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

Percentage Accurate: 99.8% → 99.8%

Time: 6.0s

Alternatives: 7

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, 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 99.8%

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

   1. associate-*l* 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 77.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot 3\right)\\ t_1 := x \cdot \left(3 \cdot y\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 6 \cdot 10^{-45}:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq 4:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+76}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (* x 3.0))) (t_1 (* x (* 3.0 y))))
   (if (<= t_0 -5e+48)
     t_1
     (if (<= t_0 6e-45)
       (- z)
       (if (<= t_0 4.0) t_1 (if (<= t_0 2e+76) (- z) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x * 3.0);
	double t_1 = x * (3.0 * y);
	double tmp;
	if (t_0 <= -5e+48) {
		tmp = t_1;
	} else if (t_0 <= 6e-45) {
		tmp = -z;
	} else if (t_0 <= 4.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+76) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y * (x * 3.0d0)
    t_1 = x * (3.0d0 * y)
    if (t_0 <= (-5d+48)) then
        tmp = t_1
    else if (t_0 <= 6d-45) then
        tmp = -z
    else if (t_0 <= 4.0d0) then
        tmp = t_1
    else if (t_0 <= 2d+76) then
        tmp = -z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (x * 3.0);
	double t_1 = x * (3.0 * y);
	double tmp;
	if (t_0 <= -5e+48) {
		tmp = t_1;
	} else if (t_0 <= 6e-45) {
		tmp = -z;
	} else if (t_0 <= 4.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+76) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x * 3.0)
	t_1 = x * (3.0 * y)
	tmp = 0
	if t_0 <= -5e+48:
		tmp = t_1
	elif t_0 <= 6e-45:
		tmp = -z
	elif t_0 <= 4.0:
		tmp = t_1
	elif t_0 <= 2e+76:
		tmp = -z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x * 3.0))
	t_1 = Float64(x * Float64(3.0 * y))
	tmp = 0.0
	if (t_0 <= -5e+48)
		tmp = t_1;
	elseif (t_0 <= 6e-45)
		tmp = Float64(-z);
	elseif (t_0 <= 4.0)
		tmp = t_1;
	elseif (t_0 <= 2e+76)
		tmp = Float64(-z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x * 3.0);
	t_1 = x * (3.0 * y);
	tmp = 0.0;
	if (t_0 <= -5e+48)
		tmp = t_1;
	elseif (t_0 <= 6e-45)
		tmp = -z;
	elseif (t_0 <= 4.0)
		tmp = t_1;
	elseif (t_0 <= 2e+76)
		tmp = -z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+48], t$95$1, If[LessEqual[t$95$0, 6e-45], (-z), If[LessEqual[t$95$0, 4.0], t$95$1, If[LessEqual[t$95$0, 2e+76], (-z), t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x 3) y) < -4.99999999999999973e48 or 6.00000000000000022e-45 < (*.f64 (*.f64 x 3) y) < 4

   1. Initial program 99.8%

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

      1. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 98.3%

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

      2. pow3 98.3%

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

   5. Applied egg-rr 98.3%

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

   6. Taylor expanded in z around 0 82.2%

      \[\leadsto \color{blue}{3 \cdot \left({1}^{0.3333333333333333} \cdot \left(x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 82.2%

         \[\leadsto \color{blue}{\left({1}^{0.3333333333333333} \cdot \left(x \cdot y\right)\right) \cdot 3} \]

      2. pow-base-1 82.2%

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

      3. *-lft-identity 82.2%

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

      4. rem-square-sqrt 81.6%

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

      5. unpow2 81.6%

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

      6. associate-*r* 81.6%

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

      7. unpow2 81.6%

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

      8. rem-square-sqrt 82.3%

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

   8. Simplified 82.3%

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

   if -4.99999999999999973e48 < (*.f64 (*.f64 x 3) y) < 6.00000000000000022e-45 or 4 < (*.f64 (*.f64 x 3) y) < 2.0000000000000001e76

   1. Initial program 99.9%

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

      1. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 77.7%

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

      1. neg-mul-1 77.7%

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

   6. Simplified 77.7%

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

   if 2.0000000000000001e76 < (*.f64 (*.f64 x 3) y)

   1. Initial program 99.8%

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

      1. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

      1. add-cube-cbrt 98.7%

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

      2. pow3 98.6%

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

   5. Applied egg-rr 98.6%

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

   6. Taylor expanded in z around 0 85.5%

      \[\leadsto \color{blue}{3 \cdot \left({1}^{0.3333333333333333} \cdot \left(x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 85.5%

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

      2. *-lft-identity 85.5%

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

      3. associate-*r* 85.7%

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

      4. *-commutative 85.7%

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

   8. Simplified 85.7%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 3\right) \leq -5 \cdot 10^{+48}:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot 3\right) \leq 6 \cdot 10^{-45}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \cdot \left(x \cdot 3\right) \leq 4:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot 3\right) \leq 2 \cdot 10^{+76}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \end{array} \]

Alternative 3: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+88} \lor \neg \left(x \leq 6.8 \cdot 10^{-64}\right):\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.9e+88) (not (<= x 6.8e-64))) (* 3.0 (* x y)) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.9e+88) || !(x <= 6.8e-64)) {
		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 ((x <= (-3.9d+88)) .or. (.not. (x <= 6.8d-64))) 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 ((x <= -3.9e+88) || !(x <= 6.8e-64)) {
		tmp = 3.0 * (x * y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.9e+88) or not (x <= 6.8e-64):
		tmp = 3.0 * (x * y)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.9e+88) || !(x <= 6.8e-64))
		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 ((x <= -3.9e+88) || ~((x <= 6.8e-64)))
		tmp = 3.0 * (x * y);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.9e+88], N[Not[LessEqual[x, 6.8e-64]], $MachinePrecision]], N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -3.9000000000000001e88 or 6.80000000000000024e-64 < x

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. add-sqr-sqrt 54.8%

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

      3. associate-*r* 54.9%

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

      4. fma-neg 54.9%

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

   5. Applied egg-rr 54.9%

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

   6. Taylor expanded in y around inf 68.4%

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

   if -3.9000000000000001e88 < x < 6.80000000000000024e-64

   1. Initial program 99.9%

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

      1. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 69.3%

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

      1. neg-mul-1 69.3%

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

   6. Simplified 69.3%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+88} \lor \neg \left(x \leq 6.8 \cdot 10^{-64}\right):\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 4: 70.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+74} \lor \neg \left(x \leq 1.4 \cdot 10^{-63}\right):\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.2e+74) (not (<= x 1.4e-63))) (* x (* 3.0 y)) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.2e+74) || !(x <= 1.4e-63)) {
		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 ((x <= (-4.2d+74)) .or. (.not. (x <= 1.4d-63))) 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 ((x <= -4.2e+74) || !(x <= 1.4e-63)) {
		tmp = x * (3.0 * y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.2e+74) or not (x <= 1.4e-63):
		tmp = x * (3.0 * y)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.2e+74) || !(x <= 1.4e-63))
		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 ((x <= -4.2e+74) || ~((x <= 1.4e-63)))
		tmp = x * (3.0 * y);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.2e+74], N[Not[LessEqual[x, 1.4e-63]], $MachinePrecision]], N[(x * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -4.1999999999999998e74 or 1.4000000000000001e-63 < x

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 98.4%

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

      2. pow3 98.4%

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

   5. Applied egg-rr 98.4%

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

   6. Taylor expanded in z around 0 67.4%

      \[\leadsto \color{blue}{3 \cdot \left({1}^{0.3333333333333333} \cdot \left(x \cdot y\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 67.4%

         \[\leadsto \color{blue}{\left({1}^{0.3333333333333333} \cdot \left(x \cdot y\right)\right) \cdot 3} \]

      2. pow-base-1 67.4%

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

      3. *-lft-identity 67.4%

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

      4. rem-square-sqrt 67.0%

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

      5. unpow2 67.0%

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

      6. associate-*r* 67.0%

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

      7. unpow2 67.0%

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

      8. rem-square-sqrt 67.4%

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

   8. Simplified 67.4%

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

   if -4.1999999999999998e74 < x < 1.4000000000000001e-63

   1. Initial program 99.9%

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

      1. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 68.8%

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

      1. neg-mul-1 68.8%

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

   6. Simplified 68.8%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+74} \lor \neg \left(x \leq 1.4 \cdot 10^{-63}\right):\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

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 99.8%

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

   1. associate-*l* 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 99.8%

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

5. Final simplification 99.8%

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

Alternative 6: 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 99.8%

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

   1. associate-*l* 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 50.2%

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

   1. neg-mul-1 50.2%

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

6. Simplified 50.2%

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

7. Final simplification 50.2%

   \[\leadsto -z \]

Alternative 7: 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 99.8%

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

   1. associate-*l* 99.9%

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

3. Simplified 99.9%

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

   1. add-cube-cbrt 98.2%

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

   2. pow3 98.1%

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

5. Applied egg-rr 98.1%

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

   1. rem-cube-cbrt 99.9%

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

   2. sub-neg 99.9%

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

   3. *-commutative 99.9%

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

   4. associate-*r* 99.8%

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

   5. fma-def 99.8%

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

   6. add-sqr-sqrt 57.8%

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

   7. sqrt-unprod 62.7%

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

   8. sqr-neg 62.7%

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

   9. sqrt-unprod 20.4%

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

   10. add-sqr-sqrt 48.6%

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

7. Applied egg-rr 48.6%

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

8. Taylor expanded in x 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 2023293 
(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))