Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, A

Percentage Accurate: 99.8% → 99.8%

Time: 10.0s

Alternatives: 9

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 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 79.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+164}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-34} \lor \neg \left(z \leq 1.4 \cdot 10^{-11}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z \cdot \sin y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.9e+164)
   (* (sin y) (- z))
   (if (or (<= z -2e-34) (not (<= z 1.4e-11)))
     (* x (- 1.0 (/ (* z (sin y)) x)))
     (* x (cos y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.9e+164) {
		tmp = sin(y) * -z;
	} else if ((z <= -2e-34) || !(z <= 1.4e-11)) {
		tmp = x * (1.0 - ((z * sin(y)) / x));
	} else {
		tmp = 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 <= (-1.9d+164)) then
        tmp = sin(y) * -z
    else if ((z <= (-2d-34)) .or. (.not. (z <= 1.4d-11))) then
        tmp = x * (1.0d0 - ((z * sin(y)) / x))
    else
        tmp = x * cos(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.9e+164) {
		tmp = Math.sin(y) * -z;
	} else if ((z <= -2e-34) || !(z <= 1.4e-11)) {
		tmp = x * (1.0 - ((z * Math.sin(y)) / x));
	} else {
		tmp = x * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.9e+164:
		tmp = math.sin(y) * -z
	elif (z <= -2e-34) or not (z <= 1.4e-11):
		tmp = x * (1.0 - ((z * math.sin(y)) / x))
	else:
		tmp = x * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.9e+164)
		tmp = Float64(sin(y) * Float64(-z));
	elseif ((z <= -2e-34) || !(z <= 1.4e-11))
		tmp = Float64(x * Float64(1.0 - Float64(Float64(z * sin(y)) / x)));
	else
		tmp = Float64(x * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.9e+164)
		tmp = sin(y) * -z;
	elseif ((z <= -2e-34) || ~((z <= 1.4e-11)))
		tmp = x * (1.0 - ((z * sin(y)) / x));
	else
		tmp = x * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.9e+164], N[(N[Sin[y], $MachinePrecision] * (-z)), $MachinePrecision], If[Or[LessEqual[z, -2e-34], N[Not[LessEqual[z, 1.4e-11]], $MachinePrecision]], N[(x * N[(1.0 - N[(N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.90000000000000011e164

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 82.7%

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \sin y\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 82.7%

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

      2. neg-mul-1 82.7%

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

   5. Simplified 82.7%

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

   if -1.90000000000000011e164 < z < -1.99999999999999986e-34 or 1.4e-11 < z

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. log1p-expm1-u 99.8%

         \[\leadsto x \cdot \cos y - z \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin y\right)\right)} \]

   4. Applied egg-rr 99.8%

      \[\leadsto x \cdot \cos y - z \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin y\right)\right)} \]

   5. Taylor expanded in y around 0 91.2%

      \[\leadsto x \cdot \color{blue}{1} - z \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin y\right)\right) \]

   6. Taylor expanded in x around inf 82.5%

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

      1. associate-*r/ 82.5%

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

      2. neg-mul-1 82.5%

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

      3. distribute-lft-neg-in 82.5%

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

      4. *-commutative 82.5%

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

   8. Simplified 82.5%

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

   if -1.99999999999999986e-34 < z < 1.4e-11

   1. Initial program 99.9%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 92.3%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+164}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-34} \lor \neg \left(z \leq 1.4 \cdot 10^{-11}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z \cdot \sin y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+164}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-34} \lor \neg \left(z \leq 1.1 \cdot 10^{-11}\right):\\ \;\;\;\;x \cdot \left(1 - z \cdot \frac{\sin y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.8e+164)
   (* (sin y) (- z))
   (if (or (<= z -1.7e-34) (not (<= z 1.1e-11)))
     (* x (- 1.0 (* z (/ (sin y) x))))
     (* x (cos y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.8e+164) {
		tmp = sin(y) * -z;
	} else if ((z <= -1.7e-34) || !(z <= 1.1e-11)) {
		tmp = x * (1.0 - (z * (sin(y) / x)));
	} else {
		tmp = 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 <= (-1.8d+164)) then
        tmp = sin(y) * -z
    else if ((z <= (-1.7d-34)) .or. (.not. (z <= 1.1d-11))) then
        tmp = x * (1.0d0 - (z * (sin(y) / x)))
    else
        tmp = x * cos(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.8e+164) {
		tmp = Math.sin(y) * -z;
	} else if ((z <= -1.7e-34) || !(z <= 1.1e-11)) {
		tmp = x * (1.0 - (z * (Math.sin(y) / x)));
	} else {
		tmp = x * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.8e+164:
		tmp = math.sin(y) * -z
	elif (z <= -1.7e-34) or not (z <= 1.1e-11):
		tmp = x * (1.0 - (z * (math.sin(y) / x)))
	else:
		tmp = x * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.8e+164)
		tmp = Float64(sin(y) * Float64(-z));
	elseif ((z <= -1.7e-34) || !(z <= 1.1e-11))
		tmp = Float64(x * Float64(1.0 - Float64(z * Float64(sin(y) / x))));
	else
		tmp = Float64(x * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.8e+164)
		tmp = sin(y) * -z;
	elseif ((z <= -1.7e-34) || ~((z <= 1.1e-11)))
		tmp = x * (1.0 - (z * (sin(y) / x)));
	else
		tmp = x * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.8e+164], N[(N[Sin[y], $MachinePrecision] * (-z)), $MachinePrecision], If[Or[LessEqual[z, -1.7e-34], N[Not[LessEqual[z, 1.1e-11]], $MachinePrecision]], N[(x * N[(1.0 - N[(z * N[(N[Sin[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.79999999999999995e164

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 82.7%

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \sin y\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 82.7%

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

      2. neg-mul-1 82.7%

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

   5. Simplified 82.7%

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

   if -1.79999999999999995e164 < z < -1.7e-34 or 1.1000000000000001e-11 < z

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. log1p-expm1-u 99.8%

         \[\leadsto x \cdot \cos y - z \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin y\right)\right)} \]

   4. Applied egg-rr 99.8%

      \[\leadsto x \cdot \cos y - z \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin y\right)\right)} \]

   5. Taylor expanded in y around 0 91.2%

      \[\leadsto x \cdot \color{blue}{1} - z \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin y\right)\right) \]

   6. Taylor expanded in x around inf 82.5%

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

      1. mul-1-neg 82.5%

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

      2. unsub-neg 82.5%

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

      3. associate-/l* 82.3%

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

   8. Simplified 82.3%

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

   if -1.7e-34 < z < 1.1000000000000001e-11

   1. Initial program 99.9%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 92.3%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+164}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-34} \lor \neg \left(z \leq 1.1 \cdot 10^{-11}\right):\\ \;\;\;\;x \cdot \left(1 - z \cdot \frac{\sin y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+163}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{\frac{x}{\sin y}}\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z \cdot \frac{\sin y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -9.5e+163)
   (* (sin y) (- z))
   (if (<= z -3.6e-33)
     (* x (- 1.0 (/ z (/ x (sin y)))))
     (if (<= z 9.5e-12) (* x (cos y)) (* x (- 1.0 (* z (/ (sin y) x))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -9.5e+163) {
		tmp = sin(y) * -z;
	} else if (z <= -3.6e-33) {
		tmp = x * (1.0 - (z / (x / sin(y))));
	} else if (z <= 9.5e-12) {
		tmp = x * cos(y);
	} else {
		tmp = x * (1.0 - (z * (sin(y) / 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) :: tmp
    if (z <= (-9.5d+163)) then
        tmp = sin(y) * -z
    else if (z <= (-3.6d-33)) then
        tmp = x * (1.0d0 - (z / (x / sin(y))))
    else if (z <= 9.5d-12) then
        tmp = x * cos(y)
    else
        tmp = x * (1.0d0 - (z * (sin(y) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -9.5e+163) {
		tmp = Math.sin(y) * -z;
	} else if (z <= -3.6e-33) {
		tmp = x * (1.0 - (z / (x / Math.sin(y))));
	} else if (z <= 9.5e-12) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x * (1.0 - (z * (Math.sin(y) / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -9.5e+163:
		tmp = math.sin(y) * -z
	elif z <= -3.6e-33:
		tmp = x * (1.0 - (z / (x / math.sin(y))))
	elif z <= 9.5e-12:
		tmp = x * math.cos(y)
	else:
		tmp = x * (1.0 - (z * (math.sin(y) / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -9.5e+163)
		tmp = Float64(sin(y) * Float64(-z));
	elseif (z <= -3.6e-33)
		tmp = Float64(x * Float64(1.0 - Float64(z / Float64(x / sin(y)))));
	elseif (z <= 9.5e-12)
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x * Float64(1.0 - Float64(z * Float64(sin(y) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -9.5e+163)
		tmp = sin(y) * -z;
	elseif (z <= -3.6e-33)
		tmp = x * (1.0 - (z / (x / sin(y))));
	elseif (z <= 9.5e-12)
		tmp = x * cos(y);
	else
		tmp = x * (1.0 - (z * (sin(y) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -9.5e+163], N[(N[Sin[y], $MachinePrecision] * (-z)), $MachinePrecision], If[LessEqual[z, -3.6e-33], N[(x * N[(1.0 - N[(z / N[(x / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.5e-12], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 - N[(z * N[(N[Sin[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -9.50000000000000053e163

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 82.7%

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \sin y\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 82.7%

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

      2. neg-mul-1 82.7%

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

   5. Simplified 82.7%

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

   if -9.50000000000000053e163 < z < -3.60000000000000034e-33

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. log1p-expm1-u 99.8%

         \[\leadsto x \cdot \cos y - z \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin y\right)\right)} \]

   4. Applied egg-rr 99.8%

      \[\leadsto x \cdot \cos y - z \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin y\right)\right)} \]

   5. Taylor expanded in y around 0 95.6%

      \[\leadsto x \cdot \color{blue}{1} - z \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin y\right)\right) \]

   6. Taylor expanded in x around inf 89.6%

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

      1. mul-1-neg 89.6%

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

      2. unsub-neg 89.6%

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

      3. associate-/l* 89.3%

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

   8. Simplified 89.3%

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

      1. clear-num 89.3%

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

      2. un-div-inv 89.4%

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

   10. Applied egg-rr 89.4%

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

   if -3.60000000000000034e-33 < z < 9.4999999999999995e-12

   1. Initial program 99.9%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 92.3%

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

   if 9.4999999999999995e-12 < z

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. log1p-expm1-u 99.8%

         \[\leadsto x \cdot \cos y - z \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin y\right)\right)} \]

   4. Applied egg-rr 99.8%

      \[\leadsto x \cdot \cos y - z \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin y\right)\right)} \]

   5. Taylor expanded in y around 0 87.8%

      \[\leadsto x \cdot \color{blue}{1} - z \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin y\right)\right) \]

   6. Taylor expanded in x around inf 77.1%

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

      1. mul-1-neg 77.1%

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

      2. unsub-neg 77.1%

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

      3. associate-/l* 76.8%

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

   8. Simplified 76.8%

      \[\leadsto \color{blue}{x \cdot \left(1 - z \cdot \frac{\sin y}{x}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+163}:\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{\frac{x}{\sin y}}\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z \cdot \frac{\sin y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -94000 \lor \neg \left(z \leq 9.5 \cdot 10^{+45}\right):\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -94000.0) (not (<= z 9.5e+45)))
   (* (sin y) (- z))
   (* x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -94000.0) || !(z <= 9.5e+45)) {
		tmp = sin(y) * -z;
	} else {
		tmp = 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 <= (-94000.0d0)) .or. (.not. (z <= 9.5d+45))) then
        tmp = sin(y) * -z
    else
        tmp = x * cos(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -94000.0) || !(z <= 9.5e+45)) {
		tmp = Math.sin(y) * -z;
	} else {
		tmp = x * Math.cos(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -94000.0) or not (z <= 9.5e+45):
		tmp = math.sin(y) * -z
	else:
		tmp = x * math.cos(y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -94000.0) || !(z <= 9.5e+45))
		tmp = Float64(sin(y) * Float64(-z));
	else
		tmp = Float64(x * cos(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -94000.0) || ~((z <= 9.5e+45)))
		tmp = sin(y) * -z;
	else
		tmp = x * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -94000.0], N[Not[LessEqual[z, 9.5e+45]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] * (-z)), $MachinePrecision], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -94000 or 9.4999999999999998e45 < z

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 70.5%

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \sin y\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 70.5%

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

      2. neg-mul-1 70.5%

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

   5. Simplified 70.5%

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

   if -94000 < z < 9.4999999999999998e45

   1. Initial program 99.9%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 89.4%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -94000 \lor \neg \left(z \leq 9.5 \cdot 10^{+45}\right):\\ \;\;\;\;\sin y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0142 \lor \neg \left(y \leq 0.00041\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5 + 0.16666666666666666 \cdot \left(y \cdot z\right)\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.0142) (not (<= y 0.00041)))
   (* x (cos y))
   (+ x (* y (- (* y (+ (* x -0.5) (* 0.16666666666666666 (* y z)))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0142) || !(y <= 0.00041)) {
		tmp = x * cos(y);
	} else {
		tmp = x + (y * ((y * ((x * -0.5) + (0.16666666666666666 * (y * z)))) - 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 ((y <= (-0.0142d0)) .or. (.not. (y <= 0.00041d0))) then
        tmp = x * cos(y)
    else
        tmp = x + (y * ((y * ((x * (-0.5d0)) + (0.16666666666666666d0 * (y * z)))) - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.0142) || !(y <= 0.00041)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (y * ((y * ((x * -0.5) + (0.16666666666666666 * (y * z)))) - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.0142) or not (y <= 0.00041):
		tmp = x * math.cos(y)
	else:
		tmp = x + (y * ((y * ((x * -0.5) + (0.16666666666666666 * (y * z)))) - z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.0142) || !(y <= 0.00041))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(y * Float64(Float64(y * Float64(Float64(x * -0.5) + Float64(0.16666666666666666 * Float64(y * z)))) - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.0142) || ~((y <= 0.00041)))
		tmp = x * cos(y);
	else
		tmp = x + (y * ((y * ((x * -0.5) + (0.16666666666666666 * (y * z)))) - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.0142], N[Not[LessEqual[y, 0.00041]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(y * N[(N[(x * -0.5), $MachinePrecision] + N[(0.16666666666666666 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.014200000000000001 or 4.0999999999999999e-4 < y

   1. Initial program 99.7%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 55.1%

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

   if -0.014200000000000001 < y < 4.0999999999999999e-4

   1. Initial program 100.0%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{x + y \cdot \left(y \cdot \left(-0.5 \cdot x + 0.16666666666666666 \cdot \left(y \cdot z\right)\right) - z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0142 \lor \neg \left(y \leq 0.00041\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(y \cdot \left(x \cdot -0.5 + 0.16666666666666666 \cdot \left(y \cdot z\right)\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.0% accurate, 14.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{-204}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-207}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.1e-204) x (if (<= x 1.25e-207) (* y (- z)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.1e-204) {
		tmp = x;
	} else if (x <= 1.25e-207) {
		tmp = y * -z;
	} else {
		tmp = 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) :: tmp
    if (x <= (-5.1d-204)) then
        tmp = x
    else if (x <= 1.25d-207) then
        tmp = y * -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.1e-204) {
		tmp = x;
	} else if (x <= 1.25e-207) {
		tmp = y * -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.1e-204:
		tmp = x
	elif x <= 1.25e-207:
		tmp = y * -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.1e-204)
		tmp = x;
	elseif (x <= 1.25e-207)
		tmp = Float64(y * Float64(-z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.1e-204)
		tmp = x;
	elseif (x <= 1.25e-207)
		tmp = y * -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.1e-204], x, If[LessEqual[x, 1.25e-207], N[(y * (-z)), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -5.10000000000000027e-204 or 1.25000000000000004e-207 < x

   1. Initial program 99.8%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 43.4%

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

   if -5.10000000000000027e-204 < x < 1.25000000000000004e-207

   1. Initial program 99.9%

      \[x \cdot \cos y - z \cdot \sin y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 55.0%

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

      1. mul-1-neg 55.0%

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

      2. unsub-neg 55.0%

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

      3. *-commutative 55.0%

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

   5. Simplified 55.0%

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

   6. Taylor expanded in x around 0 45.3%

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

      1. mul-1-neg 45.3%

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

   8. Applied egg-rr 45.3%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{-204}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-207}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.7% 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 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]
2. Add Preprocessing

3. Taylor expanded in y around 0 53.8%

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

   1. mul-1-neg 53.8%

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

   2. unsub-neg 53.8%

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

   3. *-commutative 53.8%

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

5. Simplified 53.8%

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

6. Final simplification 53.8%

   \[\leadsto x - y \cdot z \]
7. Add Preprocessing

Alternative 9: 38.7% 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 99.8%

   \[x \cdot \cos y - z \cdot \sin y \]
2. Add Preprocessing

3. Taylor expanded in y around 0 39.3%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024148 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, A"
  :precision binary64
  (- (* x (cos y)) (* z (sin y))))