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

Percentage Accurate: 99.8% → 99.8%

Time: 7.5s

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, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot y, 3, 0 - z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (* x y) 3.0 (- 0.0 z)))

C

double code(double x, double y, double z) {
	return fma((x * y), 3.0, (0.0 - z));
}

Julia

function code(x, y, z)
	return fma(Float64(x * y), 3.0, Float64(0.0 - z))
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] * 3.0 + N[(0.0 - z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. sub-neg N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot y, 3, \color{blue}{0 - z}\right) \]

   8. --lowering--.f64 99.9

      \[\leadsto \mathsf{fma}\left(x \cdot y, 3, \color{blue}{0 - z}\right) \]

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot y, 3, 0 - z\right)} \]
5. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 99.9

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(x \cdot y, 3, 0 - z\right) \]
8. Add Preprocessing

Alternative 2: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot 3\right)\\ t_1 := x \cdot \left(y \cdot 3\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+117}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+86}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (* x 3.0))) (t_1 (* x (* y 3.0))))
   (if (<= t_0 -2e+117) t_1 (if (<= t_0 4e+86) (- 0.0 z) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x * 3.0);
	double t_1 = x * (y * 3.0);
	double tmp;
	if (t_0 <= -2e+117) {
		tmp = t_1;
	} else if (t_0 <= 4e+86) {
		tmp = 0.0 - z;
	} else {
		tmp = t_1;
	}
	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 * (y * 3.0d0)
    if (t_0 <= (-2d+117)) then
        tmp = t_1
    else if (t_0 <= 4d+86) then
        tmp = 0.0d0 - z
    else
        tmp = t_1
    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 * (y * 3.0);
	double tmp;
	if (t_0 <= -2e+117) {
		tmp = t_1;
	} else if (t_0 <= 4e+86) {
		tmp = 0.0 - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x * 3.0)
	t_1 = x * (y * 3.0)
	tmp = 0
	if t_0 <= -2e+117:
		tmp = t_1
	elif t_0 <= 4e+86:
		tmp = 0.0 - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x * 3.0))
	t_1 = Float64(x * Float64(y * 3.0))
	tmp = 0.0
	if (t_0 <= -2e+117)
		tmp = t_1;
	elseif (t_0 <= 4e+86)
		tmp = Float64(0.0 - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x * 3.0);
	t_1 = x * (y * 3.0);
	tmp = 0.0;
	if (t_0 <= -2e+117)
		tmp = t_1;
	elseif (t_0 <= 4e+86)
		tmp = 0.0 - z;
	else
		tmp = t_1;
	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[(y * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+117], t$95$1, If[LessEqual[t$95$0, 4e+86], N[(0.0 - z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 3 binary64)) y) < -2.0000000000000001e117 or 4.0000000000000001e86 < (*.f64 (*.f64 x #s(literal 3 binary64)) y)

   1. Initial program 99.8%

      \[\left(x \cdot 3\right) \cdot y - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right) + 0} \]

      2. *-commutative N/A

         \[\leadsto 3 \cdot \color{blue}{\left(y \cdot x\right)} + 0 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(3 \cdot y\right) \cdot x} + 0 \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(-3\right)\right)} \cdot y\right) \cdot x + 0 \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-3 \cdot y\right)\right)} \cdot x + 0 \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(-3 \cdot y\right)\right)} + 0 \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(-3 \cdot y\right), 0\right)} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(\mathsf{neg}\left(-3\right)\right) \cdot y}, 0\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{3} \cdot y, 0\right) \]

      10. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{3 \cdot y + 0}, 0\right) \]

      11. accelerator-lowering-fma.f64 84.7

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(3, y, 0\right)}, 0\right) \]

   5. Simplified 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(3, y, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left(3 \cdot y + 0\right)} \]

      2. +-rgt-identity N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 84.8

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

   7. Applied egg-rr 84.8%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 84.7

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

   9. Applied egg-rr 84.7%

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

   if -2.0000000000000001e117 < (*.f64 (*.f64 x #s(literal 3 binary64)) y) < 4.0000000000000001e86

   1. Initial program 99.9%

      \[\left(x \cdot 3\right) \cdot y - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 75.5

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

   5. Simplified 75.5%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. neg-lowering-neg.f64 75.5

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

   7. Applied egg-rr 75.5%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 3\right) \leq -2 \cdot 10^{+117}:\\ \;\;\;\;x \cdot \left(y \cdot 3\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot 3\right) \leq 4 \cdot 10^{+86}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot 3\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+117}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+86}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (* x 3.0))))
   (if (<= t_0 -2e+117) t_0 (if (<= t_0 4e+86) (- 0.0 z) (* (* x y) 3.0)))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x * 3.0);
	double tmp;
	if (t_0 <= -2e+117) {
		tmp = t_0;
	} else if (t_0 <= 4e+86) {
		tmp = 0.0 - z;
	} else {
		tmp = (x * y) * 3.0;
	}
	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) :: tmp
    t_0 = y * (x * 3.0d0)
    if (t_0 <= (-2d+117)) then
        tmp = t_0
    else if (t_0 <= 4d+86) then
        tmp = 0.0d0 - z
    else
        tmp = (x * y) * 3.0d0
    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 tmp;
	if (t_0 <= -2e+117) {
		tmp = t_0;
	} else if (t_0 <= 4e+86) {
		tmp = 0.0 - z;
	} else {
		tmp = (x * y) * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x * 3.0)
	tmp = 0
	if t_0 <= -2e+117:
		tmp = t_0
	elif t_0 <= 4e+86:
		tmp = 0.0 - z
	else:
		tmp = (x * y) * 3.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x * 3.0))
	tmp = 0.0
	if (t_0 <= -2e+117)
		tmp = t_0;
	elseif (t_0 <= 4e+86)
		tmp = Float64(0.0 - z);
	else
		tmp = Float64(Float64(x * y) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x * 3.0);
	tmp = 0.0;
	if (t_0 <= -2e+117)
		tmp = t_0;
	elseif (t_0 <= 4e+86)
		tmp = 0.0 - z;
	else
		tmp = (x * y) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+117], t$95$0, If[LessEqual[t$95$0, 4e+86], N[(0.0 - z), $MachinePrecision], N[(N[(x * y), $MachinePrecision] * 3.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 3 binary64)) y) < -2.0000000000000001e117

   1. Initial program 99.8%

      \[\left(x \cdot 3\right) \cdot y - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right) + 0} \]

      2. *-commutative N/A

         \[\leadsto 3 \cdot \color{blue}{\left(y \cdot x\right)} + 0 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(3 \cdot y\right) \cdot x} + 0 \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(-3\right)\right)} \cdot y\right) \cdot x + 0 \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-3 \cdot y\right)\right)} \cdot x + 0 \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(-3 \cdot y\right)\right)} + 0 \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(-3 \cdot y\right), 0\right)} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(\mathsf{neg}\left(-3\right)\right) \cdot y}, 0\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{3} \cdot y, 0\right) \]

      10. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{3 \cdot y + 0}, 0\right) \]

      11. accelerator-lowering-fma.f64 88.4

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(3, y, 0\right)}, 0\right) \]

   5. Simplified 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(3, y, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left(3 \cdot y + 0\right)} \]

      2. +-rgt-identity N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 88.5

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

   7. Applied egg-rr 88.5%

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

   if -2.0000000000000001e117 < (*.f64 (*.f64 x #s(literal 3 binary64)) y) < 4.0000000000000001e86

   1. Initial program 99.9%

      \[\left(x \cdot 3\right) \cdot y - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 75.5

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

   5. Simplified 75.5%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. neg-lowering-neg.f64 75.5

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

   7. Applied egg-rr 75.5%

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

   if 4.0000000000000001e86 < (*.f64 (*.f64 x #s(literal 3 binary64)) y)

   1. Initial program 99.8%

      \[\left(x \cdot 3\right) \cdot y - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right) + 0} \]

      2. *-commutative N/A

         \[\leadsto 3 \cdot \color{blue}{\left(y \cdot x\right)} + 0 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(3 \cdot y\right) \cdot x} + 0 \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(-3\right)\right)} \cdot y\right) \cdot x + 0 \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-3 \cdot y\right)\right)} \cdot x + 0 \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(-3 \cdot y\right)\right)} + 0 \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(-3 \cdot y\right), 0\right)} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(\mathsf{neg}\left(-3\right)\right) \cdot y}, 0\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{3} \cdot y, 0\right) \]

      10. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{3 \cdot y + 0}, 0\right) \]

      11. accelerator-lowering-fma.f64 80.5

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(3, y, 0\right)}, 0\right) \]

   5. Simplified 80.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(3, y, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left(3 \cdot y + 0\right)} \]

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 80.5

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

   7. Applied egg-rr 80.5%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 3\right) \leq -2 \cdot 10^{+117}:\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot 3\right) \leq 4 \cdot 10^{+86}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot 3\right)\\ \mathbf{if}\;t\_0 \leq -2.35 \cdot 10^{+113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 3.4 \cdot 10^{+86}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (* x 3.0))))
   (if (<= t_0 -2.35e+113) t_0 (if (<= t_0 3.4e+86) (- 0.0 z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x * 3.0);
	double tmp;
	if (t_0 <= -2.35e+113) {
		tmp = t_0;
	} else if (t_0 <= 3.4e+86) {
		tmp = 0.0 - 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) :: tmp
    t_0 = y * (x * 3.0d0)
    if (t_0 <= (-2.35d+113)) then
        tmp = t_0
    else if (t_0 <= 3.4d+86) then
        tmp = 0.0d0 - 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 tmp;
	if (t_0 <= -2.35e+113) {
		tmp = t_0;
	} else if (t_0 <= 3.4e+86) {
		tmp = 0.0 - z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x * 3.0)
	tmp = 0
	if t_0 <= -2.35e+113:
		tmp = t_0
	elif t_0 <= 3.4e+86:
		tmp = 0.0 - z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x * 3.0))
	tmp = 0.0
	if (t_0 <= -2.35e+113)
		tmp = t_0;
	elseif (t_0 <= 3.4e+86)
		tmp = Float64(0.0 - z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x * 3.0);
	tmp = 0.0;
	if (t_0 <= -2.35e+113)
		tmp = t_0;
	elseif (t_0 <= 3.4e+86)
		tmp = 0.0 - 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]}, If[LessEqual[t$95$0, -2.35e+113], t$95$0, If[LessEqual[t$95$0, 3.4e+86], N[(0.0 - z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 3 binary64)) y) < -2.3499999999999999e113 or 3.3999999999999998e86 < (*.f64 (*.f64 x #s(literal 3 binary64)) y)

   1. Initial program 99.8%

      \[\left(x \cdot 3\right) \cdot y - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right) + 0} \]

      2. *-commutative N/A

         \[\leadsto 3 \cdot \color{blue}{\left(y \cdot x\right)} + 0 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(3 \cdot y\right) \cdot x} + 0 \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(-3\right)\right)} \cdot y\right) \cdot x + 0 \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-3 \cdot y\right)\right)} \cdot x + 0 \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(-3 \cdot y\right)\right)} + 0 \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(-3 \cdot y\right), 0\right)} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(\mathsf{neg}\left(-3\right)\right) \cdot y}, 0\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{3} \cdot y, 0\right) \]

      10. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{3 \cdot y + 0}, 0\right) \]

      11. accelerator-lowering-fma.f64 84.7

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(3, y, 0\right)}, 0\right) \]

   5. Simplified 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(3, y, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left(3 \cdot y + 0\right)} \]

      2. +-rgt-identity N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 84.8

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

   7. Applied egg-rr 84.8%

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

   if -2.3499999999999999e113 < (*.f64 (*.f64 x #s(literal 3 binary64)) y) < 3.3999999999999998e86

   1. Initial program 99.9%

      \[\left(x \cdot 3\right) \cdot y - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. neg-sub0 N/A

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

      3. --lowering--.f64 75.5

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

   5. Simplified 75.5%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. neg-lowering-neg.f64 75.5

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

   7. Applied egg-rr 75.5%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot 3\right) \leq -2.35 \cdot 10^{+113}:\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot 3\right) \leq 3.4 \cdot 10^{+86}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y * (x * 3.0d0)) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x \cdot 3\right) \cdot y - z \]
2. Add Preprocessing

3. Final simplification 99.9%

   \[\leadsto y \cdot \left(x \cdot 3\right) - z \]
4. Add Preprocessing

Alternative 7: 50.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 0.0 - 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 = 0.0d0 - z
end function

Java

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

Python

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

Julia

function code(x, y, z)
	return Float64(0.0 - z)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(0.0 - z), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(x \cdot 3\right) \cdot y - z \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

   2. neg-sub0 N/A

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

   3. --lowering--.f64 53.9

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

5. Simplified 53.9%

   \[\leadsto \color{blue}{0 - z} \]
6. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

   2. neg-lowering-neg.f64 53.9

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

7. Applied egg-rr 53.9%

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

8. Final simplification 53.9%

   \[\leadsto 0 - z \]
9. Add Preprocessing

Alternative 8: 2.3% accurate, 14.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 99.9%

   \[\left(x \cdot 3\right) \cdot y - z \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

   2. neg-sub0 N/A

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

   3. --lowering--.f64 53.9

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

5. Simplified 53.9%

   \[\leadsto \color{blue}{0 - z} \]
6. Step-by-step derivation

   1. flip-- N/A

      \[\leadsto \color{blue}{\frac{0 \cdot 0 - z \cdot z}{0 + z}} \]

   2. +-lft-identity N/A

      \[\leadsto \frac{0 \cdot 0 - z \cdot z}{\color{blue}{z}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} - z \cdot z}{z} \]

   4. sub-div N/A

      \[\leadsto \color{blue}{\frac{0}{z} - \frac{z \cdot z}{z}} \]

   5. frac-sub N/A

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

   6. mul0-lft N/A

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

   7. cube-mult N/A

      \[\leadsto \frac{0 - \color{blue}{{z}^{3}}}{z \cdot z} \]

   8. sqr-pow N/A

      \[\leadsto \frac{0 - \color{blue}{{z}^{\left(\frac{3}{2}\right)} \cdot {z}^{\left(\frac{3}{2}\right)}}}{z \cdot z} \]

   9. pow-prod-down N/A

      \[\leadsto \frac{0 - \color{blue}{{\left(z \cdot z\right)}^{\left(\frac{3}{2}\right)}}}{z \cdot z} \]

   10. sqr-neg N/A

       \[\leadsto \frac{0 - {\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}}^{\left(\frac{3}{2}\right)}}{z \cdot z} \]

   11. sub0-neg N/A

       \[\leadsto \frac{0 - {\left(\color{blue}{\left(0 - z\right)} \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{z \cdot z} \]

   12. sub0-neg N/A

       \[\leadsto \frac{0 - {\left(\left(0 - z\right) \cdot \color{blue}{\left(0 - z\right)}\right)}^{\left(\frac{3}{2}\right)}}{z \cdot z} \]

   13. pow-prod-down N/A

       \[\leadsto \frac{0 - \color{blue}{{\left(0 - z\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(0 - z\right)}^{\left(\frac{3}{2}\right)}}}{z \cdot z} \]

   14. sqr-pow N/A

       \[\leadsto \frac{0 - \color{blue}{{\left(0 - z\right)}^{3}}}{z \cdot z} \]

   15. sub0-neg N/A

       \[\leadsto \frac{0 - {\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}}^{3}}{z \cdot z} \]

   16. cube-neg N/A

       \[\leadsto \frac{0 - \color{blue}{\left(\mathsf{neg}\left({z}^{3}\right)\right)}}{z \cdot z} \]

   17. sub0-neg N/A

       \[\leadsto \frac{0 - \color{blue}{\left(0 - {z}^{3}\right)}}{z \cdot z} \]

   18. metadata-eval N/A

       \[\leadsto \frac{0 - \left(\color{blue}{{0}^{3}} - {z}^{3}\right)}{z \cdot z} \]

   19. sub-div N/A

       \[\leadsto \color{blue}{\frac{0}{z \cdot z} - \frac{{0}^{3} - {z}^{3}}{z \cdot z}} \]

   20. div0 N/A

       \[\leadsto \color{blue}{0} - \frac{{0}^{3} - {z}^{3}}{z \cdot z} \]

   21. +-rgt-identity N/A

       \[\leadsto 0 - \frac{{0}^{3} - {z}^{3}}{\color{blue}{z \cdot z + 0}} \]

   22. mul0-lft N/A

       \[\leadsto 0 - \frac{{0}^{3} - {z}^{3}}{z \cdot z + \color{blue}{0 \cdot z}} \]

   23. +-lft-identity N/A

       \[\leadsto 0 - \frac{{0}^{3} - {z}^{3}}{\color{blue}{0 + \left(z \cdot z + 0 \cdot z\right)}} \]

7. Applied egg-rr 2.0%

   \[\leadsto \color{blue}{z} \]
8. Add Preprocessing

Developer Target 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]

Reproduce

herbie shell --seed 2024198 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* x (* 3 y)) z))

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