-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 77.8%

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

      1. mul-1-neg 77.8%

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

   6. Simplified 77.8%

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

   if -1.25000000000000011e105 < z < 1.99999999999999985e-76

   1. Initial program 99.8%

      \[\left(x \cdot 3\right) \cdot y - z \]
   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. add-sqr-sqrt 36.1%

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

      2. sqrt-unprod 82.7%

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

      3. sqr-neg 82.7%

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

      4. sqrt-unprod 48.4%

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

      5. add-sqr-sqrt 79.1%

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

      6. fma-neg 79.1%

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

      7. add-cube-cbrt 79.1%

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

      8. distribute-rgt-neg-in 79.1%

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

      9. associate-*l* 79.0%

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

      10. add-cube-cbrt 77.9%

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

      11. associate-*l* 77.9%

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

      12. prod-diff 77.9%

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

   5. Applied egg-rr 55.1%

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

   7. Simplified 55.2%

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

   8. Taylor expanded in x around inf 79.3%

      \[\leadsto \color{blue}{y \cdot \left(\sqrt[3]{9} \cdot \left(\sqrt[3]{3} \cdot x\right)\right)} \]

   10. Applied egg-rr 79.6%

       \[\leadsto \color{blue}{0 - x \cdot \left(y \cdot -3\right)} \]
   11. Step-by-step derivation

       1. neg-sub0 79.6%

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

       2. distribute-rgt-neg-in 79.6%

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

       3. distribute-rgt-neg-in 79.6%

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

       4. metadata-eval 79.6%

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

   12. Simplified 79.6%

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

   13. Taylor expanded in x around 0 79.5%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+105}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-76}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3: 69.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+105}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-76}:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.7e+105) (- z) (if (<= z 2e-76) (* x (* 3.0 y)) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.7e+105) {
		tmp = -z;
	} else if (z <= 2e-76) {
		tmp = x * (3.0 * y);
	} 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 (z <= (-1.7d+105)) then
        tmp = -z
    else if (z <= 2d-76) then
        tmp = x * (3.0d0 * y)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.7e+105) {
		tmp = -z;
	} else if (z <= 2e-76) {
		tmp = x * (3.0 * y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.7e+105:
		tmp = -z
	elif z <= 2e-76:
		tmp = x * (3.0 * y)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.7e+105)
		tmp = Float64(-z);
	elseif (z <= 2e-76)
		tmp = Float64(x * Float64(3.0 * y));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.7e+105)
		tmp = -z;
	elseif (z <= 2e-76)
		tmp = x * (3.0 * y);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.7e+105], (-z), If[LessEqual[z, 2e-76], N[(x * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7e105 or 1.99999999999999985e-76 < z

   1. Initial program 99.1%

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

      1. associate-*l* 100.0%

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

      2. fma-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 77.8%

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

      1. mul-1-neg 77.8%

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

   6. Simplified 77.8%

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

   if -1.7e105 < z < 1.99999999999999985e-76

   1. Initial program 99.8%

      \[\left(x \cdot 3\right) \cdot y - z \]
   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. add-sqr-sqrt 36.1%

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

      2. sqrt-unprod 82.7%

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

      3. sqr-neg 82.7%

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

      4. sqrt-unprod 48.4%

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

      5. add-sqr-sqrt 79.1%

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

      6. fma-neg 79.1%

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

      7. add-cube-cbrt 79.1%

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

      8. distribute-rgt-neg-in 79.1%

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

      9. associate-*l* 79.0%

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

      10. add-cube-cbrt 77.9%

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

      11. associate-*l* 77.9%

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

      12. prod-diff 77.9%

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

   5. Applied egg-rr 55.1%

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

   7. Simplified 55.2%

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

   8. Taylor expanded in x around inf 79.3%

      \[\leadsto \color{blue}{y \cdot \left(\sqrt[3]{9} \cdot \left(\sqrt[3]{3} \cdot x\right)\right)} \]

   10. Applied egg-rr 79.6%

       \[\leadsto \color{blue}{0 - x \cdot \left(y \cdot -3\right)} \]
   11. Step-by-step derivation

       1. neg-sub0 79.6%

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

       2. distribute-rgt-neg-in 79.6%

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

       3. distribute-rgt-neg-in 79.6%

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

       4. metadata-eval 79.6%

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

   12. Simplified 79.6%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+105}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-76}:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 4: 69.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+105}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-76}:\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.25e+105) (- z) (if (<= z 2e-76) (* y (* x 3.0)) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.25e+105) {
		tmp = -z;
	} else if (z <= 2e-76) {
		tmp = y * (x * 3.0);
	} 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 (z <= (-1.25d+105)) then
        tmp = -z
    else if (z <= 2d-76) then
        tmp = y * (x * 3.0d0)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.25e+105) {
		tmp = -z;
	} else if (z <= 2e-76) {
		tmp = y * (x * 3.0);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.25e+105:
		tmp = -z
	elif z <= 2e-76:
		tmp = y * (x * 3.0)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.25e+105)
		tmp = Float64(-z);
	elseif (z <= 2e-76)
		tmp = Float64(y * Float64(x * 3.0));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.25e+105)
		tmp = -z;
	elseif (z <= 2e-76)
		tmp = y * (x * 3.0);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.25e+105], (-z), If[LessEqual[z, 2e-76], N[(y * N[(x * 3.0), $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25000000000000011e105 or 1.99999999999999985e-76 < z

   1. Initial program 99.1%

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

      1. associate-*l* 100.0%

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

      2. fma-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 77.8%

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

      1. mul-1-neg 77.8%

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

   6. Simplified 77.8%

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

   if -1.25000000000000011e105 < z < 1.99999999999999985e-76

   1. Initial program 99.8%

      \[\left(x \cdot 3\right) \cdot y - z \]
   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. add-sqr-sqrt 36.1%

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

      2. sqrt-unprod 82.7%

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

      3. sqr-neg 82.7%

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

      4. sqrt-unprod 48.4%

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

      5. add-sqr-sqrt 79.1%

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

      6. fma-neg 79.1%

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

      7. add-cube-cbrt 79.1%

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

      8. distribute-rgt-neg-in 79.1%

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

      9. associate-*l* 79.0%

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

      10. add-cube-cbrt 77.9%

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

      11. associate-*l* 77.9%

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

      12. prod-diff 77.9%

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

   5. Applied egg-rr 55.1%

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

   7. Simplified 55.2%

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

   8. Taylor expanded in x around inf 79.3%

      \[\leadsto \color{blue}{y \cdot \left(\sqrt[3]{9} \cdot \left(\sqrt[3]{3} \cdot x\right)\right)} \]

   10. Applied egg-rr 79.6%

       \[\leadsto \color{blue}{0 - x \cdot \left(y \cdot -3\right)} \]
   11. Step-by-step derivation

       1. neg-sub0 79.6%

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

       2. associate-*r* 79.5%

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

       3. *-commutative 79.5%

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

       4. distribute-rgt-neg-in 79.5%

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

       5. metadata-eval 79.5%

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

       6. associate-*l* 79.6%

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

   12. Simplified 79.6%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+105}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-76}:\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in x around 0 99.8%

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

3. Final simplification 99.8%

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

Alternative 6: 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]

Derivation

1. Initial program 99.4%

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

   1. fma-udef 99.9%

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

   2. associate-*l* 99.4%

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

   3. unsub-neg 99.4%

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

   4. associate-*l* 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 7: 50.3% 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 99.4%

   \[\left(x \cdot 3\right) \cdot y - z \]
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 48.5%

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

   1. mul-1-neg 48.5%

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

6. Simplified 48.5%

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

7. Final simplification 48.5%

   \[\leadsto -z \]

Alternative 8: 2.2% 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 99.4%

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

   1. add-sqr-sqrt 50.5%

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

   2. sqrt-unprod 64.9%

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

   3. sqr-neg 64.9%

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

   4. sqrt-unprod 29.9%

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

   5. add-sqr-sqrt 52.6%

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

   6. fma-neg 52.6%

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

   7. add-cube-cbrt 52.6%

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

   8. distribute-rgt-neg-in 52.6%

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

   9. associate-*l* 52.5%

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

   10. add-cube-cbrt 51.9%

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

   11. associate-*l* 51.9%

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

   12. prod-diff 51.9%

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

5. Applied egg-rr 36.2%

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

7. Simplified 36.3%

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

8. Taylor expanded in y around 0 2.0%

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

9. Final simplification 2.0%

   \[\leadsto z \]

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 2023230 
(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))