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

Percentage Accurate: 99.8% → 99.8%

Time: 6.2s

Alternatives: 6

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(0.5 * N[(y * N[Power[z, 0.5], $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%

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

3. Simplified 99.8%

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

   1. +-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. add-sqr-sqrt 99.6%

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

   4. associate-*l* 99.6%

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

   5. fma-def 99.6%

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

   6. pow1/2 99.6%

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

   7. sqrt-pow1 99.7%

      \[\leadsto 0.5 \cdot \mathsf{fma}\left(\color{blue}{{z}^{\left(\frac{0.5}{2}\right)}}, \sqrt{\sqrt{z}} \cdot y, x\right) \]

   8. metadata-eval 99.7%

      \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{\color{blue}{0.25}}, \sqrt{\sqrt{z}} \cdot y, x\right) \]

   9. pow1/2 99.7%

      \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{0.25}, \sqrt{\color{blue}{{z}^{0.5}}} \cdot y, x\right) \]

   10. sqrt-pow1 99.7%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{0.25}, \color{blue}{{z}^{\left(\frac{0.5}{2}\right)}} \cdot y, x\right) \]

   11. metadata-eval 99.7%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{0.25}, {z}^{\color{blue}{0.25}} \cdot y, x\right) \]

5. Applied egg-rr 99.7%

   \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left({z}^{0.25}, {z}^{0.25} \cdot y, x\right)} \]
6. Step-by-step derivation

   1. fma-udef 99.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left({z}^{0.25} \cdot \left({z}^{0.25} \cdot y\right) + x\right)} \]

   2. *-commutative 99.6%

      \[\leadsto 0.5 \cdot \left({z}^{0.25} \cdot \color{blue}{\left(y \cdot {z}^{0.25}\right)} + x\right) \]

   3. *-commutative 99.6%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(y \cdot {z}^{0.25}\right) \cdot {z}^{0.25}} + x\right) \]

   4. associate-*l* 99.7%

      \[\leadsto 0.5 \cdot \left(\color{blue}{y \cdot \left({z}^{0.25} \cdot {z}^{0.25}\right)} + x\right) \]

   5. fma-def 99.7%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(y, {z}^{0.25} \cdot {z}^{0.25}, x\right)} \]

   6. pow-sqr 99.9%

      \[\leadsto 0.5 \cdot \mathsf{fma}\left(y, \color{blue}{{z}^{\left(2 \cdot 0.25\right)}}, x\right) \]

   7. metadata-eval 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 73.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-35} \lor \neg \left(t_0 \leq 5 \cdot 10^{-105}\right) \land \left(t_0 \leq 5 \cdot 10^{+21} \lor \neg \left(t_0 \leq 2 \cdot 10^{+44}\right)\right):\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))))
   (if (or (<= t_0 -1e-35)
           (and (not (<= t_0 5e-105))
                (or (<= t_0 5e+21) (not (<= t_0 2e+44)))))
     (* 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 <= -1e-35) || (!(t_0 <= 5e-105) && ((t_0 <= 5e+21) || !(t_0 <= 2e+44)))) {
		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 <= (-1d-35)) .or. (.not. (t_0 <= 5d-105)) .and. (t_0 <= 5d+21) .or. (.not. (t_0 <= 2d+44))) 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 <= -1e-35) || (!(t_0 <= 5e-105) && ((t_0 <= 5e+21) || !(t_0 <= 2e+44)))) {
		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 <= -1e-35) or (not (t_0 <= 5e-105) and ((t_0 <= 5e+21) or not (t_0 <= 2e+44))):
		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 <= -1e-35) || (!(t_0 <= 5e-105) && ((t_0 <= 5e+21) || !(t_0 <= 2e+44))))
		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 <= -1e-35) || (~((t_0 <= 5e-105)) && ((t_0 <= 5e+21) || ~((t_0 <= 2e+44)))))
		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, -1e-35], And[N[Not[LessEqual[t$95$0, 5e-105]], $MachinePrecision], Or[LessEqual[t$95$0, 5e+21], N[Not[LessEqual[t$95$0, 2e+44]], $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)) < -1.00000000000000001e-35 or 4.99999999999999963e-105 < (*.f64 y (sqrt.f64 z)) < 5e21 or 2.0000000000000002e44 < (*.f64 y (sqrt.f64 z))

   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 0 76.9%

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

   if -1.00000000000000001e-35 < (*.f64 y (sqrt.f64 z)) < 4.99999999999999963e-105 or 5e21 < (*.f64 y (sqrt.f64 z)) < 2.0000000000000002e44

   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 85.2%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -1 \cdot 10^{-35} \lor \neg \left(y \cdot \sqrt{z} \leq 5 \cdot 10^{-105}\right) \land \left(y \cdot \sqrt{z} \leq 5 \cdot 10^{+21} \lor \neg \left(y \cdot \sqrt{z} \leq 2 \cdot 10^{+44}\right)\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 3: 73.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ t_1 := 0.5 \cdot t_0\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-105}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+21}:\\ \;\;\;\;0.5 \cdot \sqrt{y \cdot \left(y \cdot z\right)}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))) (t_1 (* 0.5 t_0)))
   (if (<= t_0 -1e-35)
     t_1
     (if (<= t_0 5e-105)
       (* 0.5 x)
       (if (<= t_0 5e+21)
         (* 0.5 (sqrt (* y (* y z))))
         (if (<= t_0 2e+44) (* 0.5 x) t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = y * sqrt(z);
	double t_1 = 0.5 * t_0;
	double tmp;
	if (t_0 <= -1e-35) {
		tmp = t_1;
	} else if (t_0 <= 5e-105) {
		tmp = 0.5 * x;
	} else if (t_0 <= 5e+21) {
		tmp = 0.5 * sqrt((y * (y * z)));
	} else if (t_0 <= 2e+44) {
		tmp = 0.5 * x;
	} 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 * sqrt(z)
    t_1 = 0.5d0 * t_0
    if (t_0 <= (-1d-35)) then
        tmp = t_1
    else if (t_0 <= 5d-105) then
        tmp = 0.5d0 * x
    else if (t_0 <= 5d+21) then
        tmp = 0.5d0 * sqrt((y * (y * z)))
    else if (t_0 <= 2d+44) then
        tmp = 0.5d0 * x
    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 * Math.sqrt(z);
	double t_1 = 0.5 * t_0;
	double tmp;
	if (t_0 <= -1e-35) {
		tmp = t_1;
	} else if (t_0 <= 5e-105) {
		tmp = 0.5 * x;
	} else if (t_0 <= 5e+21) {
		tmp = 0.5 * Math.sqrt((y * (y * z)));
	} else if (t_0 <= 2e+44) {
		tmp = 0.5 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.sqrt(z)
	t_1 = 0.5 * t_0
	tmp = 0
	if t_0 <= -1e-35:
		tmp = t_1
	elif t_0 <= 5e-105:
		tmp = 0.5 * x
	elif t_0 <= 5e+21:
		tmp = 0.5 * math.sqrt((y * (y * z)))
	elif t_0 <= 2e+44:
		tmp = 0.5 * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * sqrt(z))
	t_1 = Float64(0.5 * t_0)
	tmp = 0.0
	if (t_0 <= -1e-35)
		tmp = t_1;
	elseif (t_0 <= 5e-105)
		tmp = Float64(0.5 * x);
	elseif (t_0 <= 5e+21)
		tmp = Float64(0.5 * sqrt(Float64(y * Float64(y * z))));
	elseif (t_0 <= 2e+44)
		tmp = Float64(0.5 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * sqrt(z);
	t_1 = 0.5 * t_0;
	tmp = 0.0;
	if (t_0 <= -1e-35)
		tmp = t_1;
	elseif (t_0 <= 5e-105)
		tmp = 0.5 * x;
	elseif (t_0 <= 5e+21)
		tmp = 0.5 * sqrt((y * (y * z)));
	elseif (t_0 <= 2e+44)
		tmp = 0.5 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-35], t$95$1, If[LessEqual[t$95$0, 5e-105], N[(0.5 * x), $MachinePrecision], If[LessEqual[t$95$0, 5e+21], N[(0.5 * N[Sqrt[N[(y * N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+44], N[(0.5 * x), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (sqrt.f64 z)) < -1.00000000000000001e-35 or 2.0000000000000002e44 < (*.f64 y (sqrt.f64 z))

   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 0 80.6%

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

   if -1.00000000000000001e-35 < (*.f64 y (sqrt.f64 z)) < 4.99999999999999963e-105 or 5e21 < (*.f64 y (sqrt.f64 z)) < 2.0000000000000002e44

   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 85.2%

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

   if 4.99999999999999963e-105 < (*.f64 y (sqrt.f64 z)) < 5e21

   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 60.1%

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

      1. add-sqr-sqrt 59.6%

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

      2. sqrt-unprod 60.1%

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

      3. pow1/2 60.1%

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

      4. *-commutative 60.1%

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

      5. *-commutative 60.1%

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

      6. swap-sqr 52.4%

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

      7. add-sqr-sqrt 52.5%

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

   6. Applied egg-rr 52.5%

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

      1. unpow1/2 52.5%

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

      2. *-commutative 52.5%

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

      3. associate-*l* 60.2%

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

   8. Simplified 60.2%

      \[\leadsto 0.5 \cdot \color{blue}{\sqrt{y \cdot \left(y \cdot z\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -1 \cdot 10^{-35}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{elif}\;y \cdot \sqrt{z} \leq 5 \cdot 10^{-105}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;y \cdot \sqrt{z} \leq 5 \cdot 10^{+21}:\\ \;\;\;\;0.5 \cdot \sqrt{y \cdot \left(y \cdot z\right)}\\ \mathbf{elif}\;y \cdot \sqrt{z} \leq 2 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * (z ** 0.5d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(0.5 * N[(x + N[(y * N[Power[z, 0.5], $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. Step-by-step derivation

   1. +-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. add-sqr-sqrt 99.6%

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

   4. associate-*l* 99.6%

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

   5. fma-def 99.6%

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

   6. pow1/2 99.6%

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

   7. sqrt-pow1 99.7%

      \[\leadsto 0.5 \cdot \mathsf{fma}\left(\color{blue}{{z}^{\left(\frac{0.5}{2}\right)}}, \sqrt{\sqrt{z}} \cdot y, x\right) \]

   8. metadata-eval 99.7%

      \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{\color{blue}{0.25}}, \sqrt{\sqrt{z}} \cdot y, x\right) \]

   9. pow1/2 99.7%

      \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{0.25}, \sqrt{\color{blue}{{z}^{0.5}}} \cdot y, x\right) \]

   10. sqrt-pow1 99.7%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{0.25}, \color{blue}{{z}^{\left(\frac{0.5}{2}\right)}} \cdot y, x\right) \]

   11. metadata-eval 99.7%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{0.25}, {z}^{\color{blue}{0.25}} \cdot y, x\right) \]

5. Applied egg-rr 99.7%

   \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left({z}^{0.25}, {z}^{0.25} \cdot y, x\right)} \]
6. Step-by-step derivation

   1. fma-udef 99.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left({z}^{0.25} \cdot \left({z}^{0.25} \cdot y\right) + x\right)} \]

   2. *-commutative 99.6%

      \[\leadsto 0.5 \cdot \left({z}^{0.25} \cdot \color{blue}{\left(y \cdot {z}^{0.25}\right)} + x\right) \]

   3. *-commutative 99.6%

      \[\leadsto 0.5 \cdot \left(\color{blue}{\left(y \cdot {z}^{0.25}\right) \cdot {z}^{0.25}} + x\right) \]

   4. associate-*l* 99.7%

      \[\leadsto 0.5 \cdot \left(\color{blue}{y \cdot \left({z}^{0.25} \cdot {z}^{0.25}\right)} + x\right) \]

   5. fma-def 99.7%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(y, {z}^{0.25} \cdot {z}^{0.25}, x\right)} \]

   6. pow-sqr 99.9%

      \[\leadsto 0.5 \cdot \mathsf{fma}\left(y, \color{blue}{{z}^{\left(2 \cdot 0.25\right)}}, x\right) \]

   7. metadata-eval 99.9%

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

7. Simplified 99.9%

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

   1. fma-udef 99.8%

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

   2. pow1/2 99.8%

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

   3. +-commutative 99.8%

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

   4. flip-+ 52.9%

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

   5. div-sub 52.8%

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

   6. sub-neg 52.8%

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

   7. distribute-rgt-neg-out 52.8%

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

   8. +-commutative 52.8%

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

   9. fma-udef 52.8%

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

   10. swap-sqr 46.6%

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

   11. add-sqr-sqrt 46.7%

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

   12. associate-*r* 52.8%

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

   13. sub-neg 52.8%

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

   14. distribute-rgt-neg-out 52.8%

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

   15. +-commutative 52.8%

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

   16. fma-udef 52.8%

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

   17. div-sub 52.8%

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

9. Applied egg-rr 99.7%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(y, \sqrt{z}, x\right)}}} \]
10. Step-by-step derivation

    1. remove-double-div 99.8%

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

    2. fma-udef 99.8%

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

    3. *-commutative 99.8%

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

    4. fma-def 99.8%

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

    5. pow1/2 99.9%

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

    6. metadata-eval 99.9%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{\color{blue}{\left(0.25 + 0.25\right)}}, y, x\right) \]

    7. pow-prod-up 99.7%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left(\color{blue}{{z}^{0.25} \cdot {z}^{0.25}}, y, x\right) \]

    8. fma-def 99.7%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\left({z}^{0.25} \cdot {z}^{0.25}\right) \cdot y + x\right)} \]

    9. associate-*r* 99.6%

       \[\leadsto 0.5 \cdot \left(\color{blue}{{z}^{0.25} \cdot \left({z}^{0.25} \cdot y\right)} + x\right) \]

    10. fma-def 99.7%

        \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left({z}^{0.25}, {z}^{0.25} \cdot y, x\right)} \]

    11. *-commutative 99.7%

        \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{0.25}, \color{blue}{y \cdot {z}^{0.25}}, x\right) \]

11. Applied egg-rr 99.7%

    \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left({z}^{0.25}, y \cdot {z}^{0.25}, x\right)} \]
12. Step-by-step derivation

    1. fma-udef 99.6%

       \[\leadsto 0.5 \cdot \color{blue}{\left({z}^{0.25} \cdot \left(y \cdot {z}^{0.25}\right) + x\right)} \]

    2. +-commutative 99.6%

       \[\leadsto 0.5 \cdot \color{blue}{\left(x + {z}^{0.25} \cdot \left(y \cdot {z}^{0.25}\right)\right)} \]

    3. *-commutative 99.6%

       \[\leadsto 0.5 \cdot \left(x + \color{blue}{\left(y \cdot {z}^{0.25}\right) \cdot {z}^{0.25}}\right) \]

    4. associate-*l* 99.7%

       \[\leadsto 0.5 \cdot \left(x + \color{blue}{y \cdot \left({z}^{0.25} \cdot {z}^{0.25}\right)}\right) \]

    5. pow-sqr 99.8%

       \[\leadsto 0.5 \cdot \left(x + y \cdot \color{blue}{{z}^{\left(2 \cdot 0.25\right)}}\right) \]

    6. metadata-eval 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

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

Alternative 5: 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 6: 51.6% 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 52.0%

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

5. Final simplification 52.0%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023290 
(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)))))