Diagrams.Solve.Polynomial:quartForm from diagrams-solve-0.1, D

Percentage Accurate: 100.0% → 100.0%

Time: 3.7s

Alternatives: 3

Speedup: 1.8×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{2} - \frac{z}{8} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (/ (* x y) 2.0) (/ z 8.0)))

C

double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

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) / 2.0d0) - (z / 8.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

Python

def code(x, y, z):
	return ((x * y) / 2.0) - (z / 8.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * y) / 2.0) - Float64(z / 8.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * y) / 2.0) - (z / 8.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision] - N[(z / 8.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{2} - \frac{z}{8} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (/ (* x y) 2.0) (/ z 8.0)))

C

double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

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) / 2.0d0) - (z / 8.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

Python

def code(x, y, z):
	return ((x * y) / 2.0) - (z / 8.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * y) / 2.0) - Float64(z / 8.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * y) / 2.0) - (z / 8.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision] - N[(z / 8.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y \cdot 0.5, x, z \cdot -0.125\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (* y 0.5) x (* z -0.125)))

C

double code(double x, double y, double z) {
	return fma((y * 0.5), x, (z * -0.125));
}

Julia

function code(x, y, z)
	return fma(Float64(y * 0.5), x, Float64(z * -0.125))
end

Wolfram

code[x_, y_, z_] := N[(N[(y * 0.5), $MachinePrecision] * x + N[(z * -0.125), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{x \cdot y}{2} - \frac{z}{8} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{\frac{x \cdot y}{2} + \left(\mathsf{neg}\left(\frac{z}{8}\right)\right)} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{x \cdot \frac{y}{2}} + \left(\mathsf{neg}\left(\frac{z}{8}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\frac{y}{2} \cdot x} + \left(\mathsf{neg}\left(\frac{z}{8}\right)\right) \]

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{2}, x, \mathsf{neg}\left(\frac{z}{8}\right)\right)} \]

   5. div-inv N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \frac{1}{2}}, x, \mathsf{neg}\left(\frac{z}{8}\right)\right) \]

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

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \frac{1}{2}}, x, \mathsf{neg}\left(\frac{z}{8}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(y \cdot \color{blue}{\frac{1}{2}}, x, \mathsf{neg}\left(\frac{z}{8}\right)\right) \]

   8. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{fma}\left(y \cdot \frac{1}{2}, x, \color{blue}{\frac{z}{\mathsf{neg}\left(8\right)}}\right) \]

   9. div-inv N/A

      \[\leadsto \mathsf{fma}\left(y \cdot \frac{1}{2}, x, \color{blue}{z \cdot \frac{1}{\mathsf{neg}\left(8\right)}}\right) \]

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

       \[\leadsto \mathsf{fma}\left(y \cdot \frac{1}{2}, x, \color{blue}{z \cdot \frac{1}{\mathsf{neg}\left(8\right)}}\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(y \cdot \frac{1}{2}, x, z \cdot \frac{1}{\color{blue}{-8}}\right) \]

   12. metadata-eval 100.0

       \[\leadsto \mathsf{fma}\left(y \cdot 0.5, x, z \cdot \color{blue}{-0.125}\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot 0.5, x, z \cdot -0.125\right)} \]
5. Add Preprocessing

Alternative 2: 79.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y \cdot 0.5\right)\\ \mathbf{if}\;y \cdot x \leq -2 \cdot 10^{-71}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{-35}:\\ \;\;\;\;z \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* y 0.5))))
   (if (<= (* y x) -2e-71) t_0 (if (<= (* y x) 5e-35) (* z -0.125) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y * 0.5);
	double tmp;
	if ((y * x) <= -2e-71) {
		tmp = t_0;
	} else if ((y * x) <= 5e-35) {
		tmp = z * -0.125;
	} 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 = x * (y * 0.5d0)
    if ((y * x) <= (-2d-71)) then
        tmp = t_0
    else if ((y * x) <= 5d-35) then
        tmp = z * (-0.125d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (y * 0.5);
	double tmp;
	if ((y * x) <= -2e-71) {
		tmp = t_0;
	} else if ((y * x) <= 5e-35) {
		tmp = z * -0.125;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y * 0.5)
	tmp = 0
	if (y * x) <= -2e-71:
		tmp = t_0
	elif (y * x) <= 5e-35:
		tmp = z * -0.125
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y * 0.5))
	tmp = 0.0
	if (Float64(y * x) <= -2e-71)
		tmp = t_0;
	elseif (Float64(y * x) <= 5e-35)
		tmp = Float64(z * -0.125);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (y * 0.5);
	tmp = 0.0;
	if ((y * x) <= -2e-71)
		tmp = t_0;
	elseif ((y * x) <= 5e-35)
		tmp = z * -0.125;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -2e-71], t$95$0, If[LessEqual[N[(y * x), $MachinePrecision], 5e-35], N[(z * -0.125), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.9999999999999998e-71 or 4.99999999999999964e-35 < (*.f64 x y)

   1. Initial program 100.0%

      \[\frac{x \cdot y}{2} - \frac{z}{8} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(x \cdot y\right)} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(x \cdot y\right) + 0} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, x \cdot y, 0\right)} \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot y\right)\right)\right)}, 0\right) \]

      4. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(x \cdot y\right)\right) + 0\right)}\right), 0\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot y\right)\right)\right)\right) + \left(\mathsf{neg}\left(0\right)\right)}, 0\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{x \cdot y} + \left(\mathsf{neg}\left(0\right)\right), 0\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot y + \color{blue}{0}, 0\right) \]

      8. accelerator-lowering-fma.f64 81.2

         \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\mathsf{fma}\left(x, y, 0\right)}, 0\right) \]

   5. Simplified 81.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \mathsf{fma}\left(x, y, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(x \cdot y + 0\right)} \]

      2. +-rgt-identity N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot y\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(y \cdot x\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot x} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot \frac{1}{2}\right)} \cdot x \]

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

         \[\leadsto \color{blue}{\left(y \cdot \frac{1}{2}\right) \cdot x} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot y\right)} \cdot x \]

      8. *-lowering-*.f64 81.2

         \[\leadsto \color{blue}{\left(0.5 \cdot y\right)} \cdot x \]

   7. Applied egg-rr 81.2%

      \[\leadsto \color{blue}{\left(0.5 \cdot y\right) \cdot x} \]

   if -1.9999999999999998e-71 < (*.f64 x y) < 4.99999999999999964e-35

   1. Initial program 100.0%

      \[\frac{x \cdot y}{2} - \frac{z}{8} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot z} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{-1}{8} \cdot z + 0} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot \frac{-1}{8}} + 0 \]

      3. accelerator-lowering-fma.f64 89.4

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, -0.125, 0\right)} \]

   5. Simplified 89.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, -0.125, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{z \cdot \frac{-1}{8}} \]

      2. *-lowering-*.f64 89.4

         \[\leadsto \color{blue}{z \cdot -0.125} \]

   7. Applied egg-rr 89.4%

      \[\leadsto \color{blue}{z \cdot -0.125} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -2 \cdot 10^{-71}:\\ \;\;\;\;x \cdot \left(y \cdot 0.5\right)\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{-35}:\\ \;\;\;\;z \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ z \cdot -0.125 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* z -0.125))

C

double code(double x, double y, double z) {
	return z * -0.125;
}

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 * (-0.125d0)
end function

Java

public static double code(double x, double y, double z) {
	return z * -0.125;
}

Python

def code(x, y, z):
	return z * -0.125

Julia

function code(x, y, z)
	return Float64(z * -0.125)
end

MATLAB

function tmp = code(x, y, z)
	tmp = z * -0.125;
end

Wolfram

code[x_, y_, z_] := N[(z * -0.125), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{x \cdot y}{2} - \frac{z}{8} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-1}{8} \cdot z} \]
4. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot z + 0} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{z \cdot \frac{-1}{8}} + 0 \]

   3. accelerator-lowering-fma.f64 48.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, -0.125, 0\right)} \]

5. Simplified 48.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, -0.125, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{z \cdot \frac{-1}{8}} \]

   2. *-lowering-*.f64 48.0

      \[\leadsto \color{blue}{z \cdot -0.125} \]

7. Applied egg-rr 48.0%

   \[\leadsto \color{blue}{z \cdot -0.125} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))