Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3

Percentage Accurate: 99.8% → 99.8%

Time: 7.2s

Alternatives: 8

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x \cdot \cos y + z \cdot \sin y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (cos y)) (* z (sin y))))

C

double code(double x, double y, double z) {
	return (x * cos(y)) + (z * sin(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * cos(y)) + (z * sin(y))
end function

Java

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

Python

def code(x, y, z):
	return (x * math.cos(y)) + (z * math.sin(y))

Julia

function code(x, y, z)
	return Float64(Float64(x * cos(y)) + Float64(z * sin(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * cos(y)) + (z * sin(y));
end

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \cos y + z \cdot \sin y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (cos y)) (* z (sin y))))

C

double code(double x, double y, double z) {
	return (x * cos(y)) + (z * sin(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * cos(y)) + (z * sin(y))
end function

Java

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

Python

def code(x, y, z):
	return (x * math.cos(y)) + (z * math.sin(y))

Julia

function code(x, y, z)
	return Float64(Float64(x * cos(y)) + Float64(z * sin(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * cos(y)) + (z * sin(y));
end

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \cos y + z \cdot \sin y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (cos y)) (* z (sin y))))

C

double code(double x, double y, double z) {
	return (x * cos(y)) + (z * sin(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * cos(y)) + (z * sin(y))
end function

Java

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

Python

def code(x, y, z):
	return (x * math.cos(y)) + (z * math.sin(y))

Julia

function code(x, y, z)
	return Float64(Float64(x * cos(y)) + Float64(z * sin(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * cos(y)) + (z * sin(y));
end

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot \cos y + z \cdot \sin y \]

2. Final simplification 99.9%

   \[\leadsto x \cdot \cos y + z \cdot \sin y \]

Alternative 2: 81.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-96} \lor \neg \left(z \leq 5.9 \cdot 10^{-220}\right):\\ \;\;\;\;x + z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.5e-96) (not (<= z 5.9e-220)))
   (+ x (* z (sin y)))
   (+ (* x (cos y)) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.5e-96) || !(z <= 5.9e-220)) {
		tmp = x + (z * sin(y));
	} else {
		tmp = (x * cos(y)) + (y * 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 ((z <= (-6.5d-96)) .or. (.not. (z <= 5.9d-220))) then
        tmp = x + (z * sin(y))
    else
        tmp = (x * cos(y)) + (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.5e-96) || !(z <= 5.9e-220)) {
		tmp = x + (z * Math.sin(y));
	} else {
		tmp = (x * Math.cos(y)) + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.5e-96) or not (z <= 5.9e-220):
		tmp = x + (z * math.sin(y))
	else:
		tmp = (x * math.cos(y)) + (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.5e-96) || !(z <= 5.9e-220))
		tmp = Float64(x + Float64(z * sin(y)));
	else
		tmp = Float64(Float64(x * cos(y)) + Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6.5e-96) || ~((z <= 5.9e-220)))
		tmp = x + (z * sin(y));
	else
		tmp = (x * cos(y)) + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.5e-96], N[Not[LessEqual[z, 5.9e-220]], $MachinePrecision]], N[(x + N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.50000000000000001e-96 or 5.8999999999999996e-220 < z

   1. Initial program 99.9%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in y around 0 90.6%

      \[\leadsto \color{blue}{x} + z \cdot \sin y \]

   if -6.50000000000000001e-96 < z < 5.8999999999999996e-220

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in y around 0 79.7%

      \[\leadsto x \cdot \cos y + \color{blue}{y \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-96} \lor \neg \left(z \leq 5.9 \cdot 10^{-220}\right):\\ \;\;\;\;x + z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y + y \cdot z\\ \end{array} \]

Alternative 3: 86.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-95} \lor \neg \left(z \leq 2.5 \cdot 10^{-107}\right):\\ \;\;\;\;x + z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x \cdot \cos y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3e-95) (not (<= z 2.5e-107)))
   (+ x (* z (sin y)))
   (/ 1.0 (/ 1.0 (* x (cos y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3e-95) || !(z <= 2.5e-107)) {
		tmp = x + (z * sin(y));
	} else {
		tmp = 1.0 / (1.0 / (x * cos(y)));
	}
	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 ((z <= (-3d-95)) .or. (.not. (z <= 2.5d-107))) then
        tmp = x + (z * sin(y))
    else
        tmp = 1.0d0 / (1.0d0 / (x * cos(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3e-95) || !(z <= 2.5e-107)) {
		tmp = x + (z * Math.sin(y));
	} else {
		tmp = 1.0 / (1.0 / (x * Math.cos(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3e-95) or not (z <= 2.5e-107):
		tmp = x + (z * math.sin(y))
	else:
		tmp = 1.0 / (1.0 / (x * math.cos(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3e-95) || !(z <= 2.5e-107))
		tmp = Float64(x + Float64(z * sin(y)));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(x * cos(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3e-95) || ~((z <= 2.5e-107)))
		tmp = x + (z * sin(y));
	else
		tmp = 1.0 / (1.0 / (x * cos(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3e-95], N[Not[LessEqual[z, 2.5e-107]], $MachinePrecision]], N[(x + N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3e-95 or 2.49999999999999985e-107 < z

   1. Initial program 99.9%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in y around 0 92.1%

      \[\leadsto \color{blue}{x} + z \cdot \sin y \]

   if -3e-95 < z < 2.49999999999999985e-107

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in y around 0 79.3%

      \[\leadsto x \cdot \cos y + \color{blue}{y \cdot z} \]
   3. Step-by-step derivation

      1. flip-+ 47.8%

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

      2. clear-num 47.6%

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

      3. *-commutative 47.6%

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

      4. pow2 47.6%

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

      5. *-commutative 47.6%

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

      6. pow2 47.6%

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

   4. Applied egg-rr 47.6%

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

   5. Taylor expanded in x around inf 94.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\cos y \cdot x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-95} \lor \neg \left(z \leq 2.5 \cdot 10^{-107}\right):\\ \;\;\;\;x + z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x \cdot \cos y}}\\ \end{array} \]

Alternative 4: 74.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0074 \lor \neg \left(y \leq 0.044\right):\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z + x \cdot \left(1 + -0.5 \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.0074) (not (<= y 0.044)))
   (* z (sin y))
   (+ (* y z) (* x (+ 1.0 (* -0.5 (* y y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0074) || !(y <= 0.044)) {
		tmp = z * sin(y);
	} else {
		tmp = (y * z) + (x * (1.0 + (-0.5 * (y * y))));
	}
	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 ((y <= (-0.0074d0)) .or. (.not. (y <= 0.044d0))) then
        tmp = z * sin(y)
    else
        tmp = (y * z) + (x * (1.0d0 + ((-0.5d0) * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0074) || !(y <= 0.044)) {
		tmp = z * Math.sin(y);
	} else {
		tmp = (y * z) + (x * (1.0 + (-0.5 * (y * y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.0074) or not (y <= 0.044):
		tmp = z * math.sin(y)
	else:
		tmp = (y * z) + (x * (1.0 + (-0.5 * (y * y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.0074) || !(y <= 0.044))
		tmp = Float64(z * sin(y));
	else
		tmp = Float64(Float64(y * z) + Float64(x * Float64(1.0 + Float64(-0.5 * Float64(y * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.0074) || ~((y <= 0.044)))
		tmp = z * sin(y);
	else
		tmp = (y * z) + (x * (1.0 + (-0.5 * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.0074], N[Not[LessEqual[y, 0.044]], $MachinePrecision]], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(N[(y * z), $MachinePrecision] + N[(x * N[(1.0 + N[(-0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0074000000000000003 or 0.043999999999999997 < y

   1. Initial program 99.7%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in x around 0 57.4%

      \[\leadsto \color{blue}{z \cdot \sin y} \]

   if -0.0074000000000000003 < y < 0.043999999999999997

   1. Initial program 100.0%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in y around 0 100.0%

      \[\leadsto x \cdot \cos y + \color{blue}{y \cdot z} \]

   3. Taylor expanded in y around 0 99.8%

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

      1. unpow2 99.8%

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

   5. Simplified 99.8%

      \[\leadsto x \cdot \color{blue}{\left(1 + -0.5 \cdot \left(y \cdot y\right)\right)} + y \cdot z \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0074 \lor \neg \left(y \leq 0.044\right):\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;y \cdot z + x \cdot \left(1 + -0.5 \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 5: 76.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x + z \cdot \sin y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (* z (sin y))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot \cos y + z \cdot \sin y \]

2. Taylor expanded in y around 0 82.6%

   \[\leadsto \color{blue}{x} + z \cdot \sin y \]

3. Final simplification 82.6%

   \[\leadsto x + z \cdot \sin y \]

Alternative 6: 41.4% accurate, 29.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+153}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+191}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.05e+153) (* y z) (if (<= z 5.5e+191) x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.05e+153) {
		tmp = y * z;
	} else if (z <= 5.5e+191) {
		tmp = x;
	} else {
		tmp = y * 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 (z <= (-1.05d+153)) then
        tmp = y * z
    else if (z <= 5.5d+191) then
        tmp = x
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.05e+153) {
		tmp = y * z;
	} else if (z <= 5.5e+191) {
		tmp = x;
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.05e+153:
		tmp = y * z
	elif z <= 5.5e+191:
		tmp = x
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.05e+153)
		tmp = Float64(y * z);
	elseif (z <= 5.5e+191)
		tmp = x;
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.05e+153)
		tmp = y * z;
	elseif (z <= 5.5e+191)
		tmp = x;
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.05e+153], N[(y * z), $MachinePrecision], If[LessEqual[z, 5.5e+191], x, N[(y * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05000000000000008e153 or 5.5000000000000002e191 < z

   1. Initial program 99.8%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in y around 0 57.4%

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

   3. Taylor expanded in y around inf 47.0%

      \[\leadsto \color{blue}{y \cdot z} \]

   if -1.05000000000000008e153 < z < 5.5000000000000002e191

   1. Initial program 99.9%

      \[x \cdot \cos y + z \cdot \sin y \]

   2. Taylor expanded in y around 0 70.1%

      \[\leadsto x \cdot \cos y + \color{blue}{y \cdot z} \]
   3. Step-by-step derivation

      1. flip-+ 43.3%

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

      2. clear-num 43.2%

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

      3. *-commutative 43.2%

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

      4. pow2 43.2%

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

      5. *-commutative 43.2%

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

      6. pow2 43.2%

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

   4. Applied egg-rr 43.2%

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

   5. Taylor expanded in y around 0 54.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
   6. Step-by-step derivation

      1. clear-num 54.1%

         \[\leadsto \color{blue}{\frac{x}{1}} \]

      2. div-inv 54.1%

         \[\leadsto \color{blue}{x \cdot \frac{1}{1}} \]

      3. metadata-eval 54.1%

         \[\leadsto x \cdot \color{blue}{1} \]

   7. Applied egg-rr 54.1%

      \[\leadsto \color{blue}{x \cdot 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+153}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+191}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]

Alternative 7: 52.2% accurate, 41.4× speedup

Math

\[\begin{array}{l} \\ x + y \cdot z \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x \cdot \cos y + z \cdot \sin y \]

2. Taylor expanded in y around 0 59.3%

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

3. Final simplification 59.3%

   \[\leadsto x + y \cdot z \]

Alternative 8: 39.1% accurate, 207.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = x
end function

Java

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

Python

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

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.9%

   \[x \cdot \cos y + z \cdot \sin y \]

2. Taylor expanded in y around 0 67.2%

   \[\leadsto x \cdot \cos y + \color{blue}{y \cdot z} \]
3. Step-by-step derivation

   1. flip-+ 42.2%

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

   2. clear-num 42.0%

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

   3. *-commutative 42.0%

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

   4. pow2 42.0%

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

   5. *-commutative 42.0%

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

   6. pow2 42.0%

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

4. Applied egg-rr 42.0%

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

5. Taylor expanded in y around 0 44.3%

   \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
6. Step-by-step derivation

   1. clear-num 44.4%

      \[\leadsto \color{blue}{\frac{x}{1}} \]

   2. div-inv 44.4%

      \[\leadsto \color{blue}{x \cdot \frac{1}{1}} \]

   3. metadata-eval 44.4%

      \[\leadsto x \cdot \color{blue}{1} \]

7. Applied egg-rr 44.4%

   \[\leadsto \color{blue}{x \cdot 1} \]

8. Final simplification 44.4%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023171 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))