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

Percentage Accurate: 99.8% → 99.8%

Time: 5.1s

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

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (fma x (* 3.0 y) (- z)))

C

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

Julia

x, y = sort([x, y])
function code(x, y, z)
	return fma(x, Float64(3.0 * y), Float64(-z))
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(x * N[(3.0 * y), $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. Final simplification 99.8%

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

Alternative 2: 71.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6.3 \cdot 10^{-36} \lor \neg \left(y \leq 1.1 \cdot 10^{-29}\right) \land \left(y \leq 0.00062 \lor \neg \left(y \leq 5.2 \cdot 10^{+58}\right)\right):\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6.3e-36)
         (and (not (<= y 1.1e-29)) (or (<= y 0.00062) (not (<= y 5.2e+58)))))
   (* y (* x 3.0))
   (- z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.3e-36) || (!(y <= 1.1e-29) && ((y <= 0.00062) || !(y <= 5.2e+58)))) {
		tmp = y * (x * 3.0);
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 ((y <= (-6.3d-36)) .or. (.not. (y <= 1.1d-29)) .and. (y <= 0.00062d0) .or. (.not. (y <= 5.2d+58))) then
        tmp = y * (x * 3.0d0)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.3e-36) || (!(y <= 1.1e-29) && ((y <= 0.00062) || !(y <= 5.2e+58)))) {
		tmp = y * (x * 3.0);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (y <= -6.3e-36) or (not (y <= 1.1e-29) and ((y <= 0.00062) or not (y <= 5.2e+58))):
		tmp = y * (x * 3.0)
	else:
		tmp = -z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((y <= -6.3e-36) || (!(y <= 1.1e-29) && ((y <= 0.00062) || !(y <= 5.2e+58))))
		tmp = Float64(y * Float64(x * 3.0));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -6.3e-36) || (~((y <= 1.1e-29)) && ((y <= 0.00062) || ~((y <= 5.2e+58)))))
		tmp = y * (x * 3.0);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[y, -6.3e-36], And[N[Not[LessEqual[y, 1.1e-29]], $MachinePrecision], Or[LessEqual[y, 0.00062], N[Not[LessEqual[y, 5.2e+58]], $MachinePrecision]]]], N[(y * N[(x * 3.0), $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if y < -6.299999999999999e-36 or 1.09999999999999995e-29 < y < 6.2e-4 or 5.19999999999999976e58 < y

   1. Initial program 99.8%

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

      1. add-cbrt-cube 49.7%

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

      2. pow3 49.7%

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

      3. *-commutative 49.7%

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

      4. associate-*l* 49.7%

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

      5. fma-neg 49.7%

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

      6. add-sqr-sqrt 24.5%

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

      7. sqrt-unprod 40.5%

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

      8. sqr-neg 40.5%

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

      9. sqrt-unprod 19.6%

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

      10. add-sqr-sqrt 37.9%

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

   3. Applied egg-rr 37.9%

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

   4. Taylor expanded in x around inf 29.2%

      \[\leadsto \sqrt[3]{\color{blue}{27 \cdot \left({y}^{3} \cdot {x}^{3}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 29.2%

         \[\leadsto \sqrt[3]{\color{blue}{\left(27 \cdot {y}^{3}\right) \cdot {x}^{3}}} \]

      2. metadata-eval 29.2%

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

      3. cube-prod 29.2%

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

      4. *-commutative 29.2%

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

      5. cube-prod 38.6%

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

      6. associate-*r* 38.6%

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

      7. *-commutative 38.6%

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

   6. Simplified 38.6%

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

      1. rem-cbrt-cube 67.1%

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

   8. Applied egg-rr 67.1%

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

   if -6.299999999999999e-36 < y < 1.09999999999999995e-29 or 6.2e-4 < y < 5.19999999999999976e58

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 70.6%

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

      1. mul-1-neg 70.6%

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

   4. Simplified 70.6%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.3 \cdot 10^{-36} \lor \neg \left(y \leq 1.1 \cdot 10^{-29}\right) \land \left(y \leq 0.00062 \lor \neg \left(y \leq 5.2 \cdot 10^{+58}\right)\right):\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3: 71.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot 3\right)\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-29}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+58}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (* x 3.0))))
   (if (<= y -7.5e-39)
     t_0
     (if (<= y 2.25e-29)
       (- z)
       (if (<= y 8e-5) t_0 (if (<= y 7.2e+58) (- z) (* 3.0 (* x y))))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = y * (x * 3.0);
	double tmp;
	if (y <= -7.5e-39) {
		tmp = t_0;
	} else if (y <= 2.25e-29) {
		tmp = -z;
	} else if (y <= 8e-5) {
		tmp = t_0;
	} else if (y <= 7.2e+58) {
		tmp = -z;
	} else {
		tmp = 3.0 * (x * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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) :: tmp
    t_0 = y * (x * 3.0d0)
    if (y <= (-7.5d-39)) then
        tmp = t_0
    else if (y <= 2.25d-29) then
        tmp = -z
    else if (y <= 8d-5) then
        tmp = t_0
    else if (y <= 7.2d+58) then
        tmp = -z
    else
        tmp = 3.0d0 * (x * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = y * (x * 3.0);
	double tmp;
	if (y <= -7.5e-39) {
		tmp = t_0;
	} else if (y <= 2.25e-29) {
		tmp = -z;
	} else if (y <= 8e-5) {
		tmp = t_0;
	} else if (y <= 7.2e+58) {
		tmp = -z;
	} else {
		tmp = 3.0 * (x * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = y * (x * 3.0)
	tmp = 0
	if y <= -7.5e-39:
		tmp = t_0
	elif y <= 2.25e-29:
		tmp = -z
	elif y <= 8e-5:
		tmp = t_0
	elif y <= 7.2e+58:
		tmp = -z
	else:
		tmp = 3.0 * (x * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(y * Float64(x * 3.0))
	tmp = 0.0
	if (y <= -7.5e-39)
		tmp = t_0;
	elseif (y <= 2.25e-29)
		tmp = Float64(-z);
	elseif (y <= 8e-5)
		tmp = t_0;
	elseif (y <= 7.2e+58)
		tmp = Float64(-z);
	else
		tmp = Float64(3.0 * Float64(x * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = y * (x * 3.0);
	tmp = 0.0;
	if (y <= -7.5e-39)
		tmp = t_0;
	elseif (y <= 2.25e-29)
		tmp = -z;
	elseif (y <= 8e-5)
		tmp = t_0;
	elseif (y <= 7.2e+58)
		tmp = -z;
	else
		tmp = 3.0 * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.5e-39], t$95$0, If[LessEqual[y, 2.25e-29], (-z), If[LessEqual[y, 8e-5], t$95$0, If[LessEqual[y, 7.2e+58], (-z), N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.49999999999999971e-39 or 2.2499999999999999e-29 < y < 8.00000000000000065e-5

   1. Initial program 99.9%

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

      1. add-cbrt-cube 42.3%

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

      2. pow3 42.4%

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

      3. *-commutative 42.4%

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

      4. associate-*l* 42.4%

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

      5. fma-neg 42.4%

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

      6. add-sqr-sqrt 20.7%

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

      7. sqrt-unprod 32.1%

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

      8. sqr-neg 32.1%

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

      9. sqrt-unprod 15.8%

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

      10. add-sqr-sqrt 32.2%

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

   3. Applied egg-rr 32.2%

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

   4. Taylor expanded in x around inf 26.2%

      \[\leadsto \sqrt[3]{\color{blue}{27 \cdot \left({y}^{3} \cdot {x}^{3}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 26.3%

         \[\leadsto \sqrt[3]{\color{blue}{\left(27 \cdot {y}^{3}\right) \cdot {x}^{3}}} \]

      2. metadata-eval 26.3%

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

      3. cube-prod 26.3%

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

      4. *-commutative 26.3%

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

      5. cube-prod 32.9%

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

      6. associate-*r* 32.9%

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

      7. *-commutative 32.9%

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

   6. Simplified 32.9%

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

      1. rem-cbrt-cube 64.0%

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

   8. Applied egg-rr 64.0%

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

   if -7.49999999999999971e-39 < y < 2.2499999999999999e-29 or 8.00000000000000065e-5 < y < 7.19999999999999993e58

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 70.4%

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

      1. mul-1-neg 70.4%

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

   4. Simplified 70.4%

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

   if 7.19999999999999993e58 < y

   1. Initial program 99.8%

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

      1. add-cbrt-cube 61.5%

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

      2. pow3 61.5%

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

      3. *-commutative 61.5%

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

      4. associate-*l* 61.5%

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

      5. fma-neg 61.5%

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

      6. add-sqr-sqrt 30.7%

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

      7. sqrt-unprod 54.3%

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

      8. sqr-neg 54.3%

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

      9. sqrt-unprod 26.0%

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

      10. add-sqr-sqrt 47.0%

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

   3. Applied egg-rr 47.0%

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

   4. Taylor expanded in x around inf 33.7%

      \[\leadsto \sqrt[3]{\color{blue}{27 \cdot \left({y}^{3} \cdot {x}^{3}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 33.7%

         \[\leadsto \sqrt[3]{\color{blue}{\left(27 \cdot {y}^{3}\right) \cdot {x}^{3}}} \]

      2. metadata-eval 33.7%

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

      3. cube-prod 33.8%

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

      4. *-commutative 33.8%

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

      5. cube-prod 47.9%

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

      6. associate-*r* 48.0%

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

      7. *-commutative 48.0%

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

   6. Simplified 48.0%

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

      1. rem-cbrt-cube 71.1%

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

      2. associate-*l* 71.0%

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

      3. *-commutative 71.0%

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

   8. Applied egg-rr 71.0%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-39}:\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-29}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-5}:\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+58}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 4: 70.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-40}:\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-50}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 0.005:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+59}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -1.5e-40)
   (* y (* x 3.0))
   (if (<= y 8e-50)
     (- z)
     (if (<= y 0.005)
       (* x (* 3.0 y))
       (if (<= y 8e+59) (- z) (* 3.0 (* x y)))))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.5e-40) {
		tmp = y * (x * 3.0);
	} else if (y <= 8e-50) {
		tmp = -z;
	} else if (y <= 0.005) {
		tmp = x * (3.0 * y);
	} else if (y <= 8e+59) {
		tmp = -z;
	} else {
		tmp = 3.0 * (x * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 (y <= (-1.5d-40)) then
        tmp = y * (x * 3.0d0)
    else if (y <= 8d-50) then
        tmp = -z
    else if (y <= 0.005d0) then
        tmp = x * (3.0d0 * y)
    else if (y <= 8d+59) then
        tmp = -z
    else
        tmp = 3.0d0 * (x * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.5e-40) {
		tmp = y * (x * 3.0);
	} else if (y <= 8e-50) {
		tmp = -z;
	} else if (y <= 0.005) {
		tmp = x * (3.0 * y);
	} else if (y <= 8e+59) {
		tmp = -z;
	} else {
		tmp = 3.0 * (x * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if y <= -1.5e-40:
		tmp = y * (x * 3.0)
	elif y <= 8e-50:
		tmp = -z
	elif y <= 0.005:
		tmp = x * (3.0 * y)
	elif y <= 8e+59:
		tmp = -z
	else:
		tmp = 3.0 * (x * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (y <= -1.5e-40)
		tmp = Float64(y * Float64(x * 3.0));
	elseif (y <= 8e-50)
		tmp = Float64(-z);
	elseif (y <= 0.005)
		tmp = Float64(x * Float64(3.0 * y));
	elseif (y <= 8e+59)
		tmp = Float64(-z);
	else
		tmp = Float64(3.0 * Float64(x * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.5e-40)
		tmp = y * (x * 3.0);
	elseif (y <= 8e-50)
		tmp = -z;
	elseif (y <= 0.005)
		tmp = x * (3.0 * y);
	elseif (y <= 8e+59)
		tmp = -z;
	else
		tmp = 3.0 * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, -1.5e-40], N[(y * N[(x * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e-50], (-z), If[LessEqual[y, 0.005], N[(x * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+59], (-z), N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.5000000000000001e-40

   1. Initial program 99.9%

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

      1. add-cbrt-cube 43.4%

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

      2. pow3 43.4%

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

      3. *-commutative 43.4%

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

      4. associate-*l* 43.4%

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

      5. fma-neg 43.4%

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

      6. add-sqr-sqrt 19.7%

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

      7. sqrt-unprod 32.2%

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

      8. sqr-neg 32.2%

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

      9. sqrt-unprod 17.2%

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

      10. add-sqr-sqrt 33.6%

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

   3. Applied egg-rr 33.6%

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

   4. Taylor expanded in x around inf 27.0%

      \[\leadsto \sqrt[3]{\color{blue}{27 \cdot \left({y}^{3} \cdot {x}^{3}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 27.0%

         \[\leadsto \sqrt[3]{\color{blue}{\left(27 \cdot {y}^{3}\right) \cdot {x}^{3}}} \]

      2. metadata-eval 27.0%

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

      3. cube-prod 27.0%

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

      4. *-commutative 27.0%

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

      5. cube-prod 34.3%

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

      6. associate-*r* 34.3%

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

      7. *-commutative 34.3%

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

   6. Simplified 34.3%

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

      1. rem-cbrt-cube 63.1%

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

   8. Applied egg-rr 63.1%

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

   if -1.5000000000000001e-40 < y < 8.00000000000000006e-50 or 0.0050000000000000001 < y < 7.99999999999999977e59

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 71.2%

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

      1. mul-1-neg 71.2%

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

   4. Simplified 71.2%

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

   if 8.00000000000000006e-50 < y < 0.0050000000000000001

   1. Initial program 99.5%

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

      1. add-cbrt-cube 24.8%

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

      2. pow3 24.9%

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

      3. *-commutative 24.9%

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

      4. associate-*l* 24.9%

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

      5. fma-neg 24.9%

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

      6. add-sqr-sqrt 23.3%

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

      7. sqrt-unprod 24.2%

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

      8. sqr-neg 24.2%

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

      9. sqrt-unprod 1.5%

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

      10. add-sqr-sqrt 14.0%

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

   3. Applied egg-rr 14.0%

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

   4. Taylor expanded in x around inf 14.1%

      \[\leadsto \sqrt[3]{\color{blue}{27 \cdot \left({y}^{3} \cdot {x}^{3}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 14.3%

         \[\leadsto \sqrt[3]{\color{blue}{\left(27 \cdot {y}^{3}\right) \cdot {x}^{3}}} \]

      2. metadata-eval 14.3%

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

      3. cube-prod 14.3%

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

      4. *-commutative 14.3%

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

      5. cube-prod 14.3%

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

      6. associate-*r* 14.3%

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

      7. *-commutative 14.3%

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

   6. Simplified 14.3%

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

      1. rem-cbrt-cube 70.4%

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

      2. *-commutative 70.4%

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

      3. associate-*r* 70.9%

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

   8. Applied egg-rr 70.9%

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

   if 7.99999999999999977e59 < y

   1. Initial program 99.8%

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

      1. add-cbrt-cube 61.5%

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

      2. pow3 61.5%

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

      3. *-commutative 61.5%

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

      4. associate-*l* 61.5%

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

      5. fma-neg 61.5%

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

      6. add-sqr-sqrt 30.7%

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

      7. sqrt-unprod 54.3%

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

      8. sqr-neg 54.3%

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

      9. sqrt-unprod 26.0%

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

      10. add-sqr-sqrt 47.0%

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

   3. Applied egg-rr 47.0%

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

   4. Taylor expanded in x around inf 33.7%

      \[\leadsto \sqrt[3]{\color{blue}{27 \cdot \left({y}^{3} \cdot {x}^{3}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 33.7%

         \[\leadsto \sqrt[3]{\color{blue}{\left(27 \cdot {y}^{3}\right) \cdot {x}^{3}}} \]

      2. metadata-eval 33.7%

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

      3. cube-prod 33.8%

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

      4. *-commutative 33.8%

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

      5. cube-prod 47.9%

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

      6. associate-*r* 48.0%

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

      7. *-commutative 48.0%

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

   6. Simplified 48.0%

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

      1. rem-cbrt-cube 71.1%

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

      2. associate-*l* 71.0%

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

      3. *-commutative 71.0%

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

   8. Applied egg-rr 71.0%

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot 3} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-40}:\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-50}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 0.005:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+59}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (- (* 3.0 (* x y)) z))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return (3.0 * (x * y)) - z

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(Float64(3.0 * Float64(x * y)) - z)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = (3.0 * (x * y)) - z;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(3.0 * N[(x * y), $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 99.8%

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

3. Final simplification 99.8%

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

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (- (* x (* 3.0 y)) z))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return (x * (3.0 * y)) - z

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(Float64(x * Float64(3.0 * y)) - z)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = (x * (3.0 * y)) - z;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(x * N[(3.0 * y), $MachinePrecision]), $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-neg 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 7: 50.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ -z \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (- z))

C

assert(x < y);
double code(double x, double y, double z) {
	return -z;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return -z

Julia

x, y = sort([x, y])
function code(x, y, z)
	return Float64(-z)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = -z;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := (-z)

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 53.2%

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

   1. mul-1-neg 53.2%

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

4. Simplified 53.2%

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

5. Final simplification 53.2%

   \[\leadsto -z \]

Alternative 8: 2.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ z \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 z)

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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

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

Python

[x, y] = sort([x, y])
def code(x, y, z):
	return z

Julia

x, y = sort([x, y])
function code(x, y, z)
	return z
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z)
	tmp = z;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
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. add-cbrt-cube 47.1%

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

   2. pow3 47.1%

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

   3. *-commutative 47.1%

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

   4. associate-*l* 47.1%

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

   5. fma-neg 47.1%

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

   6. add-sqr-sqrt 23.7%

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

   7. sqrt-unprod 34.1%

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

   8. sqr-neg 34.1%

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

   9. sqrt-unprod 12.1%

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

   10. add-sqr-sqrt 27.2%

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

3. Applied egg-rr 27.2%

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

4. Taylor expanded in x around 0 2.3%

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

5. Final simplification 2.3%

   \[\leadsto z \]

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 2023221 
(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))