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

Percentage Accurate: 99.8% → 99.8%

Time: 6.6s

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, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{y \cdot 0.5}{{z}^{-0.5}} + 0.5 \cdot x \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. +-commutative 99.5%

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

   2. distribute-lft-in 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.8%

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

5. Applied egg-rr 99.8%

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

   1. add-cube-cbrt 99.1%

      \[\leadsto \color{blue}{\left(\sqrt[3]{y \cdot \left(\sqrt{z} \cdot 0.5\right)} \cdot \sqrt[3]{y \cdot \left(\sqrt{z} \cdot 0.5\right)}\right) \cdot \sqrt[3]{y \cdot \left(\sqrt{z} \cdot 0.5\right)}} + 0.5 \cdot x \]

   2. pow3 99.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{y \cdot \left(\sqrt{z} \cdot 0.5\right)}\right)}^{3}} + 0.5 \cdot x \]

   3. add-sqr-sqrt 99.0%

      \[\leadsto {\left(\sqrt[3]{y \cdot \color{blue}{\left(\sqrt{\sqrt{z} \cdot 0.5} \cdot \sqrt{\sqrt{z} \cdot 0.5}\right)}}\right)}^{3} + 0.5 \cdot x \]

   4. sqrt-unprod 99.1%

      \[\leadsto {\left(\sqrt[3]{y \cdot \color{blue}{\sqrt{\left(\sqrt{z} \cdot 0.5\right) \cdot \left(\sqrt{z} \cdot 0.5\right)}}}\right)}^{3} + 0.5 \cdot x \]

   5. swap-sqr 99.1%

      \[\leadsto {\left(\sqrt[3]{y \cdot \sqrt{\color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right) \cdot \left(0.5 \cdot 0.5\right)}}}\right)}^{3} + 0.5 \cdot x \]

   6. add-sqr-sqrt 99.1%

      \[\leadsto {\left(\sqrt[3]{y \cdot \sqrt{\color{blue}{z} \cdot \left(0.5 \cdot 0.5\right)}}\right)}^{3} + 0.5 \cdot x \]

   7. metadata-eval 99.1%

      \[\leadsto {\left(\sqrt[3]{y \cdot \sqrt{z \cdot \color{blue}{0.25}}}\right)}^{3} + 0.5 \cdot x \]

7. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{y \cdot \sqrt{z \cdot 0.25}}\right)}^{3}} + 0.5 \cdot x \]
8. Step-by-step derivation

   1. unpow3 99.1%

      \[\leadsto \color{blue}{\left(\sqrt[3]{y \cdot \sqrt{z \cdot 0.25}} \cdot \sqrt[3]{y \cdot \sqrt{z \cdot 0.25}}\right) \cdot \sqrt[3]{y \cdot \sqrt{z \cdot 0.25}}} + 0.5 \cdot x \]

   2. add-cube-cbrt 99.8%

      \[\leadsto \color{blue}{y \cdot \sqrt{z \cdot 0.25}} + 0.5 \cdot x \]

   3. *-commutative 99.8%

      \[\leadsto \color{blue}{\sqrt{z \cdot 0.25} \cdot y} + 0.5 \cdot x \]

   4. *-commutative 99.8%

      \[\leadsto \sqrt{\color{blue}{0.25 \cdot z}} \cdot y + 0.5 \cdot x \]

   5. sqrt-prod 99.8%

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

   6. metadata-eval 99.8%

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

   7. metadata-eval 99.8%

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

   8. associate-/r/ 99.8%

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

   9. metadata-eval 99.8%

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

   10. sqrt-div 99.7%

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

   11. associate-/r/ 99.8%

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

   12. div-inv 99.5%

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

   13. clear-num 99.5%

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

   14. inv-pow 99.5%

       \[\leadsto 0.5 \cdot \frac{y}{\sqrt{\color{blue}{{z}^{-1}}}} + 0.5 \cdot x \]

   15. sqrt-pow1 99.5%

       \[\leadsto 0.5 \cdot \frac{y}{\color{blue}{{z}^{\left(\frac{-1}{2}\right)}}} + 0.5 \cdot x \]

   16. metadata-eval 99.5%

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

9. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{0.5 \cdot \frac{y}{{z}^{-0.5}}} + 0.5 \cdot x \]
10. Step-by-step derivation

    1. *-commutative 99.5%

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

    2. associate-*l/ 99.8%

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

11. Simplified 99.8%

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

12. Final simplification 99.8%

    \[\leadsto \frac{y \cdot 0.5}{{z}^{-0.5}} + 0.5 \cdot x \]

Alternative 2: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-84} \lor \neg \left(t_0 \leq 10^{-92}\right):\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))))
   (if (or (<= t_0 -4e-84) (not (<= t_0 1e-92))) (* 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 <= -4e-84) || !(t_0 <= 1e-92)) {
		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 <= (-4d-84)) .or. (.not. (t_0 <= 1d-92))) 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 <= -4e-84) || !(t_0 <= 1e-92)) {
		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 <= -4e-84) or not (t_0 <= 1e-92):
		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 <= -4e-84) || !(t_0 <= 1e-92))
		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 <= -4e-84) || ~((t_0 <= 1e-92)))
		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, -4e-84], N[Not[LessEqual[t$95$0, 1e-92]], $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)) < -4.0000000000000001e-84 or 9.99999999999999988e-93 < (*.f64 y (sqrt.f64 z))

   1. Initial program 99.2%

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

      1. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 79.6%

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

   if -4.0000000000000001e-84 < (*.f64 y (sqrt.f64 z)) < 9.99999999999999988e-93

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 89.2%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -4 \cdot 10^{-84} \lor \neg \left(y \cdot \sqrt{z} \leq 10^{-92}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 3: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-84}:\\ \;\;\;\;0.5 \cdot t_0\\ \mathbf{elif}\;t_0 \leq 10^{-92}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))))
   (if (<= t_0 -4e-84)
     (* 0.5 t_0)
     (if (<= t_0 1e-92) (* 0.5 x) (* y (* 0.5 (sqrt z)))))))

C

double code(double x, double y, double z) {
	double t_0 = y * sqrt(z);
	double tmp;
	if (t_0 <= -4e-84) {
		tmp = 0.5 * t_0;
	} else if (t_0 <= 1e-92) {
		tmp = 0.5 * x;
	} else {
		tmp = y * (0.5 * sqrt(z));
	}
	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 <= (-4d-84)) then
        tmp = 0.5d0 * t_0
    else if (t_0 <= 1d-92) then
        tmp = 0.5d0 * x
    else
        tmp = y * (0.5d0 * sqrt(z))
    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 <= -4e-84) {
		tmp = 0.5 * t_0;
	} else if (t_0 <= 1e-92) {
		tmp = 0.5 * x;
	} else {
		tmp = y * (0.5 * Math.sqrt(z));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.sqrt(z)
	tmp = 0
	if t_0 <= -4e-84:
		tmp = 0.5 * t_0
	elif t_0 <= 1e-92:
		tmp = 0.5 * x
	else:
		tmp = y * (0.5 * math.sqrt(z))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * sqrt(z))
	tmp = 0.0
	if (t_0 <= -4e-84)
		tmp = Float64(0.5 * t_0);
	elseif (t_0 <= 1e-92)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(y * Float64(0.5 * sqrt(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * sqrt(z);
	tmp = 0.0;
	if (t_0 <= -4e-84)
		tmp = 0.5 * t_0;
	elseif (t_0 <= 1e-92)
		tmp = 0.5 * x;
	else
		tmp = y * (0.5 * sqrt(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-84], N[(0.5 * t$95$0), $MachinePrecision], If[LessEqual[t$95$0, 1e-92], N[(0.5 * x), $MachinePrecision], N[(y * N[(0.5 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (sqrt.f64 z)) < -4.0000000000000001e-84

   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 79.9%

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

   if -4.0000000000000001e-84 < (*.f64 y (sqrt.f64 z)) < 9.99999999999999988e-93

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 89.2%

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

   if 9.99999999999999988e-93 < (*.f64 y (sqrt.f64 z))

   1. Initial program 98.8%

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

      1. metadata-eval 98.8%

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

   3. Simplified 98.8%

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

      1. +-commutative 98.8%

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

      2. distribute-lft-in 98.8%

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

      3. *-commutative 98.8%

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

      4. associate-*l* 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*l* 98.8%

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

      4. *-commutative 98.8%

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

      5. add-sqr-sqrt 98.5%

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

      6. sqrt-unprod 65.6%

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

      7. sqrt-prod 54.9%

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

      8. associate-*r* 66.3%

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

      9. distribute-lft-in 66.3%

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

      10. +-commutative 66.3%

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

      11. flip-+ 48.2%

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

      12. add-sqr-sqrt 48.2%

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

      13. sqrt-prod 49.1%

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

      14. sqrt-prod 50.1%

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

      15. associate-*r* 50.1%

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

      16. add-sqr-sqrt 50.2%

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

   7. Applied egg-rr 98.6%

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

   8. Taylor expanded in x around 0 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-*l* 80.1%

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

   10. Simplified 80.1%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -4 \cdot 10^{-84}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{elif}\;y \cdot \sqrt{z} \leq 10^{-92}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \sqrt{z}\right)\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. +-commutative 99.5%

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

   2. distribute-lft-in 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

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 99.5%

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

   1. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 6: 50.8% 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 99.5%

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

   1. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around inf 44.8%

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

5. Final simplification 44.8%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023252 
(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)))))