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

Percentage Accurate: 99.8% → 99.8%

Time: 8.8s

Alternatives: 5

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))

C

double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * sqrt(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 = (1.0d0 / 2.0d0) * (x + (y * sqrt(z)))
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * Math.sqrt(z)));
}

Python

def code(x, y, z):
	return (1.0 / 2.0) * (x + (y * math.sqrt(z)))

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / 2.0) * Float64(x + Float64(y * sqrt(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / 2.0) * (x + (y * sqrt(z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(x + N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))

C

double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * sqrt(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 = (1.0d0 / 2.0d0) * (x + (y * sqrt(z)))
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * Math.sqrt(z)));
}

Python

def code(x, y, z):
	return (1.0 / 2.0) * (x + (y * math.sqrt(z)))

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / 2.0) * Float64(x + Float64(y * sqrt(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / 2.0) * (x + (y * sqrt(z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(x + N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
2. Step-by-step derivation

   1. metadata-eval 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto 0.5 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right) \]

Alternative 2: 75.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+53} \lor \neg \left(t_0 \leq 2 \cdot 10^{-33}\right):\\ \;\;\;\;0.5 \cdot \frac{y}{{z}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))))
   (if (or (<= t_0 -2e+53) (not (<= t_0 2e-33)))
     (* 0.5 (/ y (pow z -0.5)))
     (* 0.5 x))))

C

double code(double x, double y, double z) {
	double t_0 = y * sqrt(z);
	double tmp;
	if ((t_0 <= -2e+53) || !(t_0 <= 2e-33)) {
		tmp = 0.5 * (y / pow(z, -0.5));
	} else {
		tmp = 0.5 * x;
	}
	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 * sqrt(z)
    if ((t_0 <= (-2d+53)) .or. (.not. (t_0 <= 2d-33))) then
        tmp = 0.5d0 * (y / (z ** (-0.5d0)))
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * Math.sqrt(z);
	double tmp;
	if ((t_0 <= -2e+53) || !(t_0 <= 2e-33)) {
		tmp = 0.5 * (y / Math.pow(z, -0.5));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.sqrt(z)
	tmp = 0
	if (t_0 <= -2e+53) or not (t_0 <= 2e-33):
		tmp = 0.5 * (y / math.pow(z, -0.5))
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * sqrt(z))
	tmp = 0.0
	if ((t_0 <= -2e+53) || !(t_0 <= 2e-33))
		tmp = Float64(0.5 * Float64(y / (z ^ -0.5)));
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * sqrt(z);
	tmp = 0.0;
	if ((t_0 <= -2e+53) || ~((t_0 <= 2e-33)))
		tmp = 0.5 * (y / (z ^ -0.5));
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e+53], N[Not[LessEqual[t$95$0, 2e-33]], $MachinePrecision]], N[(0.5 * N[(y / N[Power[z, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (sqrt.f64 z)) < -2e53 or 2.0000000000000001e-33 < (*.f64 y (sqrt.f64 z))

   1. Initial program 99.7%

      \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 83.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y \cdot \sqrt{z}\right)} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 46.0%

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

      2. sqrt-unprod 28.2%

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

      3. pow1/2 28.2%

         \[\leadsto 0.5 \cdot \color{blue}{{\left(\left(y \cdot \sqrt{z}\right) \cdot \left(y \cdot \sqrt{z}\right)\right)}^{0.5}} \]

      4. *-commutative 28.2%

         \[\leadsto 0.5 \cdot {\left(\color{blue}{\left(\sqrt{z} \cdot y\right)} \cdot \left(y \cdot \sqrt{z}\right)\right)}^{0.5} \]

      5. *-commutative 28.2%

         \[\leadsto 0.5 \cdot {\left(\left(\sqrt{z} \cdot y\right) \cdot \color{blue}{\left(\sqrt{z} \cdot y\right)}\right)}^{0.5} \]

      6. swap-sqr 24.1%

         \[\leadsto 0.5 \cdot {\color{blue}{\left(\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \left(y \cdot y\right)\right)}}^{0.5} \]

      7. add-sqr-sqrt 24.1%

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

   6. Applied egg-rr 24.1%

      \[\leadsto 0.5 \cdot \color{blue}{{\left(z \cdot \left(y \cdot y\right)\right)}^{0.5}} \]
   7. Step-by-step derivation

      1. unpow1/2 24.1%

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

   8. Simplified 24.1%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{z \cdot \left(y \cdot y\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 28.2%

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

      2. sqrt-unprod 40.7%

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

      3. sqrt-prod 45.9%

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

      4. associate-*r* 46.0%

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

      5. add-sqr-sqrt 83.1%

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

      6. *-commutative 83.1%

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

      7. *-un-lft-identity 83.1%

         \[\leadsto 0.5 \cdot \left(y \cdot \color{blue}{\left(1 \cdot \sqrt{z}\right)}\right) \]

      8. associate-*r* 83.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(y \cdot 1\right) \cdot \sqrt{z}\right)} \]

      9. metadata-eval 83.1%

         \[\leadsto 0.5 \cdot \left(\left(y \cdot \color{blue}{\frac{1}{1}}\right) \cdot \sqrt{z}\right) \]

      10. div-inv 83.1%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\frac{y}{1}} \cdot \sqrt{z}\right) \]

      11. associate-/r/ 82.9%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{y}{\frac{1}{\sqrt{z}}}} \]

      12. pow1/2 82.9%

          \[\leadsto 0.5 \cdot \frac{y}{\frac{1}{\color{blue}{{z}^{0.5}}}} \]

      13. pow-flip 83.1%

          \[\leadsto 0.5 \cdot \frac{y}{\color{blue}{{z}^{\left(-0.5\right)}}} \]

      14. metadata-eval 83.1%

          \[\leadsto 0.5 \cdot \frac{y}{{z}^{\color{blue}{-0.5}}} \]

   10. Applied egg-rr 83.1%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{y}{{z}^{-0.5}}} \]

   if -2e53 < (*.f64 y (sqrt.f64 z)) < 2.0000000000000001e-33

   1. Initial program 99.9%

      \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 77.3%

      \[\leadsto 0.5 \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -2 \cdot 10^{+53} \lor \neg \left(y \cdot \sqrt{z} \leq 2 \cdot 10^{-33}\right):\\ \;\;\;\;0.5 \cdot \frac{y}{{z}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 3: 75.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+38} \lor \neg \left(t_0 \leq 8.2 \cdot 10^{-32}\right):\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))))
   (if (or (<= t_0 -2e+38) (not (<= t_0 8.2e-32))) (* 0.5 t_0) (* 0.5 x))))

C

double code(double x, double y, double z) {
	double t_0 = y * sqrt(z);
	double tmp;
	if ((t_0 <= -2e+38) || !(t_0 <= 8.2e-32)) {
		tmp = 0.5 * t_0;
	} else {
		tmp = 0.5 * x;
	}
	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 * sqrt(z)
    if ((t_0 <= (-2d+38)) .or. (.not. (t_0 <= 8.2d-32))) then
        tmp = 0.5d0 * t_0
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * Math.sqrt(z);
	double tmp;
	if ((t_0 <= -2e+38) || !(t_0 <= 8.2e-32)) {
		tmp = 0.5 * t_0;
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.sqrt(z)
	tmp = 0
	if (t_0 <= -2e+38) or not (t_0 <= 8.2e-32):
		tmp = 0.5 * t_0
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * sqrt(z))
	tmp = 0.0
	if ((t_0 <= -2e+38) || !(t_0 <= 8.2e-32))
		tmp = Float64(0.5 * t_0);
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * sqrt(z);
	tmp = 0.0;
	if ((t_0 <= -2e+38) || ~((t_0 <= 8.2e-32)))
		tmp = 0.5 * t_0;
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e+38], N[Not[LessEqual[t$95$0, 8.2e-32]], $MachinePrecision]], N[(0.5 * t$95$0), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (sqrt.f64 z)) < -1.99999999999999995e38 or 8.1999999999999995e-32 < (*.f64 y (sqrt.f64 z))

   1. Initial program 99.7%

      \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 83.1%

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

   if -1.99999999999999995e38 < (*.f64 y (sqrt.f64 z)) < 8.1999999999999995e-32

   1. Initial program 99.9%

      \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
   2. Step-by-step derivation

      1. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 77.3%

      \[\leadsto 0.5 \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -2 \cdot 10^{+38} \lor \neg \left(y \cdot \sqrt{z} \leq 8.2 \cdot 10^{-32}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(x + y \cdot \sqrt{z}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 (+ x (* y (sqrt z)))))

C

double code(double x, double y, double z) {
	return 0.5 * (x + (y * sqrt(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 = 0.5d0 * (x + (y * sqrt(z)))
end function

Java

public static double code(double x, double y, double z) {
	return 0.5 * (x + (y * Math.sqrt(z)));
}

Python

def code(x, y, z):
	return 0.5 * (x + (y * math.sqrt(z)))

Julia

function code(x, y, z)
	return Float64(0.5 * Float64(x + Float64(y * sqrt(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.5 * (x + (y * sqrt(z)));
end

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
2. Step-by-step derivation

   1. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto 0.5 \cdot \left(x + y \cdot \sqrt{z}\right) \]

Alternative 5: 50.2% accurate, 36.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 x))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.5d0 * x
end function

Java

public static double code(double x, double y, double z) {
	return 0.5 * x;
}

Python

def code(x, y, z):
	return 0.5 * x

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = 0.5 * x;
end

Wolfram

code[x_, y_, z_] := N[(0.5 * x), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
2. Step-by-step derivation

   1. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around inf 45.3%

   \[\leadsto 0.5 \cdot \color{blue}{x} \]

5. Final simplification 45.3%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023287 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B"
  :precision binary64
  (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))