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

Percentage Accurate: 99.8% → 99.8%

Time: 21.5s

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, 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 2: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 1.86 \cdot 10^{+55}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x - y \cdot \frac{y}{\frac{x}{z}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.5e-11)
   (* 0.5 x)
   (if (<= x 1.86e+55)
     (* 0.5 (* y (sqrt z)))
     (* 0.5 (- x (* y (/ y (/ x z))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.5e-11) {
		tmp = 0.5 * x;
	} else if (x <= 1.86e+55) {
		tmp = 0.5 * (y * sqrt(z));
	} else {
		tmp = 0.5 * (x - (y * (y / (x / 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) :: tmp
    if (x <= (-2.5d-11)) then
        tmp = 0.5d0 * x
    else if (x <= 1.86d+55) then
        tmp = 0.5d0 * (y * sqrt(z))
    else
        tmp = 0.5d0 * (x - (y * (y / (x / z))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.5e-11) {
		tmp = 0.5 * x;
	} else if (x <= 1.86e+55) {
		tmp = 0.5 * (y * Math.sqrt(z));
	} else {
		tmp = 0.5 * (x - (y * (y / (x / z))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.5e-11:
		tmp = 0.5 * x
	elif x <= 1.86e+55:
		tmp = 0.5 * (y * math.sqrt(z))
	else:
		tmp = 0.5 * (x - (y * (y / (x / z))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.5e-11)
		tmp = Float64(0.5 * x);
	elseif (x <= 1.86e+55)
		tmp = Float64(0.5 * Float64(y * sqrt(z)));
	else
		tmp = Float64(0.5 * Float64(x - Float64(y * Float64(y / Float64(x / z)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.5e-11)
		tmp = 0.5 * x;
	elseif (x <= 1.86e+55)
		tmp = 0.5 * (y * sqrt(z));
	else
		tmp = 0.5 * (x - (y * (y / (x / z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.5e-11], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 1.86e+55], N[(0.5 * N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x - N[(y * N[(y / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.50000000000000009e-11

   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 80.4%

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

   if -2.50000000000000009e-11 < x < 1.86e55

   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 82.7%

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

   if 1.86e55 < x

   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. Step-by-step derivation

      1. cancel-sign-sub 99.9%

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

      2. distribute-lft-neg-in 99.9%

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

      3. flip-- 30.6%

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

      4. sqr-neg 30.6%

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

      5. swap-sqr 30.3%

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

      6. rem-square-sqrt 30.3%

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

      7. sub-neg 30.3%

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

   5. Applied egg-rr 30.3%

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

      1. div-sub 30.3%

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

      2. associate-/l* 74.0%

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

      3. associate-*r* 77.8%

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

      4. frac-2neg 77.8%

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

      5. sub-neg 77.8%

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

      6. +-commutative 77.8%

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

      7. distribute-rgt-neg-in 77.8%

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

      8. fma-def 77.8%

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

      9. distribute-lft-neg-in 77.8%

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

      10. neg-mul-1 77.8%

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

      11. times-frac 89.0%

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

      12. metadata-eval 89.0%

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

      13. frac-2neg 89.0%

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

      14. /-rgt-identity 89.0%

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

      15. sub-neg 89.0%

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

      16. +-commutative 89.0%

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

   7. Applied egg-rr 89.0%

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

   8. Taylor expanded in y around 0 87.9%

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

   9. Taylor expanded in y around 0 87.3%

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

       1. associate-/l* 89.3%

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

   11. Simplified 89.3%

       \[\leadsto 0.5 \cdot \left(\frac{x}{1} - y \cdot \color{blue}{\frac{y}{\frac{x}{z}}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-11}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 1.86 \cdot 10^{+55}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x - y \cdot \frac{y}{\frac{x}{z}}\right)\\ \end{array} \]

Alternative 3: 53.3% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(0.5 * N[(x - N[(N[(y / x), $MachinePrecision] * N[(y * 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. Step-by-step derivation

   1. cancel-sign-sub 99.8%

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

   2. distribute-lft-neg-in 99.8%

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

   3. flip-- 53.9%

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

   4. sqr-neg 53.9%

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

   5. swap-sqr 49.9%

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

   6. rem-square-sqrt 49.9%

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

   7. sub-neg 49.9%

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

5. Applied egg-rr 49.9%

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

   1. div-sub 49.9%

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

   2. associate-*r* 53.8%

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

   3. *-commutative 53.8%

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

   4. associate-/l* 64.6%

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

   5. frac-sub 41.4%

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

   6. frac-2neg 41.4%

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

   7. neg-mul-1 41.4%

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

   8. distribute-lft-neg-in 41.4%

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

   9. times-frac 39.8%

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

7. Applied egg-rr 53.7%

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

8. Taylor expanded in y around 0 26.8%

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

9. Taylor expanded in x around 0 44.3%

   \[\leadsto 0.5 \cdot \color{blue}{\left(x + -1 \cdot \frac{{y}^{2} \cdot z}{x}\right)} \]
10. Step-by-step derivation

    1. *-lft-identity 44.3%

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

    2. fma-def 44.3%

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

    3. mul-1-neg 44.3%

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

    4. fma-neg 44.3%

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

    5. *-lft-identity 44.3%

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

    6. unpow2 44.3%

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

    7. associate-*r* 45.9%

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

    8. *-lft-identity 45.9%

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

    9. associate-*l/ 45.9%

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

    10. associate-*r* 48.6%

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

    11. *-commutative 48.6%

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

    12. associate-*r/ 48.6%

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

    13. *-rgt-identity 48.6%

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

11. Simplified 48.6%

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

12. Final simplification 48.6%

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

Alternative 4: 53.3% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(0.5 * N[(x - N[(y * N[(y / N[(x / z), $MachinePrecision]), $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. cancel-sign-sub 99.8%

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

   2. distribute-lft-neg-in 99.8%

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

   3. flip-- 53.9%

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

   4. sqr-neg 53.9%

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

   5. swap-sqr 49.9%

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

   6. rem-square-sqrt 49.9%

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

   7. sub-neg 49.9%

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

5. Applied egg-rr 49.9%

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

   1. div-sub 49.9%

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

   2. associate-/l* 67.2%

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

   3. associate-*r* 72.1%

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

   4. frac-2neg 72.1%

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

   5. sub-neg 72.1%

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

   6. +-commutative 72.1%

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

   7. distribute-rgt-neg-in 72.1%

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

   8. fma-def 72.1%

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

   9. distribute-lft-neg-in 72.1%

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

   10. neg-mul-1 72.1%

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

   11. times-frac 87.0%

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

   12. metadata-eval 87.0%

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

   13. frac-2neg 87.0%

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

   14. /-rgt-identity 87.0%

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

   15. sub-neg 87.0%

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

   16. +-commutative 87.0%

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

7. Applied egg-rr 87.0%

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

8. Taylor expanded in y around 0 85.5%

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

9. Taylor expanded in y around 0 48.6%

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

    1. associate-/l* 49.5%

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

11. Simplified 49.5%

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

12. Final simplification 49.5%

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

Alternative 5: 50.5% 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 47.5%

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

5. Final simplification 47.5%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023297 
(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)))))