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

Percentage Accurate: 99.8% → 99.8%

Time: 11.2s

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 99.8%

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

Alternative 2: 78.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-25} \lor \neg \left(z \leq 8.5 \cdot 10^{-60}\right):\\ \;\;\;\;x \cdot \left(1 + z \cdot \frac{\sin y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.3e-25) (not (<= z 8.5e-60)))
   (* x (+ 1.0 (* z (/ (sin y) x))))
   (* x (cos y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.3e-25) || !(z <= 8.5e-60)) {
		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 <= (-3.3d-25)) .or. (.not. (z <= 8.5d-60))) 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 <= -3.3e-25) || !(z <= 8.5e-60)) {
		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 <= -3.3e-25) or not (z <= 8.5e-60):
		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 <= -3.3e-25) || !(z <= 8.5e-60))
		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 <= -3.3e-25) || ~((z <= 8.5e-60)))
		tmp = x * (1.0 + (z * (sin(y) / x)));
	else
		tmp = x * cos(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.3e-25], N[Not[LessEqual[z, 8.5e-60]], $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 2 regimes

2. if z < -3.2999999999999998e-25 or 8.50000000000000044e-60 < z

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \cos y, z \cdot \sin y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 87.7%

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

      1. associate-/l* 87.6%

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

   7. Simplified 87.6%

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

   8. Taylor expanded in y around 0 76.5%

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

   if -3.2999999999999998e-25 < z < 8.50000000000000044e-60

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \cos y, z \cdot \sin y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 92.9%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-25} \lor \neg \left(z \leq 8.5 \cdot 10^{-60}\right):\\ \;\;\;\;x \cdot \left(1 + z \cdot \frac{\sin y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \cos y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00185 \lor \neg \left(y \leq 0.00039\right):\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + -0.5 \cdot \left(x \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -0.00185) (not (<= y 0.00039)))
   (* z (sin y))
   (+ x (* y (+ z (* -0.5 (* x y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.00185) || !(y <= 0.00039)) {
		tmp = z * sin(y);
	} else {
		tmp = x + (y * (z + (-0.5 * (x * 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 <= (-0.00185d0)) .or. (.not. (y <= 0.00039d0))) then
        tmp = z * sin(y)
    else
        tmp = x + (y * (z + ((-0.5d0) * (x * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -0.00185) || !(y <= 0.00039)) {
		tmp = z * Math.sin(y);
	} else {
		tmp = x + (y * (z + (-0.5 * (x * y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -0.00185) or not (y <= 0.00039):
		tmp = z * math.sin(y)
	else:
		tmp = x + (y * (z + (-0.5 * (x * y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -0.00185) || !(y <= 0.00039))
		tmp = Float64(z * sin(y));
	else
		tmp = Float64(x + Float64(y * Float64(z + Float64(-0.5 * Float64(x * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -0.00185) || ~((y <= 0.00039)))
		tmp = z * sin(y);
	else
		tmp = x + (y * (z + (-0.5 * (x * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -0.00185], N[Not[LessEqual[y, 0.00039]], $MachinePrecision]], N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z + N[(-0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0018500000000000001 or 3.89999999999999993e-4 < y

   1. Initial program 99.6%

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

      1. fma-define 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \cos y, z \cdot \sin y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 55.0%

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

   if -0.0018500000000000001 < y < 3.89999999999999993e-4

   1. Initial program 100.0%

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

      1. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \cos y, z \cdot \sin y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 100.0%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00185 \lor \neg \left(y \leq 0.00039\right):\\ \;\;\;\;z \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + -0.5 \cdot \left(x \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -215000000 \lor \neg \left(y \leq 1.4 \cdot 10^{+15}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5 + -0.16666666666666666 \cdot \left(y \cdot z\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -215000000.0) (not (<= y 1.4e+15)))
   (* x (cos y))
   (+ x (* y (+ z (* y (+ (* x -0.5) (* -0.16666666666666666 (* y z)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -215000000.0) || !(y <= 1.4e+15)) {
		tmp = x * cos(y);
	} else {
		tmp = x + (y * (z + (y * ((x * -0.5) + (-0.16666666666666666 * (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 ((y <= (-215000000.0d0)) .or. (.not. (y <= 1.4d+15))) then
        tmp = x * cos(y)
    else
        tmp = x + (y * (z + (y * ((x * (-0.5d0)) + ((-0.16666666666666666d0) * (y * z))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -215000000.0) || !(y <= 1.4e+15)) {
		tmp = x * Math.cos(y);
	} else {
		tmp = x + (y * (z + (y * ((x * -0.5) + (-0.16666666666666666 * (y * z))))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -215000000.0) or not (y <= 1.4e+15):
		tmp = x * math.cos(y)
	else:
		tmp = x + (y * (z + (y * ((x * -0.5) + (-0.16666666666666666 * (y * z))))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -215000000.0) || !(y <= 1.4e+15))
		tmp = Float64(x * cos(y));
	else
		tmp = Float64(x + Float64(y * Float64(z + Float64(y * Float64(Float64(x * -0.5) + Float64(-0.16666666666666666 * Float64(y * z)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -215000000.0) || ~((y <= 1.4e+15)))
		tmp = x * cos(y);
	else
		tmp = x + (y * (z + (y * ((x * -0.5) + (-0.16666666666666666 * (y * z))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -215000000.0], N[Not[LessEqual[y, 1.4e+15]], $MachinePrecision]], N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z + N[(y * N[(N[(x * -0.5), $MachinePrecision] + N[(-0.16666666666666666 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.15e8 or 1.4e15 < y

   1. Initial program 99.7%

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

      1. fma-define 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \cos y, z \cdot \sin y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 49.1%

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

   if -2.15e8 < y < 1.4e15

   1. Initial program 100.0%

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

      1. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \cos y, z \cdot \sin y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 96.7%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -215000000 \lor \neg \left(y \leq 1.4 \cdot 10^{+15}\right):\\ \;\;\;\;x \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z + y \cdot \left(x \cdot -0.5 + -0.16666666666666666 \cdot \left(y \cdot z\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 35.0% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-268} \lor \neg \left(x \leq 3.6 \cdot 10^{-177}\right):\\ \;\;\;\;z \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.2e-268) (not (<= x 3.6e-177))) (* z (/ x z)) (* y z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.2e-268) || !(x <= 3.6e-177)) {
		tmp = z * (x / z);
	} 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 ((x <= (-1.2d-268)) .or. (.not. (x <= 3.6d-177))) then
        tmp = z * (x / z)
    else
        tmp = y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.2e-268) || !(x <= 3.6e-177)) {
		tmp = z * (x / z);
	} else {
		tmp = y * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.2e-268) or not (x <= 3.6e-177):
		tmp = z * (x / z)
	else:
		tmp = y * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.2e-268) || !(x <= 3.6e-177))
		tmp = Float64(z * Float64(x / z));
	else
		tmp = Float64(y * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.2e-268) || ~((x <= 3.6e-177)))
		tmp = z * (x / z);
	else
		tmp = y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.2e-268], N[Not[LessEqual[x, 3.6e-177]], $MachinePrecision]], N[(z * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(y * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1999999999999999e-268 or 3.59999999999999983e-177 < x

   1. Initial program 99.8%

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

      1. expm1-log1p-u 99.8%

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

   4. Applied egg-rr 99.8%

      \[\leadsto x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos y\right)\right)} + z \cdot \sin y \]
   5. Step-by-step derivation

      1. expm1-undefine 99.6%

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

      2. flip-- 99.6%

         \[\leadsto x \cdot \color{blue}{\frac{e^{\mathsf{log1p}\left(\cos y\right)} \cdot e^{\mathsf{log1p}\left(\cos y\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(\cos y\right)} + 1}} + z \cdot \sin y \]

      3. log1p-undefine 99.6%

         \[\leadsto x \cdot \frac{e^{\color{blue}{\log \left(1 + \cos y\right)}} \cdot e^{\mathsf{log1p}\left(\cos y\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(\cos y\right)} + 1} + z \cdot \sin y \]

      4. rem-exp-log 99.6%

         \[\leadsto x \cdot \frac{\color{blue}{\left(1 + \cos y\right)} \cdot e^{\mathsf{log1p}\left(\cos y\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(\cos y\right)} + 1} + z \cdot \sin y \]

      5. +-commutative 99.6%

         \[\leadsto x \cdot \frac{\color{blue}{\left(\cos y + 1\right)} \cdot e^{\mathsf{log1p}\left(\cos y\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(\cos y\right)} + 1} + z \cdot \sin y \]

      6. log1p-undefine 99.5%

         \[\leadsto x \cdot \frac{\left(\cos y + 1\right) \cdot e^{\color{blue}{\log \left(1 + \cos y\right)}} - 1 \cdot 1}{e^{\mathsf{log1p}\left(\cos y\right)} + 1} + z \cdot \sin y \]

      7. rem-exp-log 99.5%

         \[\leadsto x \cdot \frac{\left(\cos y + 1\right) \cdot \color{blue}{\left(1 + \cos y\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(\cos y\right)} + 1} + z \cdot \sin y \]

      8. +-commutative 99.5%

         \[\leadsto x \cdot \frac{\left(\cos y + 1\right) \cdot \color{blue}{\left(\cos y + 1\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(\cos y\right)} + 1} + z \cdot \sin y \]

      9. metadata-eval 99.5%

         \[\leadsto x \cdot \frac{\left(\cos y + 1\right) \cdot \left(\cos y + 1\right) - \color{blue}{1}}{e^{\mathsf{log1p}\left(\cos y\right)} + 1} + z \cdot \sin y \]

      10. log1p-undefine 99.5%

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

      11. rem-exp-log 99.6%

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

      12. +-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. difference-of-sqr-1 99.6%

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

      2. associate--l+ 99.8%

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

      3. metadata-eval 99.8%

         \[\leadsto x \cdot \frac{\left(\left(\cos y + 1\right) + 1\right) \cdot \left(\cos y + \color{blue}{0}\right)}{\left(\cos y + 1\right) + 1} + z \cdot \sin y \]

      4. +-rgt-identity 99.8%

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

      5. associate-/l* 99.8%

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

      6. associate-+l+ 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-+l+ 99.8%

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

      9. metadata-eval 99.8%

         \[\leadsto x \cdot \left(\left(\cos y + 2\right) \cdot \frac{\cos y}{\cos y + \color{blue}{2}}\right) + z \cdot \sin y \]

   8. Simplified 99.8%

      \[\leadsto x \cdot \color{blue}{\left(\left(\cos y + 2\right) \cdot \frac{\cos y}{\cos y + 2}\right)} + z \cdot \sin y \]

   9. Taylor expanded in z around inf 88.4%

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

       1. +-commutative 88.4%

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

       2. distribute-lft-in 88.4%

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

       3. associate-*l/ 88.3%

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

       4. associate-/r/ 88.4%

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

       5. distribute-lft-out 88.3%

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

       6. associate-/r/ 88.3%

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

       7. fma-define 88.3%

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

   11. Simplified 88.3%

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

   12. Taylor expanded in y around 0 39.5%

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

   if -1.1999999999999999e-268 < x < 3.59999999999999983e-177

   1. Initial program 99.9%

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

      1. fma-define 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \cos y, z \cdot \sin y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 52.9%

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

   6. Taylor expanded in x around 0 42.8%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-268} \lor \neg \left(x \leq 3.6 \cdot 10^{-177}\right):\\ \;\;\;\;z \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 32.7% accurate, 13.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+223} \lor \neg \left(z \leq 6 \cdot 10^{+129}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.9e+223) (not (<= z 6e+129))) (* y z) (* y (/ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.9e+223) || !(z <= 6e+129)) {
		tmp = y * z;
	} else {
		tmp = y * (x / 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 <= (-3.9d+223)) .or. (.not. (z <= 6d+129))) then
        tmp = y * z
    else
        tmp = y * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.9e+223) || !(z <= 6e+129)) {
		tmp = y * z;
	} else {
		tmp = y * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.9e+223) or not (z <= 6e+129):
		tmp = y * z
	else:
		tmp = y * (x / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.9e+223) || !(z <= 6e+129))
		tmp = Float64(y * z);
	else
		tmp = Float64(y * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.9e+223) || ~((z <= 6e+129)))
		tmp = y * z;
	else
		tmp = y * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.9e+223], N[Not[LessEqual[z, 6e+129]], $MachinePrecision]], N[(y * z), $MachinePrecision], N[(y * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.8999999999999999e223 or 6.0000000000000006e129 < z

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \cos y, z \cdot \sin y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 54.8%

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

   6. Taylor expanded in x around 0 32.2%

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

   if -3.8999999999999999e223 < z < 6.0000000000000006e129

   1. Initial program 99.8%

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

      1. fma-define 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \cos y, z \cdot \sin y\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 52.6%

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

   6. Taylor expanded in y around inf 44.4%

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

   7. Taylor expanded in z around 0 38.0%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+223} \lor \neg \left(z \leq 6 \cdot 10^{+129}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.5% 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. Step-by-step derivation

   1. fma-define 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \cos y, z \cdot \sin y\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 53.2%

   \[\leadsto \color{blue}{x + y \cdot z} \]
6. Add Preprocessing

Alternative 8: 17.1% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fma-define 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \cos y, z \cdot \sin y\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 53.2%

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

6. Taylor expanded in x around 0 15.7%

   \[\leadsto \color{blue}{y \cdot z} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024160 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))